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### Hypotraceable graph

A graph is a hypotraceable graph if has no Hamiltonian path (i.e., it is not a traceable graph), but has a Hamiltonian path (i.e., is a traceable graph) for every (Bondy and Murty 1976, p. 61).There are no hypotraceable graphs on ten or fewer nodes (E. Weisstein, Dec. 11, 2013). In fact, the nonexistence of hypotraceable graphs on small numbers of vertices led T. Gallai to conjecture that no such graphs exist. This conjecture was refuted when a hypotraceable graph with 40 vertices was subsequently found by Horton (Grünbaum 1974, Thomassen 1974). Thomassen (1974) then showed that there exists a hypotraceable graph with vertices for , 37, 39, 40, and all . The smallest of these is the 34-vertex Thomassen graph (left figure above; Thomassen 1974; Bondy and Murty 1976, pp. 239-240).Walter (1969) gave an example of a connected graph in which the longest paths do not have a vertex in common, a property shared by hypotraceable..

### Tait's hamiltonian graph conjecture

The conjecture that every cubic polyhedral graph is Hamiltonian. It was proposed by Tait in 1880 and refuted by Tutte (1946) with the counterexample on 46 vertices (Lederberg 1965) now known as Tutte's graph. Had the conjecture been true, it would have implied the four-color theorem.The following table summarizes some named counterexamples, illustrated above. The smallest examples known has 38 vertices (Lederberg 1965), and was apparently also discovered by D. Barnette and J. Bosák around the same time.graphreference38Barnette-Bośak-Lederberg graphLederberg (1965), Thomassen (1981), Grünbaum (2003, Fig. 17.1.5)42Faulkner-Younger graph 42Faulkner and Younger (1974)42Grinberg graph 42Faulkner and Younger (1974)44Faulkner-Younger graph 44Faulkner and Younger (1974)44Grinberg graph 44Sachs (1968), Berge (1973), Read and Wilson (1998, p. 274)46Grinberg graph 46Bondy..

### Seymour conjecture

Seymour conjectured that a graph of order with minimum vertex degree contains the th graph power of a Hamiltonian cycle, generalizing Pósa's Conjecture. Komlós et al. (1998) proved the conjecture for sufficiently large using Szemerédi's regularity lemma and a technique called the blow-up lemma.

### Hamilton decomposition

A Hamilton decomposition (also called a Hamiltonian decomposition; Bosák 1990, p. 123) of a Hamiltonian regular graph is a partition of its edge set into Hamiltonian cycles. The figure above illustrates the six distinct Hamilton decompositions of the pentatope graph .The definition is sometimes extended to a decomposition into Hamiltonian cycles for a regular graph of even degree or into Hamiltonian cycles and a single perfect matching for a regular graph of odd degree (Alspach 2010), with a decomposition of the latter type being known as a quasi-Hamilton decomposition (Bosák 1990, p. 123).For a cubic graph, a Hamilton decomposition is equivalent to a single Hamiltonian cycle. For a quartic graph, a Hamilton decomposition is equivalent to a Hamiltonian cycle , the removal of whose edges leaves a connected graph. When this connected graph exists, it gives the second of the pair of Hamiltonian cycles which together..

### P&oacute;sa's conjecture

Dirac (1952) proved that if the minimum vertex degree for a graph on nodes, then contains a Hamiltonian cycle (Bollobás 1978, Komlós et al. 1996).In 1962, Pósa conjectured that contains a square of a Hamiltonian cycle if (Erdős 1964, p. 159; Komlós et al. 1996), where a graph contains the square of a Hamiltonian cycle if there is a Hamiltonian cycle such that , for , 2, ..., .Komlós et al. (1996) proved that there exists a natural number such that if a graph has order and minimum vertex degree at least , then contains the square of a Hamiltonian cycle. This proved Pósa's conjecture (Erdős 1964) for sufficiently large . Kierstead and Quintana (1998) proved Pósa's conjecture for graphs containing a 4-clique .The conjecture was generalized by Seymour (1974) to state that if , then contains the th power of a Hamiltonian cycle (Komlós et al. 1996)...

### Planar hypotraceable graph

A planar hypotraceable graph is a hypotraceable graph that is also planar. A number of planar hypotraceable graphs are illustrated above.Using a theorem of Thomassen (1974), the Wiener-Araya graph on 42 vertices can be used to construct a planar hypotraceable graph on 162 vertices, smaller than the 186-vertex graph obtained from the 48-Zamfirescu graph using Thomassen's construction. These graphs are implemented in the Wolfram Language as GraphData["PlanarHypohamiltonian", 162] and GraphData["PlanarHypohamiltonian", 186], respectively.Jooyandeh et al. (2017) showed that there exist planar hypotraceable graphson 154 vertices, as well as on all vertex counts greater than or equal to 156.Using the 70-node Araya-Wiener graph, a 340-nodecubic planar hypotraceable graph can be constructed (Araya and Wiener 2011).Holton and Sheehan (1993) asked if there exists an integer such that a cubic planar hypotraceable..

### Barnette's conjecture

Barnette's conjecture asserts that every 3-connected bipartite cubic planar graph is Hamiltonian. The only graph on nine or fewer vertices satisfying Barnette's conditions is the cubical graph, which is indeed Hamiltonian. The skeletons of the truncated octahedron, great rhombicuboctahedron, and great rhombicosidodecahedron also satisfy the conditions and, since they are Archimedean solids, are indeed Hamiltonian. Holton et al. (1985) proved that all graphs having fewer than 66 vertices satisfy the conjecture, but the general conjecture remains open.Similarly, Barnette conjectured that all cubic, 3-connected, planar graphs with a face size of at most 6 are Hamiltonian. Aldred et al. (2000) have verified this conjecture for all graphs with fewer than 177 vertices.

### Markstr&ouml;m graph

The Markström graph is a cubic planar graph on 24 vertices which lacks cycles of length 4 and 8 but contains cycles of length 16. (In particular, it contains cycles of lengths 3, 5, 6, 7, and 9-24.)This graph is related to the open Erdős-Gyárfás conjecture, which posits that any graph with minimum vertex degree 3 contains a simple cycle whose length is a power of two. Gordon Royle and Klas Markström showed that any counterexample must have at least 17 vertices and that any cubic counterexample must have at least 30 vertices. Markström found four graphs on 24 vertices in which the only power-of-two cycles has 16 vertices; the Markström graph is the only planar graph of these four.

### Fortunate prime

Consider the Euclid numbers defined bywhere is the th prime and is the primorial. The first few values of are 3, 7, 31, 211, 2311, 30031, 510511, ... (OEIS A006862).Now let be the next prime (i.e., the smallest prime greater than ),where is the prime counting function. The first few values of are 5, 11, 37, 223, 2333, 30047, 510529, ... (OEIS A035345).Then R. F. Fortune conjectured that is prime for all . The first values of are 3, 5, 7, 13, 23, 17, 19, 23, ... (OEIS A005235), and values of up to are indeed prime (Guy 1994), a result extended to 1000 by E. W. Weisstein (Nov. 17, 2003). The indices of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, .... In numerical order with duplicates removed, the Fortunate primes are 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, ... (OEIS A046066)...

### Unistable polyhedron

A uniform-density polyhedral solid is unistable (also called monostable) if it is stable on exactly one face (Croft et al. 1991, p. 61). For example, the 19-faced polyhedron illustrated above is unistable.Whether unistability is possible with fewer faces is an unsolvedproblem.Various turtles, such as the Indian star tortoise, have unistable shapes (Rehmeyer 2007).

### Octahedral number

A figurate number which is the sum of two consecutivepyramidal numbers,(1)The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (OEIS A005900). The generating function for the octahedral numbers is(2)Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (Dickson 2005, p. 23).A related set of numbers is the number of cubes in the Haűy construction of the octahedron. Each cross section has area(3)where is an odd number, and adding all cross sections gives(4)for an odd number. Re-indexing so that gives(5)the first few values of which are 1, 7, 25, 63, 129, ... (OEIS A001845).These numbers have the generating function(6)

### Twin prime conjecture

There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true. While Hardy and Wright (1979, p. 5) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219) states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics."Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.The conjecture..

### Euler's conjecture

Define as the quantity appearing in Waring's problem, then Euler conjectured thatwhere is the floor function.

### Triangle triangle picking

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives(1)(2)where(3)is the triangle area of a triangle with vertices , , and .The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result(4)(Pfiefer..

### Multiplicative persistence

Multiply all the digits of a number by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence, and the final digit obtained is called the multiplicative digital root of .For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has an multiplicative persistence of two and a multiplicative digital root of 0. The multiplicative persistences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, ... (OEIS A031346). The smallest numbers having multiplicative persistences of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (OEIS A003001; Wells 1986, p. 78). There is no number with multiplicative persistence (Carmody 2001; updating Wells 1986, p. 78). It is conjectured that the..

### Totient valence function

is the number of integers for which the totient function , also called the multiplicity of (Guy 1994). Erdős (1958) proved that if a multiplicity occurs once, it occurs infinitely often.The values of for , 2, ... are 2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, ... (OEIS A014197), and the nonzero values are 2, 3, 4, 4, 5, 2, 6, 6, 4, 5, 2, 10, 2, 2, 7, 8, 9, ... (OEIS A058277), which occur for , 2, 4, 6, 8, 10, 12, 16, 18, 20, ... (OEIS A002202). The table below lists values for . such that 121, 2233, 4, 6445, 8, 10, 12647, 9, 14, 188515, 16, 20, 24, 3010211, 2212613, 21, 26, 28, 36, 4216617, 32, 34, 40, 48, 6018419, 27, 38, 5420525, 33, 44, 50, 6622223, 46241035, 39, 45, 52, 56, 70, 72, 78, 84, 9028229, 5830231, 6232751, 64, 68, 80, 96, 102, 12036837, 57, 63, 74, 76, 108, 114, 12640941, 55, 75, 82, 88, 100, 110, 132, 15042443, 49, 86, 9844369, 92, 13846247, 94481165, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210The smallest such that has exactly 2, 3, 4, ... solutions are given by 1,..

### Totient function

The totient function , also called Euler's totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) , where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function can be simply defined as the number of totatives of . For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so .The totient function is implemented in the WolframLanguage as EulerPhi[n].The number is called the cototient of and gives the number of positive integers that have at least one prime factor in common with . is always even for . By convention, , although the Wolfram Language defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of for , 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (OEIS A000010). The totient function is..

### Mertens theorem

Consider the Euler product(1)where is the Riemann zeta function and is the th prime. , but taking the finite product up to , premultiplying by a factor , and letting gives(2)(3)where is the Euler-Mascheroni constant (Havil 2003, p. 173). This amazing result is known as the Mertens theorem.At least for , the sequence of finite products approaches strictly from above (Rosser and Schoenfeld 1962). However, it is highly likely that the finite product is less than its limiting value for infinitely many values of , which is usually the case for any such inequality due to the presence of zeros of on the critical line . An example is Littlewood's famous proof that the sense of the inequality , where is the prime counting function and is the logarithmic integral, reverses infinitely often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend [this] result to show that [the Mertens inequality] fails for large ; we have not investigated..

### Tetrahedron triangle picking

Given a regular tetrahedron of unit volume, the mean triangle area of a triangle picked at random inside it is approximately , and the variance is .

### Matchstick graph

A matchstick graph is a simple graph which has a graph embedding that is planar, for which all distances between vertices have unit distance, and which is non-degenerate (so no vertices are coincident, no edges cross or overlap, and no vertices are coincident with edges on which they are not incident).A matchstick graph is therefore both planar and unit-distance, but a planar unit-distance graph may fail to be a matchstick graph if a single embedding cannot be constructed having both properties. Examples include the prism graphs and Moser spindle, with the sole 6-vertex connected planar unit-distance non-matchstick graph being the 3-prism graph . The numbers of connected graphs on , 2, ... vertices that are planar and unit-distance but not matchstick are 0, 0, 0, 0, 0, 1, 13, ... (E. Weisstein, Apr. 30, 2018), where the 7-vertex examples are illustrated above.The numbers of connected matchstick graphs on , 2, ... nodes are 1, 1, 2,..

### Eberhart's conjecture

If is the th prime such that is a Mersenne prime, thenIt was modified by Wagstaff (1983) to yield Wagstaff'sconjecture,where is the Euler-Mascheroni constant.

### Tetrahedron circumscribing

The (not necessarily regular) tetrahedron of least volume circumscribed around a convex body with volume is not known. If is a parallelepiped, then the smallest-volume tetrahedron containing it has volume 9/2. It is conjectured that this is the worst possible fit for the general problem, but this remains unproved.

