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Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon,(1)gives the ratio of the radii of the final to initial circles as(2)Numerically,(3)(OEIS A085365), where is the corresponding constant for polygon circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) assertion that is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757)...

Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an -gon are then related by(1)so an infinitely nested set of circumscribed polygons and circles has(2)(3)(4)Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is(5)(OEIS A051762).By writing(6)it is possible to expand the series about infinity, change the order of summation, do the sum symbolically, and obtain the quickly converging series(7)where is the Riemann zeta function.Bouwkamp (1965) produced the following infinite productformulas for the constant,(8)(9)(10)where is the sinc function (cf. Prudnikov et al. 1986, p. 757), is the Riemann zeta function, and is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence(11)where(12)(cited in Pickover..

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).It is..

A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula(1)giving a triangle area in terms of its side lengths , , and semiperimeter . Finding a Heronian triangle is therefore equivalent to solving the Diophantine equation(2)The complete set of solutions for integer Heronian triangles (the three side lengths and area can be multiplied by their least common multiple to make them all integers) were found by Euler (Buchholz 1992; Dickson 2005, p. 193), and parametric versions were given by Brahmagupta and Carmichael (1952) as(3)(4)(5)(6)(7)This produces one member of each similarity class of Heronian triangles for any integers , , and such that , , and (Buchholz 1992).The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10, 13,..

Bertelsen's number is an erroneous name erroneously given to the erroneous value of , where is the prime counting function. This value is 56 lower than the correct value of . Ore (1988, p. 69) states that the erroneous value 478 originated in Bertelsen's application of Meissel's method in 1893 (MathPages; Prime Curios!). However, the incorrect value actually first appears in Meissel (1885) rather than Bertelsen in 1893, as correctly noted by Lagarias et al. 1985. (Note that MathPages incorrectly states that Lagarias et al. attribute the result to Bertelsen.)Unfortunately, the incorrect value has continued to be propagated in modern works such as Hardy and Wright (1979, p. 9), Davis and Hersch (1981, p. 175; but actually given correctly in the table on p. 213), Sondheimer (1981), Kramer (1983), Ore (1988, p. 77), and Cormen et al. (1990)...

(1)The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEISA001223). Rankin has shown that(2)for infinitely many and for some constant (Guy 1994). At a March 2003 meeting on elementary and analytic number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that(3)(Montgomery 2003). Unfortunately, this proof turned out to be flawed.An integer is called a jumping champion if is the most frequently occurring difference between consecutive primes for some (Odlyzko et al.).

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the cardioid as one disk rolling on another.

In geometry, the term "enlargement" is a synonym for expansion.In nonstandard analysis, let be a set of urelements, and let be the superstructure with individuals in : 1. , 2. , 3. . Let be a superstructure monomorphism, with and for . Then is an enlargement of provided that for each set in , there is a hyperfinite set that contains all the standard entities of .It is the case that is an enlargement of if and only if every concurrent binary relation satisfies the following: There is an element of the range of such that for every in the domain of , the pair is in the relation .

The knight graph is a graph on vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other).The number of edges in the knight graph is (8 times the triangular numbers), so for , 2, ..., the first few values are 0, 0, 8, 24, 48, 80, 120, ... (OEIS A033996).Knight graphs are bipartite and therefore areperfect.The following table summarizes some named graph complements of knight graphs.-knight graph-queen graph-knight graph-queen graphThe knight graph is implemented in the Wolfram Language as KnightTourGraph[m, n], and precomputed properties are available in using GraphData["Knight", m, n].Closed formulas for the numbers of -graph cycles of the knight graph are given by for odd and(1)(E. Weisstein, Nov. 16, 2014).A knight's path is a