### Tetrahedral number

A figurate number of the form(1)(2)(3)where is the th triangular number and is a binomial coefficient. These numbers correspond to placing discrete points in the configuration of a tetrahedron (triangular base pyramid). Tetrahedral numbers are pyramidal numbers with , and are the sum of consecutive triangular numbers. The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (OEIS A000292). The generating function for the tetrahedral numbers is(4)Tetrahedral numbers are even, except for every fourthtetrahedral number, which is odd (Conway and Guy 1996).The only numbers which are simultaneously square and tetrahedral are , , and (giving , , and ), as proved by Meyl (1878; cited in Dickson 2005, p. 25).Numbers which are simultaneously triangular and tetrahedral satisfy the binomial coefficient equation(5)the only solutions of which are(6)(7)(8)(9)(10)(OEIS A027568; Avanesov 1966/1967; Mordell1969, p. 258; Guy 1994, p. 147).Beukers..

### Disk triangle picking

Pick three points , , and distributed independently and uniformly in a unit disk (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is(1)Using disk point picking, this can be writtenas(2)where(3)A trigonometric substitution can then be used to remove the trigonometric functions and split the integral into(4)where(5)(6)However, the easiest way to evaluate the integral is using Crofton's formula and polar coordinates to yield a mean triangle area(7)for unit-radius disks (OEIS A189511), or(8)for unit-area disks (OEIS A093587; Woolhouse 1867; Solomon 1978; Pfiefer 1989; Zinani 2003). This problem is very closely related to Sylvester's four-point problem, and can be derived as the limit as of the general polygon triangle picking problem.The distribution of areas, illustrated above, is apparently not known exactly.The probability that three random points in a disk form an acute..

### Magic square

A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, ..., arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constantIf every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square.The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown..

### Determined by spectrum

Two nonisomorphic graphs can share the same graph spectrum, i.e., have the same eigenvalues of their adjacency matrices. Such graphs are called cospectral. For example, the graph union and star graph , illustrated above, both have spectrum (Skiena 1990, p. 85). This is the smallest pair of simple graphs that are cospectral. Determining which graphs are uniquely determined by their spectra is in general a very hard problem.Only a small fraction of graphs are known to be so determined, but it is conceivablethat almost all graphs have this property (van Dam and Haemers 2002).In the Wolfram Language, graphs knownto be determined by their spectra are identified as GraphData["DeterminedBySpectrum"].The numbers of simple graphs on , 2, ... nodes that are determined by spectrum are 1, 2, 4, 11, 32, 146, 934, 10624, 223629, ... (OEIS A178925), while the corresponding numbers not determined by spectrum are 0, 0, 0, 0, 2, 10, 110, 1722,..

The quantities obtained from cubic, hexagonal, etc., lattice sums, evaluated at , are called Madelung constants.For cubic lattice sums(1)the Madelung constants expressible in closed form for even indices , a few examples of which are summarized in the following table, where is the Dirichlet beta function and is the Dirichlet eta function.OEISconstant2A0860544A016639To obtain the closed form for , break up the double sum into pieces that do not include ,(2)(3)(4)where the negative sums have been reindexed to run over positive quantities. But , so all the above terms can be combined into(5)The second of these sums can be done analytically as(6)which in the case reduces to(7)The first sum is more difficult, but in the case can be written(8)Combining these then gives the original sum as(9) is given by Benson's formula (Borwein and Bailey 2003, p. 24)(10)(11)(12)(OEIS A085469), where the prime indicates thatsummation over (0, 0, 0)..

### Tait's knot conjectures

P. G. Tait undertook a study of knots in response to Kelvin's conjecture that the atoms were composed of knotted vortex tubes of ether (Thomson 1869). He categorized knots in terms of the number of crossings in a plane projection. He also made some conjectures which remained unproven until the discovery of Jones polynomials: 1. Reduced alternating diagrams have minimal linkcrossing number, 2. Any two reduced alternating diagrams of a given knot have equal writhe,3. The flyping conjecture, which states that the number of crossings is the same for any reduced diagram of an alternating knot. Conjectures (1) and (2) were proved by Kauffman (1987), Murasugi (1987ab), and Thistlethwaite (1987, 1988) using properties of the Jones polynomial or Kauffman polynomial F (Hoste et al. 1998). Conjecture (3) was proved true by Menasco and Thistlethwaite (1991, 1993) using properties of the Jones polynomial (Hoste et al. 1998)...

### Cyclic number

A cyclic number is an -digit integer that, when multiplied by 1, 2, 3, ..., , produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913).The decimal expansions giving the first fewcyclic numbers are(1)(2)(3)(4)(OEIS A004042).The numbers of cyclic numbers for , 1, 2, ... are 0, 1, 9, 60, 467, 3617, 25883, 248881, 2165288, 19016617, 170169241, ... (OEIS A086018). It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be Artin's constant . The fraction of cyclic numbers among primes is 0.3739551.When a cyclic number is multiplied by its generator, the result is a string of 9s.This is a special case of Midy's theorem.See Yates (1973) for a table of prime period lengths for primes ...

### Cube triangle picking

The mean triangle area of a triangle picked at random inside a unit cube is , with variance .The distribution of areas, illustrated above, is apparently not known exactly.The probability that a random triangle in a cube is obtuse is approximately .

### Strongly regular graph

A -regular simple graph on nodes is strongly -regular if there exist positive integers , , and such that every vertex has neighbors (i.e., the graph is a regular graph), every adjacent pair of vertices has common neighbors, and every nonadjacent pair has common neighbors (West 2000, pp. 464-465). A graph that is not strongly regular is said to be weakly regular.The complete graph is strongly regular for all . The status of the trivial singleton graph is unclear. Opinions differ on if is a strongly regular graph, though since it has no well-defined parameter, it is preferable to consider it not to be strongly regular (A. E. Brouwer, pers. comm., Feb. 6, 2013).The graph complement of a non-empty non-complete strongly regular graph with parameters is another strongly regular graph with parameters .A number of strongly regular graphs are implemented in the WolframLanguage as GraphData["StronglyRegular"].The..

### Levy's conjecture

Levy (1963) noted that(1)(2)and from this observation, conjectured that all odd numbers are the sum of a prime plus twice a prime. This conjecture is a stronger version of the weak Goldbach conjecture and has been verified up to (Corbit 1999).The number of ways to express as for and primes and , 2, ... are 0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, ... (OEIS A046927).

### Lehmer's totient problem

Lehmer's totient problem asks if there exist any composite numbers such that , where is the totient function? No such numbers are known. However, any such an would need to be a Carmichael number, since for every element in the integers (mod ), , so and is a Carmichael number.In 1932, Lehmer showed that such an must be odd and squarefree, and that the number of distinct prime factors must satisfy . This was subsequently extended to . The best current result is and , improving the lower bound of Cohen and Hagis (1980) since there are no Carmichael numbers less than having distinct prime factors; Pinch). However, even better results are known in the special cases , in which case (Wall 1980), and , in which case and (Lieuwens 1970).

### Square point picking

Picking two independent sets of points and from a unit uniform distribution and placing them at coordinates gives points uniformly distributed over the unit square.The distribution of distances from a randomly selected point in the unit square to its center is illustrated above.The expected distance to the square's center is(1)(2)(3)(4)(Finch 2003, p. 479; OEIS A103712), where is the universal parabolic constant. The expected distance to a fixed vertex is given by(5)(6)which is exactly twice .The expected distances from the closest and farthest vertices are given by(7)(8)Pick points at randomly in a unit square and take the convex hull . Let be the expected area of , the expected perimeter, and the expected number of vertices of . Then(9)(10)(11)(12)(13)(14)(OEIS A096428 and A096429), where is the multiplicative inverse of Gauss's constant, is the gamma function, and is the Euler-Mascheroni constant (Rényi and Sulanke..

### Cram&eacute;r conjecture

The Cramér conjecture is the unproven conjecturethatwhere is the th prime.

### Sphere tetrahedron picking

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

### Legendre's conjecture

Legendre's conjecture asserts that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398). It is one of Landau's problems.Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415).The smallest primes between and for , 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEIS A007491). The numbers of primes between and for , 2, ... are given by 2, 2, 2, 3, 2, 4, 3, 4, ... (OEIS A014085).

### Completely positive matrix

A completely positive matrix is a real square matrix that can be factorized aswhere stands for the transpose of and is any (not necessarily square) matrix with nonnegative elements. The least possible number of columns () of is called the factorization index (or the cp-rank) of . The study of complete positivity originated in inequality theory and quadratic forms (Diananda 1962, Hall and Newman 1963).There are two basic problems concerning complete positivity. 1. When is a given real matrix completely positive? 2. How can the cp-rank of be calculated? These two fundamental problems still remains open. The applications of completely positive matrices can be found in block designs (Hall and Newman 1963) and economic modelling (Gray and Wilson 1980).

### Solitary number

A solitary number is a number which does not have any friends. Solitary numbers include all primes, prime powers, and numbers for which , where is the greatest common divisor of and and is the divisor function. The first few numbers satisfying are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (OEIS A014567). Numbers such as 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 can also be easily proved to be solitary (Hickerson 2002).Some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large. For example, 24 is friendly because is a friendly pair. However, there exist numbers such as , 45, 48, and 52 which are solitary but for which . It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary,..

### Smarandache function

The Smarandache function is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that gives the smallest value for a given at which (i.e., divides factorial). For example, the number 8 does not divide , , , but does divide , so .For , 2, ..., is given by 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (OEIS A002034), where it should be noted that Sloane defines , while Ashbacher (1995) and Russo (2000, p. 4) take . The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A046022), which occur at the values where . The incrementally smallest values of relative to are = 1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 3/40, 1/15, 1/16, 1/24, 1/30, ... (OEIS A094404 and A094372), which occur at , 6, 12, 20, 24, 40, 60, 80, 90, 112, 120, 180, ... (OEIS A094371).Formulas exist for immediately computing for special forms of . The simplest cases are(1)(2)(3)(4)(5)where is a prime,..

### Simplex simplex picking

Given a simplex of unit content in Euclidean -space, pick points uniformly and independently at random, and denote the expected content of their convex hull by . Exact values are known only for and 2.(1)(2)(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ...(OEIS A026741 and A026741).(3)(4)where is a harmonic number (Buchta 1984, 1986), giving the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS A093762 and A093763).Not much is known about , although(5)(Buchta 1983, 1986) and(6)(Buchta 1986).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary .

### Sierpi&324;ski number of the second kind

A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .The smallest known example is , proved in 1962 by J. Selfridge, but the fate of a number of smaller candidates remains to be determined before this number can be established as the smallest such number. As of 1996, 35 candidates remained (Ribenboim 1996, p. 358), a number which had been reduced to 17 by the beginning of 2002 (Peterson 2003).In March 2002, L. K. Helm and D. A. Norris began a distributed computing effort dubbed "seventeen or bust" to eliminate the remaining candidates. With the aid of collaborators across the globe, this number was reduced to 12 as of December 2003 (Peterson 2003, Helm and Norris). The following table summarizes numbers subsequently found to be prime by "seventeen or bust," leaving only..

### Sierpi&324;ski's composite number theorem

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers such that is composite for every . Numbers with this property are called Sierpiński numbers of the second kind, and analogous numbers with the plus sign replaced by a minus are called Riesel numbers. It is conjectured that the smallest value of for a Sierpiński number of the second kind is (although a handful of smaller candidates remain to be eliminated) and that the smallest Riesel number is .

### Carmichael's totient function conjecture

It is thought that the totient valence function , i.e., if there is an such that , then there are at least two solutions . This assertion is called Carmichael's totient function conjecture and is equivalent to the statement that there exists an such that (Ribenboim 1996, pp. 39-40).Dickson (2005, p. 137) states that the conjecture was proved by Carmichael (1907), who also developed a method of finding the solution (Carmichael 1909). The result also appears as in exercise in Carmichael (1914). However, Carmichael (1922) subsequently discovered an error in the proof, and the conjecture currently remains open. Any counterexample to the conjecture must have more than digits (Schlafly and Wagon 1994; conservatively given as in Conway and Guy 1996, p. 155). This result was extended by Ford (1999), who showed that any counterexample must have more than digits.Ford (1998ab) showed that if there is a counterexample to Carmichael's..

### Brocard's conjecture

Brocard's conjecture states thatfor , where is the prime counting function and is the th prime. For , 2, ..., the first few values are 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, ... (OEIS A050216).

### Shanks' conjecture

Let be the first prime which follows a prime gap of between consecutive primes. Shanks' conjecture holds thatWolf conjectures a slightly different formwhich agrees better with numerical evidence.

### Selfridge's conjecture

There exist infinitely many with for all , where is the th prime. Also, there exist infinitely many such that for all .

### Hypercube line picking

Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..

### Beraha constants

The th Beraha constant (or number) is given by is , where is the golden ratio, is the silver constant, and . The following table summarizes the first few Beraha numbers.approx.1420314252.6186373.24783.41493.532103.618Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of (G. Royle, pers. comm., Nov. 21, 2005). However, the roots of chromatic polynomials of planar triangulations appear to cluster around the Beraha numbers (and, technically, are conjectured to be accumulation points of roots of planar triangulation chromatic polynomials).

### Benson's formula

An equation for a lattice sum (Borwein and Bailey 2003, p. 26)(1)(2)Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to (OEIS A085469), a value known as "the" Madelung constant.No closed form for is known (Bailey et al. 2006).

### Hyperbolic octahedron

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with .It is given by the parametric equations(1)(2)(3)for and .It is an algebraic surface of degree 18 withcomplicated terms. However, it has the simple Cartesian equation(4)where is taken to mean . Cross sections through the , , or planes are therefore astroids.The first fundamental form coefficientsare(5)(6)(7)the second fundamental form coefficientsare(8)(9)(10)The area element is(11)giving the surface area as(12)The volume is given by(13)an exact expression for which is apparently not known.The Gaussian curvature is(14)while the mean curvature is given by a complicatedexpression.

### Hyperbolic knot

A hyperbolic knot is a knot that has a complement that can be given a metric of constant curvature . All hyperbolic knots are prime knots (Hoste et al. 1998).A prime knot on 10 or fewer crossings can be tested in the Wolfram Language to see if it is hyperbolic using KnotData[knot, "Hyperbolic"].Of the prime knots with 16 or fewer crossings, all but 32 are hyperbolic. Of these 32, 12 are torus knots and the remaining 20 are satellites of the trefoil knot (Hoste et al. 1998). The nonhyperbolic knots with nine or fewer crossings are all torus knots, including (the -torus knot), , , (the -torus knot), and , the first few of which are illustrated above.The following table gives the number of nonhyperbolic and hyperbolic knots of crossing starting with .typeOEIScountstorusA0517641, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ...satelliteA0517650, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ...nonhyperbolicA0524071, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 8, 11, ...hyperbolicA0524080,..

### Ball triangle picking

Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is(1)(Buchta and Müller 1984, Finch 2010). random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then (2)(3)These can be combined..

### Aspiring number

Letwhere is the divisor function and is the restricted divisor function, and define the aliquot sequence of byIf the sequence reaches a constant, the constant is known as a perfect number. A number that is not perfect but whose sequence becomes constant is known as an aspiring number. For example, beginning with 25 gives the sequence 25, 6, 6, 6, ..., so 25 is an aspiring number and 6 is a perfect number.The first few aspiring numbers are 25, 95, 119, 143, ... (OEIS A063769). It is not known if 276 is an aspiring number, though it is very unlikely for this to be the case.

### Andrica's conjecture

Andrica's conjecture states that, for the th prime number, the inequalityholds, where the discrete function is plotted above. The high-water marks for occur for , 2, and 4, with , with no larger value among the first primes. Since the Andrica function falls asymptotically as increases, a prime gap of ever increasing size is needed to make the difference large as becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven. bears a strong resemblance to the prime difference function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ... (OEIS A001223).A generalization of Andrica's conjecture considers the equationand solves for . The smallest such is (OEIS A038458), known as the Smarandache constant, which occurs for and (Perez)...

### Hard square entropy constant

Let be the number of (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ..., is given by 2, 7, 63, 1234, ... (OEIS A006506).The hard square entropy constant is defined by(OEIS A085850). It is not known if this constanthas an exact representation.The quantity arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.

### Harborth graph

The Harborth graph is the smallest known 4-regular matchstick graph. It is therefore both planar and unit-distance. It has 104 edges and 52 vertices. This graph was named after its discoverer H. Harborth, who first presented it to a general public in 1986 (Harborth 1994, Petersen 1996, Gerbracht 2006).The Harborth graph is implemented in the WolframLanguage as GraphData["HarborthGraph"].Analytic expressions for the vertices consisting of algebraic numbers of degree 22 (with large coefficients) were derived by Gerbracht (2006). As a consequence, Gerbracht (2006) also proved that the Harborth graph is rigid.

### Almost perfect number

An almost perfect number, also known as a least deficient or slightly defective (Singh 1997) number, is a positive integer for which the divisor function satisfies . The only known almost perfect numbers are the powers of 2, namely 1, 2, 4, 8, 16, 32, ... (OEIS A000079).It seems to be an open problem to show that a number is almost perfect only if it is of the form .

### Quasiperfect number

A quasiperfect number, called a "slightly excessive number" by Singh (1997), is a "least" abundant number, i.e., one such thatQuasiperfect numbers are therefore the sum of their nontrivial divisors. No quasiperfect numbers are known, although if any exist, they must be greater than and have seven or more distinct prime factors (Hagis and Cohen 1982).

### Agoh's conjecture

Let be the th Bernoulli number and considerwhere the residues of fractions are taken in the usual way so as to yield integers, for which the minimal residue is taken. Agoh's conjecture states that this quantity is iff is prime. There are no counterexamples less than (S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's conjecture would be a contradiction to Giuga's conjecture, and vice versa.For , 2, ..., the minimal integer residues (mod ) is 0, , , 0, , 0, , 0, , 0, , ... (OEIS A046094).Kellner (2002) provided a short proof of the equivalence of Giuga'sand Agoh's conjectures. The combined conjecture can be described by a sum of fractions.

### Abc conjecture

The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal , there exists a constant such that for any three relatively prime integers , , satisfying(1)the inequality(2)holds, where indicates that the product is over primes which divide the product . If this conjecture were true, it would imply Fermat's last theorem for sufficiently large powers (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are at least non-Wieferich primes for some constant (Silverman 1988, Vardi 1991).The conjecture can also be stated by defining the height and radical of the sum as(3)(4)where runs over all prime divisors of , , and . Then the abc conjecture states that for all , there exists a constant such that for all ,(5)(van Frankenhuysen 2000). van Frankenhuysen (2000) has shown that there exists an infinite sequence of sums or rational integers with large height compared..

### Grimm's conjecture

Grimm conjectured that if , , ..., are all composite numbers, then there are distinct primes such that for .

### 2

The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since . A quantity taken to the power 2 is said to be squared. The number of times a given binary number is divisible by 2 is given by the position of the first , counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.The only known solutions to the congruenceare summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below on Jan. 27 2007, finding no other solutions in this range.reference4700063497Guy (1994)3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)8365386194032363Crump (pers. comm., 2000)10991007971508067Crump (2007)63130707451134435989380140059866138830623361447484274774099906755Montgomery (1999)In general,..

Petersson considered the absolutely converging DirichletL-series(1)Writing the denominator as(2)where(3)and(4)Petersson conjectured that and are always complex conjugate, which implies(5)and(6)This conjecture was proven by Deligne (1974), which also proved the tau conjecture as a special case. Deligne was awarded the Fields medal for his proof.

### Perfect magic cube

A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each orthogonal slice sum to the same number (i.e., the magic constant ). While this terminology is standard in the published literature (Gardner 1976, Benson and Jacoby 1981, Gardner 1988, Pickover 2002), it has been suggested at various times that such cubes instead be termed Myers cubes, Myers diagonal cubes, or diagonal magic cube (Heinz).There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4 (Schroeppel 1972; Benson and Jacoby 1981, pp. 23-25; Gardner 1988). While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist (Wells 1986, p. 72), although Schroeppel (1972) and Gardner (1988) note that any such cube must have a central value of 63. (Confusingly, Benson and Jacoby (1981, p. 5)..

### Weisstein's conjecture

On July 10, 2003, Eric Weisstein computed the numbers of (0,1)-matrices all of whose eigenvalues are real and positive, obtaining counts for , 2, ... of 1, 3, 25, 543, 29281, .... Based on agreement with OEIS A003024, Weisstein then conjectured that is equal to the number of labeled acyclic digraphs on vertices.This result was subsequently proved by McKay et al. (2003, 2004).

### Tau conjecture

The tau conjecture, also known as Ramanujan's hypothesis after its proposer, states thatwhere is the tau function. This was proven by Deligne (1974) in the course of proving the more general Petersson conjecture. Deligne was awarded the Fields medal for his proof.

### Double bubble

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..

### Smith conjecture

The set of fixed points which do not move as a knot is transformed into itself is not a knot. The conjecture was proved in 1978 (Morgan and Bass 1984). According to Morgan and Bass (1984), the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it.The generalized Smith conjecture considers to be a piecewise linear -dimensional hypersphere in , and the -fold cyclic covering of branched along , and asks if is unknotted if is an (Hartley 1983). This conjecture is true for , and false for , with counterexamples in the latter case provided by Giffen (1966), Gordon (1974), and Sumners (1975).

### Checkers

Checkers is a two-player game with the most common variant played on an checkerboard with each player starts with twelve pieces of a fixed color on opposite sites of the board. The most common variant of checkers is so-called "pool checkers," also called "Spanish pool checkers," draughts or draught (in the United Kingdom and some other countries), American checkers, and straight checkers. Play proceeds alternately between players, where all pieces may initially only move and capture in a forward diagonal direction. The allowable direction of play is modified for a piece if it is "crowned" by reaching the other side of the board, after which it may move either forwards or backwards. An opponent's piece may be captured by jumping over it diagonally, and the game is won by capturing all the opponents pieces or leaving the opponent with no legal moves.The most widely available sets of checkers consist of black and..

### Taylor's condition

For a given positive integer , does there exist a weighted tree with graph vertices whose paths have weights 1, 2, ..., , where is a binomial coefficient? Taylor showed that no such tree can exist unless it is a perfect square or a perfect square plus 2. No such trees are known except , 3, 4, and 6.Székely et al. showed computationally that there are no such trees with and 11. They also showed that if there is such a tree on vertices then the maximum vertex degree is at most and that there is no path of length larger than . They conjecture that there are only finitely many such trees.

### Vizing conjecture

Let denote the domination number of a simple graph . Then Vizing (1963) conjectured thatwhere is the graph product. While the full conjecture remains open, Clark and Suen (2000) have proved the looser result

### Ulam's conjecture

Let graph have points and graph have points , where . Then if for each , the subgraphs and are isomorphic, then the graphs and are isomorphic.

### Tutte conjecture

Tutte (1971/72) conjectured that there are no 3-connected nonhamiltonian bicubic graphs. However, a counterexample was found by J. D. Horton in 1976 (Gropp 1990), and several smaller counterexamples are now known.Known small counterexamples are summarized in the following table and illustrated above.namereference50Georges graphGeorges (1989), Grünbaum (2006, 2009)54Ellingham-Horton 54-graphEllingham and Horton (1983)78Ellingham-Horton 78-graphEllingham (1981, 1982)78Owens graphOwens (1983)92Horton 92-graphHorton (1982)96Horton 96-graphBondy and Murty (1976)

### Strong perfect graph theorem

The theorem, originally conjectured by Berge (1960, 1961), that a graph is perfect iff neither the graph nor its graph complement contains an odd graph cycle of length at least five as an induced subgraph became known as the strong perfect graph conjecture (Golumbic 1980; Skiena 1990, p. 221). The conjecture can be stated more simply as the assertion that a graph is perfect iff it contains no odd graph hole and no odd graph antihole. The proposition can be stated even more succinctly as "a graph is perfect iff it is a Berge graph."This conjecture was proved in May 2002 following a remarkable sequence of results by Chudnovsky, Robertson, Seymour, and Thomas (Cornuéjols 2002, MacKenzie 2002).

### Graph spectrum

The set of graph eigenvalues of the adjacency matrix is called the spectrum of the graph. (But note that in physics, the eigenvalues of the Laplacian matrix of a graph are sometimes known as the graph's spectrum.) The spectrum of a graph with -fold degenerate eigenvalues is commonly denoted (van Dam and Haemers 2003) or (Biggs 1993, p. 8; Buekenhout and Parker 1998).The product over the elements of the spectrum of a graph is known as the characteristic polynomial of , and is given by the characteristic polynomial of the adjacency matrix of with respect to the variable .The largest absolute value of a graph's spectrum is known as its spectralradius.The spectrum of a graph may be computed in the Wolfram Language using Eigenvalues[AdjacencyMatrix[g]]. Precomputed spectra for many named graphs can be obtained using GraphData[graph, "Spectrum"].A graph whose spectrum consists entirely of integers is known as an integralgraph.The..

### Bell number

The number of ways a set of elements can be partitioned into nonempty subsets is called a Bell number and is denoted (not to be confused with the Bernoulli number, which is also commonly denoted ).For example, there are five ways the numbers can be partitioned: , , , , and , so ., and the first few Bell numbers for , 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (OEIS A000110). The numbers of digits in for , 1, ... are given by 1, 6, 116, 1928, 27665, ... (OEIS A113015).Bell numbers are implemented in the WolframLanguage as BellB[n].Though Bell numbers have traditionally been attributed to E. T. Bell as a result of the general theory he developed in his 1934 paper (Bell 1934), the first systematic study of Bell numbers was made by Ramanujan in chapter 3 of his second notebook approximately 25-30 years prior to Bell's work (B. C. Berndt, pers. comm., Jan. 4 and 13, 2010).The first few prime Bell numbers occur at indices..

### Social golfer problem

Twenty golfers wish to play in foursomes for 5 days. Is it possible for each golfer to play no more than once with any other golfer? The answer is yes, and the following table gives a solution.MonABCDEFGHIJKLMNOPQRSTTueAEIMBJOQCHNTDGLSFKPRWedAGKOBIPTCFMSDHJRELNQThuAHLPBKNSCEORDFIQGJMTFriAFJNBLMRCGPQDEKTHIOSEvent organizers for bowling, golf, bridge, or tennis frequently tackle problems of this sort, unaware of the problem complexity. In general, it is an unsolved problem. A table of known results is maintained by Harvey.

### Projective plane

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.A finite projective plane of order is formally defined as a set of points with the properties that: 1. Any two points determine a line,2. Any two lines determine a point,3. Every point has lines on it, and 4. Every line contains points. (Note that some of these properties are redundant.) A projective plane is therefore a symmetric (, , 1)..

### Erd&#337;s squarefree conjecture

The central binomial coefficient is never squarefree for . This was proved true for all sufficiently large by Sárkőzy's theorem. Goetgheluck (1988) proved the conjecture true for and Vardi (1991) for . The conjecture was proved true in its entirety by Granville and Ramare (1996).

### Trinomial coefficient

A trinomial coefficient is a coefficient of the trinomial triangle. Following the notation of Andrews (1990), the trinomial coefficient , with and , is given by the coefficient of in the expansion of . Therefore,(1)The trinomial coefficient can be given by the closed form(2)where is a Gegenbauer polynomial.Equivalently, the trinomial coefficients are defined by(3)The trinomial coefficients also have generatingfunction(4)(5)i.e.,(6)The trinomial triangle gives the triangle oftrinomial coefficients,(7)(OEIS A027907).The central column of the trinomial triangle gives the centraltrinomial coefficients.The trinomial coefficient is also given by the number of permutations of symbols, each , 0, or 1, which sum to . For example, there are seven permutations of three symbols which sum to 0, , , , , and , , , so .An alternative (but different) definition of the trinomial coefficients is as the coefficients in (Andrews 1990), which is therefore..

### Central trinomial coefficient

The th central trinomial coefficient is defined as the coefficient of in the expansion of . It is therefore the middle column of the trinomial triangle, i.e., the trinomial coefficient . The first few central trinomial coefficients for , 2, ... are 1, 3, 7, 19, 51, 141, 393, ... (OEIS A002426).The central trinomial coefficient is also gives the number of permutations of symbols, each , 0, or 1, which sum to 0. For example, there are seven such permutations of three symbols: , , , , and , , .The generating function is given by(1)(2)The central trinomial coefficients are given by the recurrenceequation(3)with , but cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160).The coefficients satisfy the congruence(4)(T. D. Noe, pers. comm., Mar. 15, 2005) and(5)for a prime, which is easy to show using Fermat's little theorem (T. D. Noe, pers. comm., Oct. 26, 2005).Sum..

### Central binomial coefficient

The th central binomial coefficient is defined as(1)(2)where is a binomial coefficient, is a factorial, and is a double factorial.These numbers have the generating function(3)The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (OEIS A000984). The numbers of decimal digits in for , 1, ... are 1, 6, 59, 601, 6019, 60204, 602057, 6020597, ... (OEIS A114501). These digits converge to the digits in the decimal expansion of (OEIS A114493).The central binomial coefficients are never prime except for .A scaled form of the central binomial coefficient is known as a Catalannumber(4)Erdős and Graham (1975) conjectured that the central binomial coefficient is never squarefree for , and this is sometimes known as the Erdős squarefree conjecture. Sárkőzy's theorem (Sárkőzy 1985) provides a partial solution which states that the binomial coefficient is never squarefree for all sufficiently..

### Dyson's conjecture

Based on a problem in particle physics, Dyson (1962abc) conjectured that the constantterm in the Laurent seriesis the multinomial coefficientThe theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).

### Borwein conjectures

Use the definition of the q-series(1)and define(2)Then P. Borwein has conjectured that (1) the polynomials , , and defined by(3)have nonnegative coefficients, (2) the polynomials , , and defined by(4)have nonnegative coefficients, (3) the polynomials , , , , and defined by(5)have nonnegative coefficients, (4) the polynomials , , and defined by(6)have nonnegative coefficients, (5) for odd and , consider the expansion(7)with(8)then if is relatively prime to and , the coefficients of are nonnegative, and (6) given and , consider(9)the generating function for partitions inside an rectangle with hook difference conditions specified by , , and . Let and be positive rational numbers and an integer such that and are integers. then if (with strict inequalities for ) and , then has nonnegative coefficients...

### Costa minimal surface

The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.Rather amazingly, the Costa surface belongs to the dihedral group of symmetries.The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon1999).On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was..

### Nirenberg's conjecture

If the Gauss map of a complete minimal surface omits a neighborhood of the sphere, then the surface is a plane. This was proven by Osserman (1959). Xavier (1981) subsequently generalized the result as follows. If the Gauss map of a complete minimal surface omits points, then the surface is a plane.

### Robbins conjecture

The conjecture that the equations for a Robbins algebra, commutativity, associativity,and the Robbins axiomwhere denotes NOT and denotes OR, imply those for a Boolean algebra. The conjecture was finally proven using a computer (McCune 1997).

### Burnside's conjecture

In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order do not exist." Of course, simple groups of prime order do exist, namely the groups for any prime . Therefore, Burnside conjectured that every finite simple group of non-prime order must have even order. The conjecture was proven true by Feit and Thompson (1963).

### Mrs. perkins's quilt

A Mrs. Perkins's quilt is a dissection of a square of side into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect square dissection, however, the smaller squares need not be all different sizes. In addition, only prime dissections are considered so that patterns which can be dissected into lower-order squares are not permitted.The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (OEIS A005670), the first few of which are illustrated above.On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly..

### Powerful number

An integer such that if , then , is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form for .The numbers of powerful numbers , , , ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896).Golomb (1970) showed that the sum over the reciprocals of the powerful numbers is given by(1)(2)(OEIS A082695), where is the Riemann zeta function.Not every natural number is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large natural number is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerful numbers, the first few being (8, 9), (288, 289), (675, 676), (9800, 9801), ... (OEIS A060355 and A118893).Erdős (1975/1965) conjectured that there do not exist three consecutive powerful numbers...

### Positive eigenvalued matrix

The numbers of positive definite matrices of given types are summarized in the following table. For example, the three positive eigenvalues (0,1)-matrices areall of which have eigenvalue 1 with degeneracy oftwo.matrix typeOEIScounts(0,1)-matrixA0030241, 3, 25, 543, 29281, ...(-1,0,1)-matrixA0855061, 5, 133, 18905, ...Weisstein's conjecture proposed that positive eigenvalued -matrices were in one-to-one correspondence with labeled acyclic digraphs on nodes, and this was subsequently proved by McKay et al. (2003, 2004). Counts of both are therefore given by the beautiful recurrence equationwith (Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).

### Pollock's conjecture

The conjecture due to Pollock (1850) that every number is the sum of at most five tetrahedral numbers (Dickson 2005, p. 23; incorrectly described as "pyramidal numbers" and incorrectly dated to 1928 in Skiena 1997, p. 43). The conjecture is almost certainly true, but has not yet been proven.The numbers that are not the sum of tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (OEIS A000797) of 241 terms, with being almost certainly the last such number.

### Goldbach partition

A pair of primes that sum to an even integer are known as a Goldbach partition (Oliveira e Silva). Letting denote the number of Goldbach partitions of without regard to order, then the number of ways of writing as a sum of two prime numbers taking the order of the two primes into account is(1)The Goldbach conjecture is then equivalent to the statement that or, equivalently, that , for every even integer .A plot of , sometimes known as Goldbach's comet, for up to 2000 is illustrated above.The following table summarizes the values of several variants of for , 4, ....partition typeOEISvalues 1 or primeA0010311, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, ... primeA0459170, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ... odd primeA0023750, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ...Various fractal properties have been observed in Goldbach'spartition (Liang et al. 2006)...

### Giuga's conjecture

If andis necessarily a prime? In other words, definingdoes there exist a composite such that ? It is known that iff for each prime divisor of , and (Giuga 1950, Borwein et al. 1996); therefore, any counterexample must be squarefree. A composite integer satisfies iff it is both a Carmichael number and a Giuga number. Giuga showed that there are no exceptions to the conjecture up to . This was later improved to (Bedocchi 1985) and (Borwein et al. 1996).Kellner (2002) provided a short proof of the equivalence of Giuga's and Agoh'sconjectures. The combined conjecture can be described by a sum of fractions.

### Generalized moore graph

A generalized Moore graph is a regular graph of degree where the counts of vertices at each distance , 1, ... from any vertex are 1, , , , , ..., with the last distance count not necessarily filled up. That is, all the levels are full except possibly the last which must have the rest. Alternatively, the girth is as great as the naive bound allows and the diameter is as little as the naive bound allows. Stated yet another way, a generalized Moore graph is a regular graph such that the average distance between pairs of vertices achieves the naive lower bound.The numbers of generalized Moore graphs with , 2, ... nodes are 0, 0, 0, 1, 1, 4, 3, 13, 21, ... (OEIS A088933).The numbers of cubic generalized Moore graphs with , 4, 6, ... nodes are 0, 1, 2, 2, 1, 2, 7, 6, 1, 1, ... (OEIS A005007).It is an open problem if there are infinitely many generalized Moore graphs of each degree...

### Gelfand's question

Sloane's1A000027234567892A002993491234683A002994826123574A097408182612465A097409321371356A097410674141257A097411121728248A097412266315149A0974135121141310A09741415196213Consider the leftmost (i.e., most significant) decimal digit of the numbers , , ..., . Then what are the patterns of digits occurring in the table for , 2, ... (King 1994)? For example,1. Will the digit 9 ever occur in the column? The answer is "yes," in particular at values , 63, 73, 83, 93, 156, 166, 176, ... (OEIS A097415. This problem appears in Avez (1966, p. 37), where it is attributed to Gelfand. 2. Will the row "23456789" ever appear for ? None does for . If so, will it have a frequency? If so, will the frequency be rational or irrational? 3. Will a row of all the same digit occur? No such example occurs for . 4. Will the decimal expansion of an 8-digit prime ever occur? (The answer is "yes," in particular at values , 11,..

### Wagstaff's conjecture

A modification of the Eberhart's conjecture proposed by Wagstaff (1983) which proposes that if is the th prime such that is a Mersenne prime, thenwhere is the Euler-Mascheroni constant.

### Palindromic number conjecture

Apply the 196-algorithm, which consists of taking any positive integer of two digits or more, reversing the digits, and adding to the original number. Now sum the two and repeat the procedure with the sum. Of the first numbers, only 251 do not produce a palindromic number in steps (Gardner 1979).It was therefore conjectured that all numbers will eventually yield a palindromic number. However, the conjecture has been proven false for bases which are a power of 2, and seems to be false for base 10 as well. Among the first numbers, numbers apparently never generate a palindromic number (Gruenberger 1984). The first few are 196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, ... (OEIS A006960).It is conjectured, but not proven, that there are an infinite number of palindromic primes. With the exception of 11, palindromic primes must have an odd number of digits...

### Full reptend prime

A prime for which has a maximal period decimal expansion of digits. Full reptend primes are sometimes also called long primes (Conway and Guy 1996, pp. 157-163 and 166-171). There is a surprising connection between full reptend primes and Fermat primes.A prime is full reptend iff 10 is a primitive root modulo , which means that(1)for and no less than this. In other words, the multiplicative order of (mod 10) is . For example, 7 is a full reptend prime since .The full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (OEIS A001913). The first few decimal expansions of these are(2)(3)(4)(5)Here, the numbers 142857, 5882352941176470, 526315789473684210, ... (OEIS A004042) corresponding to the periodic parts of these decimal expansions are called cyclic numbers. No general method is known for finding full reptend primes.The number of full reptend primes less than for , 2, ... are 1, 9, 60, 467, 3617, ... (OEIS A086018).A..

### P versus np problem

The P versus NP problem is the determination of whether all NP-problems are actually P-problems. If P and NP are not equivalent, then the solution of NP-problems requires (in the worst case) an exhaustive search, while if they are, then asymptotically faster algorithms may exist.The answer is not currently known, but determination of the status of this question would have dramatic consequences for the potential speed with which many difficult and important problems could be solved.In the Season 1 episode "Uncertainty Principle" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes uses the game minesweeper as an analogy for the P vs. NP problem.

### Friendly pair

Define the abundancy of a positive integer as(1)where is the divisor function. Then a pair of distinct numbers is a friendly pair (and is said to be a friend of ) if their abundancies are equal:(2)For example, (4320, 4680) is a friendly pair since , , and(3)(4)Another example is , which has index 5/2. The first few friendly pairs, ordered by smallest maximum element are (6, 28), (30, 140), (80, 200), (40, 224), (12, 234), (84, 270), (66, 308), ... (OEIS A050972 and A050973).Friendly triples and higher-order tuples are also possible. Friendly triples include (2160, 5400, 13104), (9360, 21600, 23400), and (4320, 4680, 26208), friendly quadruples include (6, 28, 496, 8128), (3612, 11610, 63984, 70434), (3948, 12690, 69936, 76986), and friendly quintuples include (84, 270, 1488, 1638, 24384), (30, 140, 2480, 6200, 40640), (420, 7440, 8190, 18600, 121920).Numbers that have friends are called friendly numbers, and numbers that do not have friends..

### Untouchable number

An untouchable number is a positive integer that is not the sum of the proper divisors of any number. The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114). Erdős has proven that there are infinitely many.It is thought that 5 is the only odd untouchable number. This would follow from a very slightly stronger version of the Goldbach conjecture, namely the conjecture that every even integer is the sum of two distinct primes. Suppose is an odd number greater than 7. Then by the conjecture, and so the proper divisors of , which are 1, , and , sum to , and so is not untouchable. 1, 3 and 7 are not untouchable, being the sum of the proper divisors of 2, 4, and 8, respectively. That leaves 5 as the only odd untouchable number (F. Adams-Watters, pers. comm., Aug. 4, 2006)...

### Ore's conjecture

Define the harmonic mean of the divisors of where is the divisor function (the number of divisors of ).For , 2, ..., the values of are then 1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, 27/13, 20/9, ... (OEIS A099377 and A099378).If is a perfect number, is an integer.Ore conjectured that if is odd, then is not an integer. This implies that no odd perfect numbers exist.

### Friendly number

A friendly number is a number that is a member of a friendly pair or a higher-order friendly -tuple. Numbers that are not friendly are said to be solitary. There are some numbers that can easily be proved to be solitary, but the status of numbers 10, 14, 15, 20, and many others remains unknown (Hickerson 2002). The numbers known to be friendly are given by 6, 12, 24, 28, 30, 40, 42, 56, 60, ... (OEIS A074902).Friendly numbers have a positive density.

### Oloid

Let two disks of radius intersect one another perpendicularly and have a diameter in common. If the distance between the centers of the disks is times their radius, then the distance from the center of gravity remains constant and so the object, known as a "two circle roller," rolls smoothly (Nishihara).If the distance of two centers of disk is equal to the radius, then the convex hull produces another figure that rolls smoothly and is known as the oloid (Schatz 1975, p. 122; Nishihara), illustrated above. The oloid is an octic surface (Trott 2004, pp. 1194-1196).For circles of radii , the surface area of the resulting oloid is(the same as that of a sphere with radius ), but no closed form is apparently known for the enclosed volume.

### Zarankiewicz's conjecture

Zarankiewicz's conjecture asserts that graph crossing number for a complete bipartite graph is(1)where is the floor function. The original proof by Zarankiewicz (1954) contained an error, but was subsequently solved in some special cases by Guy (1969). Zarankiewicz (1954) showed that in general, the formula provides an upper bound to the actual number.The problem addressed by the conjecture is sometimes known as the brick factory problem, since it was described by Turán (1977) as follows: "We worked near Budapest, in a brick factory. There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected to all the storage yards. The bricks were carried on small wheeled trucks to the storage yards. All we had to do was to put the bricks on the trucks at the kilns, push the trucks to the storage yards, and unload them there. We had a reasonable piece rate for the trucks, and..

### Smallest cubic crossing number graph

The smallest cubic graphs with graph crossing number have been termed "crossing number graphs" or -crossing graphs by Pegg and Exoo (2009). The -crossing graphs are implemented in the Wolfram Language as GraphData["CrossingNumberGraphNA"], with N being a number and X a letter, for example 3C for the Heawood graph or 8B for cubic symmetric graph .The following table summarizes and updates the smallest cubic graphs having given crossing number, correcting Pegg and Exoo (2009) by and by omitting two of the three unnamed 24-node graphs (CNG 8D and CNG 8E) given as having crossing number 8 (but which actually have crossing number 7), noting that the 26-node graph here called CNG 9A and labeled as "McGee + edge" (corresponding to one of two certain edge insertions in the McGee graph) actually has (not 10), and adding the edge-excised Coxeter graph as CNG 9 B. In addition, the 28-node graphs CNG 10A with crossing number..

### Guy's conjecture

Guy's conjecture, which has not yet been proven or disproven, states that the graph crossing number for a complete graph is(1)where is the floor function, which can be rewritten(2)The values for , 2, ... are then given by 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, ... (OEIS A000241).Guy (1972) proved the conjecture for , a result extended to by Pan and Richter (2007).It is known that(3)(Richter and Thomassen 1997, de Klerk et al. 2007, Pan and Richter 2007).

### Graph crossing number

Given a "good" graph (i.e., one for which all intersecting graph edges intersect in a single point and arise from four distinct graph vertices), the crossing number is the minimum possible number of crossings with which the graph can be drawn, including using curved (non-rectilinear) edges. Several notational conventions exist in the literature, with some of the more common being (e.g., Pan and Richter 2007; Clancy et al. 2019), , (e.g., Pach and Tóth 2005), and .A graph with crossing number 0 is a planar graph. However, there appears to be no term in standard use for a graph with graph crossing number 1 (the terms "almost planar" and "1-planar" are used in the literature for other concepts). Checking if a graph has crossing number 1 is straightforward using the following algorithm (M. Haythorpe, pers. comm., Apr. 16, 2019). First, confirm that the graph is nonplanar. Then, for all non-adjacent..

### Constant problem

Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to zero. The constant problem, sometimes also called the identity problem (Richardson 1968) is a very difficult unsolved problem in transcendental number theory. However, it is known that the problem is undecidable if the expression involves oscillatory functions such as sine. However, the Ferguson-Forcade algorithm is a practical algorithm for determining if there exist integers for given real numbers such thator else establishing bounds within which no relation can exist (Bailey 1988).

### Gelfond's theorem

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that is transcendental if 1. is algebraic and 2. is algebraic and irrational. This provides a partial solution to the seventh of Hilbert'sproblems. It was proved independently by Gelfond (1934ab) and Schneider (1934ab).This establishes the transcendence of Gelfond's constant (since ) and the Gelfond-Schneider constant .Gelfond's theorem is implied by Schanuel's conjecture(Chow 1999).

### Schanuel's conjecture

Let , ..., be linearly independent over the rationals , thenhas transcendence degree at least over . Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that and are algebraically independent. Macintyre (1991) proved that the truth of Schanuel's conjecture also guarantees that there are no unexpected exponential-algebraic relations on the integers (Marker 1996).At present, a proof of Schanuel's conjecture seems out of reach (Chow 1999).

### Rational number

A rational number is a number that can be expressed as a fraction where and are integers and . A rational number is said to have numerator and denominator . Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. The set of rational numbers is of measure zero on the real line, so it is "small" compared to the irrationals and the continuum.The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted . Here, the symbol derives from the German word Quotient, which can be translated as "ratio," and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671).Any rational number is trivially also an algebraicnumber.Examples of rational numbers include , 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers.The..

### Shannon capacity

Let denote the independence number of a graph . Then the Shannon capacity , sometimes also denoted , of is defined aswhere denoted the graph strong product (Shannon 1956, Alon and Lubetzky 2006). The Shannon capacity is an important information theoretical parameter because it represents the effective size of an alphabet in a communication model represented by a graph (Alon 1998). satisfiesThe Shannon capacity is in general very difficult to calculate (Brimkov et al. 2000). In fact, the Shannon capacity of the cycle graph was not determined as until 1979 (Lovász 1979), and the Shannon capacity of is perhaps one of the most notorious open problems in extremal combinatorics (Bohman 2003).Lovász (1979) showed that the Shannon capacity of the -Kneser graph is , that of a vertex-transitive self-complementary graph (which includes all Paley graphs) is , and that of the Petersen graph is 4.All graphs whose Shannon capacity is known..

A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by(1)where is the identity matrix.A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graphA complete set of Walsh..

### Jevons' number

A semiprime which English economist and logician William Stanley Jevons incorrectly believed no one else would be able to factor. According to Jevons (1874, p. 123), "Can the reader say what two numbers multiplied together will produce the number 8616460799? I think it unlikely that anyone but myself will ever know."Actually, a modern computer can factor this number in a few milliseconds as the product of two five-digit numbers:Published factorizations include those by Lehmer (1903) and Golomb (1996).

### Bimagic square

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times.The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and after squaring. Another order 8 bimagic square is shown at right.Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as..

### Bennequin's conjecture

A braid with strands and components with positive crossings and negative crossings satisfieswhere is the unknotting number. While the second part of the inequality was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronheimer and Mrowka on Milnor's conjecture (and, independently, using the slice-Bennequin inequality).

### Wang's conjecture

Wang's conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The conjecture was refuted when Berger (1966) showed that an aperiodic set of tiles existed. Berger used tiles, but the number has subsequently been greatly reduced. In fact, Culik (1996) has reduced the number of colored square tiles to 13.For purely square tiles, Culik's record still stands as of Feb. 2009. For non-square tiles, it is much more complicated due to the Penrose tiles (2 tiles), the Robertson tiling (6 tiles), and various Ammann tilings (2-5 tiles).

### Nonhamiltonian bicubic graph

Tutte (1971) conjectured that all 3-connected bicubic graphs are Hamiltonian (the Tutte conjecture). The Horton graph on 96 nodes, illustrated above, provided the first counterexample (Bondy and Murty 1976, p. 240).Horton (1982) subsequently found a counterexample on 92 nodes, and two smaller (nonisomorphic) counterexamples on 78 nodes were found by Ellingham (1981, 1982b) and Owens (1983). Ellingham and Horton (1983) subsequently found a counterexample graph on 54 nodes and 81 edges. The smallest currently known counterexample is the 50-node Georges graph (Georges 1989; Grünbaum 2006, 2009).These small known counterexamples are summarized in the following table and illustrated above.namereference50Georges graphGeorges (1989), Grünbaum (2006, 2009)54Ellingham-Horton 54-graphEllingham and Horton (1983)78Ellingham-Horton 78-graphEllingham (1981, 1982)78Owens graphOwens (1983)92Horton 92-graphHorton..

### Unfolding

An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).Examples..

### New mersenne prime conjecture

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that be a prime is that be a prime of one of the forms , , ."Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff (1989) used the statement as an inspiration for what is now called the new Mersenne conjecture, which can be stated as follows.Consider an odd natural number . If two of the following conditions hold, then so does the third: 1. or , 2. is prime (a Mersenne prime), 3. is prime (a Wagstaff prime). This conjecture has been verified for all primes .Based on the distribution and heuristics of (cf. https://www.utm.edu/research/primes/mersenne/heuristic.html) the known Mersenne and Wagstaff prime exponents, it seems quite likely that there is only a finite number of exponents satisfying the criteria of the new Mersenne conjecture. In..

### Tiling

A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.A spiral tiling using a single piece is illustrated..

### Tetrahedron tetrahedron picking

The mean tetrahedron volume of a tetrahedron with vertices chosen at random inside another tetrahedron of unit volume is given by(1)(2)(OEIS A093525; Buchta and Reitzner 1992; Mannion1994; Schneider 1997, p. 170; Buchta and Reitzner 2001; Zinani 2003).This provides a disproof of the conjecture that the solution to this problem is a rational number (1/57 had been suggested by Croft et al. 1991, p. 54), and renders obsolete Solomon's statement that "Explicit values for random points in non-spherical regions such as tetrahedrons, parallelepipeds, etc., have apparently not yet been successfully calculated" (Solomon 1978, p. 124).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary ...

### Kravitz conjecture

The first few numbers whose abundance absolute values are odd squares (excluding the trivial cases of powers of 2) are 98, 2116, 4232, 49928, 80656, 140450, 550564, 729632, ... (OEIS A188484).Kravitz conjectured that no numbers exist whose abundance is a (positive) odd square (Guy 2004). This conjecture is false with smallest counterexample(1)(2)and first few counterexamples given by 550564, 15038884, 57365476, ... (OEIS A188486).

### Cubic nonhamiltonian graph

Cubic nonhamiltonian graphs are nonhamiltonian graphs that are also cubic. The numbers of connected cubic nonhamiltonian graphs on , 12, ... are 2, 5, 35, 219, 1666, ... (OEIS A164919). The figure above shows some nonhamiltonian connected cubic graphs that are not snarks.Cubic nonhamiltonian graphs are of special interest because of Tait's Hamiltonian graph conjecture. The cubic polyhedral nonhamiltonian graphs illustrated above all provide counterexamples to this conjecture.The Horton graphs and Ellingham-Horton graphs are examples of 3-connected bicubic graphs that provided counterexamples to the Tutte conjecture.

### Prime number theorem

The prime number theorem gives an asymptotic form for the prime counting function , which counts the number of primes less than some integer . Legendre (1808) suggested that for large ,(1)with (where is sometimes called Legendre's constant), a formula which is correct in the leading term only,(2)(Nagell 1951, p. 54; Wagon 1991, pp. 28-29; Havil 2003, p. 177).In 1792, when only 15 years old, Gauss proposed that(3)Gauss later refined his estimate to(4)where(5)is the logarithmic integral. Gauss did not publish this result, which he first mentioned in an 1849 letter to Encke. It was subsequently posthumously published in 1863 (Gauss 1863; Havil 2003, pp. 174-176).Note that has the asymptotic series about of(6)(7)and taking the first three terms has been shown to be a better estimate than alone (Derbyshire 2004, pp. 116-117).The statement (4) is often known as "the" prime number theorem and was proved..

### Boy surface

The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of the Mathematisches Forschungsinstitut Oberwolfach library building on January 28, 1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured..

### P&oacute;lya conjecture

Let be a positive integer and the number of (not necessarily distinct) prime factors of (with ). Let be the number of positive integers with an odd number of prime factors, and the number of positive integers with an even number of prime factors. Pólya (1919) conjectured thatis , where is the Liouville function.The conjecture was made in 1919, and disproven by Haselgrove (1958) using a method due to Ingham (1942). Lehman (1960) found the first explicit counterexample, , and the smallest counterexample was found by Tanaka (1980). The first for which are , 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Tanaka 1980, OEIS A028488). It is unknown if changes sign infinitely often (Tanaka 1980).

### Plutarch numbers

In Moralia, the Greek biographer and philosopher Plutarch states "Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are compound statements, and on the negative side .)" These numbers are known as the Plutarch numbers. can be interpreted as the number of bracketings on ten letters (Stanley 1997, Habsieger et al. 1998). Similarly, Plutarch's second number is given by (Habsieger et al. 1998).

### Grothendieck's constant

Let be an real square matrix with such that(1)for all real numbers , , ..., and , , ..., such that . Then Grothendieck showed that there exists a constant satisfying(2)for all vectors and in a Hilbert space with norms and . The Grothendieck constant is the smallest possible value of . For example, the best values known for small are(3)(4)(5)(Krivine 1977, 1979; König 1992; Finch 2003, p. 236).Now consider the limit(6)which is related to Khinchin's constant and sometimes also denoted . Krivine (1977) showed that(7)and postulated that(8)(OEIS A088367). The conjecture was refuted in 2011 by Yury Makarychev, Mark Braverman, Konstantin Makarychev, and Assaf Naor, who showed that is strictly less than Krivine's bound (Makarychev 2011).Similarly, if the numbers and and matrix are taken as complex, then a similar set of constants may be defined. These are known to satisfy(9)(10)(11)(Krivine 1977, 1979; König 1990, 1992; Finch..

### Bicubic graph

A bicubic graph is a bipartite cubicgraph.Tutte (1971) conjectured that all 3-connected bicubic graphs are Hamiltonian (the Tutte conjecture), but a number of nonhamiltonian bicubic graphs have subsequently been discovered.The numbers of simple bicubic graphs on , 4, ... nodes are 0, 0, 1, 1, 2, 5, 13, 38, 149, ... (OEIS A006823), the first few of which are illustrated above.The following table summarizes some named bicubic graphs.graph utility graph6cubical graph8Franklin graph12Heawood graph14Möbius-Kantor graph16Pappus graph18Desargues graph20truncated octahedral graph24Levi graph30Dyck graph32great rhombicuboctahedral graph48Gray graph54Balaban 10-cage70Foster graph90great rhombicosidodecahedral graph120Tutte 12-cage126

### Nuciferous graph

Let be a simple graph with nonsingular (0,1) adjacency matrix . If all the diagonal entries of the matrix inverse are zero and all the off-diagonal entries of are nonzero, then is called a nuciferous graph (Ghorbani 2016).The path graph has adjacency matrix (and adjacency matrix inverse) given bywhich is therefore nuciferous. Initially, this was the only example known, and in fact, no others exist on 10 or fewer nodes (E. Weisstein, Mar. 18, 2016). As a result, it was conjectured by Sciriha et al. (2013) that no others exist.This conjecture was disproved by Ghorbani (2016) who found 21 Cayleygraphs examples on 24, 28, and 30 nodes.

### Fermat number

There are two definitions of the Fermat number. The less common is a number of the form obtained by setting in a Fermat polynomial, the first few of which are 3, 5, 9, 17, 33, ... (OEIS A000051).The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form . The first few for , 1, 2, ... are 3, 5, 17, 257, 65537, 4294967297, ... (OEIS A000215). The number of digits for a Fermat number is(1)(2)(3)For , 1, ..., the numbers of digits in are therefore 1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, ... (OEIS A057755). The numbers of digits in for , 1, ... are 1, 309, 381600854690147056244358827361, ... (OEIS A114484).Being a Fermat number is the necessary (but not sufficient) form a number(4)must have in order to be prime. This can be seen by noting that if is to be prime, then cannot have any odd factors or else would be a factorable number of the form(5)Therefore, for a prime , must be a power of 2. No two Fermat numbers have..

### Prime magic square

A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including..

### Kimberling sequence

Given a sequence as input to stage , form sequence as follows: 1. For , write term and then term . 2. Discard the th term. 3. Write the remaining terms in order. Starting with the positive integers, the first few iterations are thereforeThe diagonal elements form the sequence 1, 3, 5, 4, 10, 7, 15, ... (OEIS A007063).

### Holyhedron

A holyhedron is polyhedron whose faces and holes are all finite-sided polygons and that contains at least one hole whose boundary shares no point with a face boundary. D. Wilson coined the term in 1997, although no actual holyhedron was known until 1999, when a holyhedron with faces was constructed (Vinson 2000).J. H. Conway believes that the minimal number of faces should be closer to 100, and offered a prize of divided by the number of faces for a better solution. A holyhedron with 492 faces was subsequently discovered, good for a prize of (Hatch).

### Heawood conjecture

The bound for the number of colors which are sufficient for map coloring on a surface of genus ,is the best possible, where is the floor function. is called the chromatic number, and the first few values for , 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (OEIS A000934).The fact that is also necessary was proved by Ringel and Youngs (1968) with two exceptions: the sphere (and plane), and the Klein bottle. When the four-color theorem was proved in 1976, the Klein bottle was left as the only exception, in that the Heawood formula gives seven, but the correct bound is six (as demonstrated by the Franklin graph). The four most difficult cases to prove in the Heawood conjecture were , 83, 158, and 257.

### Bold conjecture

A pair of vertices of a graph is called an -critical pair if , where denotes the graph obtained by adding the edge to and is the clique number of . The -critical pairs are never edges in . A maximal stable set of is called a forced color class of if meets every -clique of , and -critical pairs within form a connected graph.In 1993, G. Bacsó conjectured that if is a uniquely -colorable perfect graph, then has at least one forced color class. This conjecture is called the bold conjecture, and implies the strong perfect graph theorem. However, a counterexample of the conjecture was subsequently found by Sakuma (1997).

### Percolation threshold

In the field of percolation theory, the term percolation threshold is used to denote the probability which "marks the arrival" (Grimmett 1999) of an infinite connected component (i.e., of a percolation) within a particular model. The percolation threshold is commonly denoted and is sometimes called the critical phenomenon of the model.Special attention is paid to probabilities both below and above the percolation threshold; a percolation model for which is called a subcritical percolation while a model satisfying is called a supercritical percolation. Because of this distinction, the value is also sometimes called the phase transition of the model as it marks the exact point of transition between the subcritical phase and the supercritical phase . Note that by definition, subcritical percolation models are necessarily devoid of infinite connected components, whereas supercritical models always contain at least one..

### Baudet's conjecture

If , , ... are sets of positive integers andthen some contains arbitrarily long arithmetic progressions. The conjecture was proved by van der Waerden (1927) and is now known as van der Waerden's Theorem.According to de Bruijn (1977), "We do not know when and in what context he [Baudet] stated his conjecture and what partial results he had," although van der Waerden (1971, 1998) indicates he first heard of the problem in 1926.

### Kneser's conjecture

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).

### Refined alternating sign matrix conjecture

The numerators and denominators obtained by taking the ratios of adjacent terms in the triangular array of the number of "bordered" alternating sign matrices with a 1 at the top of column are, respectively, the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1. This conjecture was proven by Zeilberger (1996).

### Alternating sign matrix conjecture

The conjecture that the number of alternating sign matrices "bordered" by s is explicitly given by the formulaThis conjecture was proved by Doron Zeilberger in 1995 (Zeilberger 1996a). This proof enlisted the aid of an army of 88 referees together with extensive computer calculations. A beautiful, shorter proof was given later that year by Kuperberg (Kuperberg 1996), and the refined alternating sign matrix conjecture was subsequently proved by Zeilberger (Zeilberger 1996b) using Kuperberg's method together with techniques from -calculus and orthogonal polynomials.

### Wolstenholme prime

A prime is called a Wolstenholme prime if the central binomial coefficient(1)or equivalently if(2)where is the th Bernoulli number and the congruence is fractional.A prime is a Wolstenholme prime if and only if(3)where the congruence is again fractional.The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to (McIntosh 2004).

### Wilson prime

A Wilson prime is a prime satisfyingwhere is the Wilson quotient, or equivalently,The first few Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) showed there are no others less than (McIntosh 2004), a limit that has subsequently been increased to (Costa et al. 2012).

### Odd perfect number

In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers.To this day, it is not known if any odd perfect numbers exist, although..

### Weird number

A "weird number" is a number that is abundant (i.e., the sum of proper divisors is greater than the number) without being pseudoperfect (i.e., no subset of the proper divisors sums to the number itself). The pseudoperfect part of the definition means that finding weird numbers is a case of the subset sum problem.Since prime numbers are deficient, prime numbers are not weird. Similarly, since multiples of 6 are pseudoperfect, no weird number is a multiple of 6.The smallest weird number is 70, which has proper divisors 1, 2, 5, 7, 10, 14, and 35. These sum to 74, which is greater that the number itself, so 70 is abundant, and no subset of them sums to 70. In contrast, the smallest abundant number is 12, which has proper divisors 1, 2, 3, 4, and 6. These sum to 16, so 12 is abundant, but the subset sum equals 12, so 12 is not weird.The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...(OEIS A006037).An infinite number of weird..

### Wagstaff prime

A Wagstaff prime is a prime number of the form for a prime number. The first few are given by , 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, and 4031399 (OEIS A000978), with and larger corresponding to probable primes. These values correspond to the primes with indices , 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).The Wagstaff primes are featured in the newMersenne prime conjecture.There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving.A Wagstaff prime can also be interpreted as a repunit prime of base , asif is odd, as it must be for the above number to be prime.Some of the largest known Wagstaff probable primes are summarized in the following..

### Twin primes

Twin primes are pairs of primes of the form (, ). The term "twin prime" was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 and A006512).All twin primes except (3, 5) are of the form .It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes,(1)converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201)...

### Double mersenne number

A double Mersenne number is a number of the formwhere is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime can be prime only for prime , a double Mersenne prime can be prime only for prime , i.e., a Mersenne prime. Double Mersenne numbers are prime for , 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).The next four , , , and have known factors summarized in the following table. The status of all other double Mersenne numbers is unknown, with being the smallest unresolved case. Since this number has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer test to be practical. The only possible method of determining the status of this number is therefore attempting to find small divisors..

### Mertens conjecture

Given the Mertens function defined by(1)where is the Möbius function, Stieltjes claimed in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes (1885) claimed that he had a proof of the general result. However, it seems likely that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens (1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim(2)for was "very probable."The Mertens conjecture has important implications, since the truth of any equalityof the form(3)for any fixed (the form of the Mertens conjecture with ) would imply the Riemann hypothesis. In fact, the statement(4)for any is equivalent to the Riemann hypothesis (Derbyshire 2004, p. 251).Mertens (1897) verified the conjecture for , and this was subsequently extended to by..

### Mersenne prime

A Mersenne prime is a Mersenne number, i.e., anumber of the formthat is prime. In order for to be prime, must itself be prime. This is true since for composite with factors and , . Therefore, can be written as , which is a binomial number that always has a factor .The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (OEIS A000668) corresponding to indices , 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS A000043).Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers.It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line through the origin to the asymptotic number of Mersenne primes with for the first 51 (known) Mersenne primes gives a best-fit line with , illustrated above. If the line is not restricted to pass through..

### Ternary

The base-3 method of counting in which only the digits 0, 1, and 2 are used. Ternary numbers arise in a number of problems in mathematics, including some problems of weighing. However, according to Knuth (1998), "no substantial application of balanced ternary notation has been made" (balanced ternary uses digits , 0, and 1 instead of 0, 1, and 2).The illustration above shows a graphical representation of the numbers 0 to 25 in ternary, and the following table gives the ternary equivalents of the first few decimal numbers. The concatenation of the ternary digits of the consecutive numbers 0, 1, 2, 3, ... gives (0), (1), (2), (1, 0), (1, 1), (1, 2), (2, 0), ... (OEIS A054635).111110221210221211022211310131112321241114112242205121512025221620161212622272117122271000822182002810019100192012910021010120202301010Ternary digits have the following multiplicationtable.0120000101220211A ternary representation can..

### Strong pseudoprime

A strong pseudoprime to a base is an odd composite number with (for odd) for which either(1)or(2)for some , 1, ..., (Riesel 1994, p. 91). Note that Guy (1994, p. 27) restricts the definition of strong pseudoprimes to only those satisfying (1).The definition is motivated by the fact that a Fermat pseudoprime to the base satisfies(3)But since is odd, it can be written , and(4)If is prime, it must divide at least one of the factors, but can't divide both because it would then divide their difference(5)Therefore,(6)so write to obtain(7)If divides exactly one of these factors but is composite, it is a strong pseudoprime. A composite number is a strong pseudoprime to at most 1/4 of all bases less than itself (Monier 1980, Rabin 1980). The strong pseudoprimes provide the basis for Miller's primality test and Rabin-Miller strong pseudoprime test.A strong pseudoprime to the base is also an Euler pseudoprime to the base (Pomerance et al. 1980)...

### Lehmer's mahler measure problem

An unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials. Lehmer (1933) conjectured that if is such an integer polynomial, then(1)(2)where , denoted by Lehmer (1933) and by Hironaka (2009), is the largest positive root of this polynomial. The roots of this polynomial, plotted in the left figure above, are very special, since 8 of the 10 lie on the unit circle in the complex plane. The roots of the polynomials (represented by half their coefficients) giving the two next smallest known Mahler measures are also illustrated above (Mossinghoff 1998, p. S11).The best current bound is that of Smyth (1971), who showed that , where is a nonzero nonreciprocal polynomial that is not a product of cyclotomic polynomials (Everest 1999), and is the real root of . Generalizations of Smyth's result have been constructed by Lloyd-Smith..

### Cubic number

A cubic number is a figurate number of the form with a positive integer. The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578). The protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time recites the cubic numbers to calm himself and prevent himself from wanting to hit someone (Haddon 2003, p. 213).The generating function giving the cubic numbersis(1)The hex pyramidal numbers are equivalent tothe cubic numbers (Conway and Guy 1996).The plots above show the first 255 (top figure) and 511 (bottom figure) cubic numbers represented in binary.Pollock (1843-1850) conjectured that every number is the sum of at most 9 cubic numbers (Dickson 2005, p. 23). As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (, proved by Dickson, Pillai, and Niven in the early twentieth century), that every "sufficiently large" integer..

### Latin square

An Latin square is a Latin rectangle with . Specifically, a Latin square consists of sets of the numbers 1 to arranged in such a way that no orthogonal (row or column) contains the same number twice. For example, the two Latin squares of order two are given by(1)the 12 Latin squares of order three are given by(2)and two of the whopping 576 Latin squares of order 4 are given by(3)The numbers of Latin squares of order , 2, ... are 1, 2, 12, 576, 161280, ... (OEIS A002860). The number of isotopically distinct Latin squares of order , 2, ... are 1, 1, 1, 2, 2, 22, 564, 1676267, ... (OEIS A040082).A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct. For example, the two Latin squares(4)are orthogonal. The number of pairs of orthogonal Latin squares of order , 2, ... are 0, 0, 36, 3456, ... (OEIS A072377).The number of Latin squares of order with first row given by is the same as the number of fixed diagonal Latin..

### Sophie germain prime

A prime is said to be a Sophie Germain prime if both and are prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (OEIS A005384). It is not known if there are an infinite number of Sophie Germain primes (Hoffman 1998, p. 190).The numbers of Sophie Germain primes less than for , 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (OEIS A092816).The largest known proven Sophie Germain prime pair as of Feb. 29, 2016 is given by where(Caldwell), each of which has decimal digits (PrimeGrid).The definition of Sophie Germain primes and the value of the largest then-known suchprime were mentioned by the characters Hal and Catherine in the 2005 film Proof.Sophie Germain primes of the form correspond to the indices of composite Mersenne numbers .Around 1825, Sophie Germain proved that the first case of Fermat's last theorem is true for such primes, i.e., if is a Sophie Germain prime, then there do not exist integers..

### Rectilinear crossing number

The rectilinear crossing number of a graph is the minimum number of crossings in a straight line drawing of in a plane. It is variously denoted , (Schaefer 2017), , or .It is sometimes claimed that the rectilinear crossing number is also known as the linear or geometric(al) crossing number, but evidence for that is slim (Schafer 2017).A disconnected graph has a rectilinear crossing number equal to the sums of the rectilinear crossing numbers of its connected components.When the (non-rectilinear) graph crossing number satisfies ,(1)(Bienstock and Dean 1993). While Bienstock and Dean don't actually prove equality for the case , they state it can be established analogously to . The result cannot be extended to , since there exist graphs with but for any (Bienstock and Dean 1993; Schaefer 2017, p. 54).G. Exoo (pers. comm., May 11-12, 2019) has written a program which can compute rectilinear crossing numbers for cubic graphs up to around..

### Heilbronn triangle problem

The Heilbronn triangle problem is to place points in a disk (square, equilateral triangle, etc.) of unit area so as to maximize the area of the smallest of the triangles determined by the points. For points, there is only a single triangle, so Heilbronn's problem degenerates into finding the largest triangle that can be constructed from points in a square. For , there are four possible triangles for each configuration, so the problem is to find the configuration of points for which the smallest of these four triangles is the maximum possible.For a unit square, the first few maxima of minimaltriangle areas are(1)(2)(3)(4)(5)(6)(7)(8)For larger values of , proofs of optimality are open, but the best known results are(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)with the configurations leading to maximum minimal triangles illustrated above (Friedman 2006; Comellas and Yebra 2002; D. Cantrell..

### Golomb ruler

An -mark Golomb ruler is a set of distinct nonnegative integers , called "marks," such that the positive differences , computed over all possible pairs of different integers , ..., with are distinct.Let be the largest integer in an -mark Golomb ruler. Then an -mark Golomb ruler is optimal if 1. There exists no other -mark Golomb rulers having smaller largest mark , and 2. The ruler is written in canonical form as the "smaller" of the equivalent rulers and , where "smaller" means the first differing entry is less than the corresponding entry in the other ruler. In such a case, is the called the "length" of the optimal -mark ruler.Thus, (0, 1, 3) is the unique optimal 3-mark Golomb ruler modulo reversal (i.e., (0, 2, 3) is considered the same ruler).For example, the set (0, 1, 3, 7) is 4-mark Golomb ruler since its differences are (, , , , , ), all of which are distinct. However, the unique optimal Golomb 4-mark ruler..

### Prime gaps

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function(1)is always even (except for ), all primes gaps are also even. The notation is commonly used to denote the smallest prime corresponding to the start of a prime gap of length , i.e., such that is prime, , , ..., are all composite, and is prime (with the additional constraint that no smaller number satisfying these properties exists).The maximal prime gap is the length of the largest prime gap that begins with a prime less than some maximum value . For , 2, ..., is given by 4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ... (OEIS A053303).Arbitrarily large prime gaps exist. For example, for any , the numbers , , ..., are all composite (Havil 2003, p. 170). However, no general method..

### Goldbach conjecture

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes. Two primes such that for a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and..

### Bernoulli number

The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function(1)These numbers arise in the series expansions of trigonometric functions, and areextremely important in number theory and analysis.There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written , while the Bernoulli numbers encountered in older literature are written (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials or with and .The Bernoulli number and polynomial should not be confused with the Bell numbers and Bell polynomial, which are also commonly denoted and , respectively.Bernoulli numbers defined by the modern definition are denoted and sometimes called "even-index" Bernoulli numbers...

### Prime arithmetic progression

An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981),..

### Gilbreath's conjecture

Let the difference of successive primes be defined by , and by(1)N. L. Gilbreath claimed that for all (Guy 1994). In 1959, the claim was verified for . In 1993, Odlyzko extended the claim to all primes up to .Gilbreath's conjecture is equivalent to the statement that, in the triangular array of the primes, iteratively taking the absolute difference of each pair of terms(2)(OEIS A036262), always gives leading term 1(after the first row).The number of terms before reaching the first greater than two in the second, third,etc., rows are given by 3, 8, 14, 14, 25, 23, 22, 25, ... (OEIS A000232).

### Aliquot sequence

Letwhere is the divisor function and is the restricted divisor function. Then the sequence of numbersis called an aliquot sequence. If the sequence for a given is bounded, it either ends at or becomes periodic. 1. If the sequence reaches a constant, the constant is known as a perfect number. A number that is not perfect, but for which the sequence becomes constant, is known as an aspiring number. 2. If the sequence reaches an alternating pair, it iscalled an amicable pair. 3. If, after iterations, the sequence yields a cycle of minimum length of the form , , ..., , then these numbers form a group of sociable numbers of order . The lengths of the aliquot sequences for , 2, ... are 1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, ... (OEIS A044050).It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. Guy (1994) cites the largest computed value as , though this has since been extended..

### Idoneal number

An idoneal number, also called a suitable number or convenient number, is a positive integer for which the fact that a number is a monomorph (i.e., is expressible in only one way as where is relatively prime to ) guarantees it to be a prime, prime power, or twice one of these. The numbers are also called Euler's idoneal numbers or suitable numbers.A positive integer is idoneal iff it cannot be written as for integer , , and with .The 65 idoneal numbers found by Gauss and Euler and conjectured to be the only such numbers (Shanks 1969) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (OEIS A000926). It is known that if any other idoneal numbers exist, there can be only one more...

### Ramanujan's square equation

In 1913, Ramanujan asked if the Diophantineequation of second ordersometimes called the Ramanujan-Nagell equation, has any solutions other than , 4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; OEIS A060728). These correspond to , 3, 5, 11, and 181 (OEIS A038198). Nagell (1948) and Skolem et al. (1959) showed there are no solutions past , thus establishing Ramanujan's question in the negative.A generalization to two variables and was considered by Euler (Engel 1998, p. 126).

### Pillai's conjecture

For every , there exist only finite many pairs of powers with and natural numbers and .

### Fermat's last theorem

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation has no integer solutions for and .The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number..

### Catalan's conjecture

The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1). In other words,(1)is the only nontrivial solution to Catalan'sDiophantine problem(2)The special case and is the case of a Mordell curve.Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000).This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in..

### Beal's conjecture

A generalization of Fermat's last theorem which states that if , where , , , , , and are any positive integers with , then , , and have a common factor. The conjecture was announced in Mauldin (1997), and a cash prize of has been offered for its proof or a counterexample (Castelvecchi 2013).This conjecture is more properly known as the Tijdeman-Zagier conjecture (Elkies 2007).

### Perfect cuboid

A perfect cuboid is a cuboid having integer side lengths,integer face diagonals(1)(2)(3)and an integer space diagonal(4)The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.No perfect cuboids are known despite an exhaustive search for all "odd sides" up to (Butler, pers. comm., Dec. 23, 2004).Solving the perfect cuboid problem is equivalent to solving the Diophantineequations(5)(6)(7)(8)A solution with integer space diagonal and two out of three face diagonals is , , and , giving , , , and , which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral polyhedron edge is , , and , giving , , , and .

### Euler's sum of powers conjecture

Euler conjectured that at least th powers are required for to provide a sum that is itself an th power. The conjecture was disproved by Lander and Parkin (1967) with the counterexample(1)Ekl (1998) defined an extended Euler conjecture that there are no solutions to the Diophantine equation(2)with and not necessarily distinct, such that . Defining(3)over all known solutions to equations, this conjecture asserts that . There are no known counterexamples to this conjecture (Ekl 1998). The following table gives the smallest known values of for small .min. soln.reference44.1.30Elkies (1988)55.1.40Lander et al. (1967)66.3.30Subba Rao (1934)77.4.41Ekl (1996)88.3.50S. Chase (Meyrignac)88.4.40N. Kuosa (Nov. 9, 2006; Meyrignac)99.5.51Ekl 1997 (Meyrignac)1010.6.62N. Kuosa (2002; Meyrignac)S. Chase found a 8.3.5 () solution that displaced the 8.5.5 () solution of Letac (1942). In 2006, N. Kuosa..

### Mordell conjecture

The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common factors (Mordell 1922). Fermat's equation has holes, so the Mordell conjecture implies that for each integer , the Fermat equation has at most a finite number of solutions.This conjecture was proved by Faltings (1984) and hence is now also known as Falting's theorem.

### Archimedes' cattle problem

Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"Solution consists..

### Khinchin's constant

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

### Littlewood conjecture

The Littlewood conjecture states that for any two real numbers ,where denotes the nearest integer function.In layman's terms, this conjecture concerns the simultaneous approximation of two real numbers by rationals, indeed saying that any two real numbers and can be simultaneously approximated at least moderately well by rationals having the same denominator (Venkatesh 2007).Though proof of the Littlewood conjecture still remains an open problem, many partial results exist. For example, Borel showed that the set of exceptional pairs of real numbers and for which the conjecture fails has Lebesgue measure zero. Much later, Einsiedler et al. (2006) proved that the set of pairs of exceptional points also has Hausdorff dimension zero.

### Bootstrap percolation

A two-dimensional binary () totalistic cellular automaton with a von Neumann neighborhood of range . It has a birth rule that at least 2 of its 4 neighbors are alive, and a survival rule that all cells survive. steps of bootstrap percolation on an grid with random initial condition of density can be implemented in the Wolfram Language asWith[{n = 10, p = 0.1, s = 20}, CellularAutomaton[ {1018, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, Table[If[Random[Real] < p, 1, 0], {s}, {s}], n ]]If the initial condition consists of a random sparse arrangement of cells with density , then the system seems to quickly converge to a steady state of rectangular islands of live cells surrounded by a sea of dead cells. However, as crosses some threshold on finite-sized grids, the behavior appears to change so that every cell becomes live. Several examples are shown above on three grids with random initial conditions and different starting densities.However, this..

### Blaschke conjecture

The only Wiedersehen surfaces are the standard round spheres. The conjecture was proven by combining the Berger-Kazdan comparison theorem with A. Weinstein's results for even and C. T. Yang's for odd. Green (1963) obtained the first proof of the Blaschke's conjecture in the two-dimensional case.

### Serre's problem

Serre's problem, also called Serre's conjecture, asserts that the implication "free module projective module" can be reversed for every module over the polynomial ring , where is a field (Serre 1955).The hard part of the proof, the one concerning finitely generated modules, was given simultaneously, and independently, by D. Quillen in Cambridge, Massachusetts and A. A. Suslin in Leningrad (St. Petersburg) in 1976. As a result, the statement is often referred to as the "Quillen-Suslin theorem."The solution to this difficult problem is part of the work for which Quillen wasawarded the Fields Medal in 1978.Quillen and Suslin received, for other contributions in algebra, the ColePrize in 1975 and 2000 respectively.

### Isomorphic factorization

Isomorphic factorization colors the edges a given graph with colors so that the colored subgraphs are isomorphic. The graph is then -splittable, with as the divisor, and the subgraph as the factor.When a complete graph is 2-split, a self-complementary graph results. Similarly, an -regular class 1 graph can be -split into graphs consisting of disconnected edges, making the edge chromatic number.The complete graph can be 3-split into identical planar graphs.Some Ramsey numbers have been bounded via isomorphic factorizations. For instance, the complete graph has the Clebsch graph as a factor, proving (Gardner 1989). That is, the complete graph on 16 points can be three-colored so that no triangle of a single color appears. (In 1955, was proven.)In addition, can be 8-split with the Petersen graph as a factor, or 5-split with a doubled cubical graph as a factor (shown by Exoo in 2005).The Hoffman-Singleton graph is 7-splittable into edges (shown..

### Happy end problem

The happy end problem, also called the "happy ending problem," is the problem of determining for the smallest number of points in general position in the plane (i.e., no three of which are collinear), such that every possible arrangement of points will always contain at least one set of points that are the vertices of a convex polygon of sides. The problem was so-named by Erdős when two investigators who first worked on the problem, Ester Klein and George Szekeres, became engaged and subsequently married (Hoffman 1998, p. 76).Since three noncollinear points always determine a triangle, .Random arrangements of points are illustrated above. Note that no convex quadrilaterals are possible for the arrangements shown in the fifth and eighth figures above, so must be greater than 4. E. Klein proved that by showing that any arrangement of five points must fall into one of the three cases (left top figure; Hoffman 1998,..

In the directed graph above, pick any vertex and follow the arrows in sequence blue-red-red three times. You will finish at the green vertex. Similarly, follow the sequence blue-blue-red three times and you will always end on the yellow vertex, no matter where you started. This is called a synchronized coloring.The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with the same outdegree and where the greatest common divisor of all cycles lengths is 1. Trahtman (2007) provided a positive solution to this problem.

### Acyclic digraph

An acyclic digraph is a directed graph containing no directed cycles, also known as a directed acyclic graph or a "DAG." Every finite acyclic digraph has at least one node of outdegree 0. The numbers of acyclic digraphs on , 2, ... vertices are 1, 2, 6, 31, 302, 5984, ... (OEIS A003087).The numbers of labeled acyclic digraphs on , 2, ... nodes are 1, 3, 25, 543, 29281, ... (OEIS A003024). Weisstein's conjecture proposed that positive eigenvalued -matrices were in one-to-one correspondence with labeled acyclic digraphs on nodes, and this was subsequently proved by McKay et al. (2004). Counts for both are therefore given by the beautiful recurrence equationwith (Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).

### Euler square

A square array made by combining objects of two types such that the first and second elements form Latin squares. Euler squares are also known as Graeco-Latin squares, Graeco-Roman squares, or Latin-Graeco squares.For many years, Euler squares were known to exist for , 4, and for every odd except . Euler's Graeco-roman squares conjecture maintained that there do not exist Euler squares of order for , 2, .... However, such squares were found to exist in 1959, refuting the conjecture. As of 1959, Euler squares are known to exist for all except and .

### Costas array

An order- Costas array is a permutation on such that the distances in each row of the triangular difference table are distinct. For example, the permutation has triangular difference table , , , and . Since each row contains no duplications, the permutation is therefore a Costas array.There is no known formula, recursion, or generating function for giving the number of Costas arrays of order . Several number-theoretic generators are known (Golomb 1984, Beard et al. 2004), but these do not generate all known Costas arrays of orders greater than .The numbers of Costas arrays for , 2, ... counting flipped and rotated matrices distinctly are 1, 2, 4, 12, 40,116, 200, 444, 760, 2160, 4368, 7852, 12828, 17252, 19612, 21104, 18276, 15096, 10240, 6464, 3536, 2052, 872, 200, 88, 56, 204, ... (OEIS A008404). Here, counts for , 25, and 26 were found by Beard et al. (2004, 2007). was verified by Rickard et al. (2006) and the case was solved by Drakakis et al. (2008).The..

### Lehmer's constant

The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.

Hadjicostas's formula is a generalization of the unitsquare double integral(1)(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni constant. It states(2)for , where is the gamma function and is the Riemann zeta function (although care must be taken at because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case gives Beukers's integral for ,(3)(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.There is an analogous formula(4)for , due to Sondow (2005), where is the Dirichlet eta function. This includes the special cases(5)(6)(7)(OEIS A094640; Sondow 2005) and(8)(9)(OEIS A103130), where is the Glaisher-Kinkelin constant (Sondow 2005)...

### Exponential integral

Min Max Re Im Let be the En-function with ,(1)(2)Then define the exponential integral by(3)where the retention of the notation is a historical artifact. Then is given by the integral(4)This function is implemented in the WolframLanguage as ExpIntegralEi[x].The exponential integral is closely related to the incomplete gamma function by(5)Therefore, for real ,(6)The exponential integral of a purely imaginarynumber can be written(7)for and where and are cosine and sine integral.Special values include(8)(OEIS A091725).The real root of the exponential integral occurs at 0.37250741078... (OEIS A091723), which is , where is Soldner's constant (Finch 2003).The quantity (OEIS A073003) is known as the Gompertz constant.The limit of the following expression can be given analytically(9)(10)(OEIS A091724), where is the Euler-Mascheroni constant.The Puiseux series of along the positive real axis is given by(11)where the denominators..

### Log gamma function

Min Max Min Max Re Im The plots above show the values of the function obtained by taking the natural logarithm of the gamma function, . Note that this introduces complicated branch cut structure inherited from the logarithm function. Min Max Re Im For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own right, and defined differently from . This special "log gamma" function is implemented in the Wolfram Language as LogGamma[z], plotted above. As can be seen, the two definitions have identical real parts, but differ markedly in their imaginary components. Most importantly, although the log gamma function and are equivalent as analytic multivalued functions, they have different branch cut structures and a different principal branch, and the log gamma function is analytic throughout the complex -plane except for a single branch cut discontinuity along the negative real axis. In particular,..

### Concurrent normals conjecture

It is conjectured that any convex body in -dimensional Euclidean space has an interior point lying on normals through distinct boundary points (Croft et al. 1991). This has been proved for and 3 by Heil (1979ab, 1985). It is known that higher dimensions always contain at least a 6-normal point, but the general conjecture remains open.

### Power series

A power series in a variable is an infinite sum of the formwhere are integers, real numbers, complex numbers, or any other quantities of a given type.Pólya conjectured that if a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural boundary (Pólya 1990, pp. 43 and 46). This conjecture was stated by G. Polya in 1916 and proved to be correct by Carlson (1921) in a result that is now regarded as a classic of early 20th century complex analysis.For any power series, one of the following is true: 1. The series converges only for . 2. The series converges absolutely for all . 3. The series converges absolutely for all in some finite open interval and diverges if or . At the points and , the series may converge absolutely, converge conditionally, or diverge. To determine the interval of convergence, apply the ratio test for absolute convergence..

If is a power series which is regular for except for poles within this circle and except for , at which points the function is assumed continuous when only points are considered, then at least a subsequence of the Padé approximants are uniformly bounded in the domain formed by removing the interiors of small circles with centers at these poles and uniformly continuous at for .

### Robertson conjecture

A conjecture due to M. S. Robertson in 1936 which treats a univalent power series containing only odd powers within the unit disk. This conjecture implies the Bieberbach conjecture and follows in turn from the Milin conjecture. de Branges' proof of the Bieberbach conjecture proceeded by proving the Milin conjecture, thus establishing the Robertson conjecture and hence implying the truth of the Bieberbach conjecture.

### Harmonic series

The series(1)is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).Progressions of the form(2)are also sometimes called harmonic series (Beyer 1987).Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,(3)(4)and since an infinite sum of 1/2's diverges, so does the harmonic series.The generalization of the harmonic series(5)is known as the Riemann zeta function.The sum of the first few terms of..

### Braun's conjecture

Let be an infinite Abelian semigroup with linear order such that is the unit element and implies for . Define a Möbius function on by andfor , 3, .... Further suppose that (the true Möbius function) for all . Then Braun's conjecture states thatfor all .

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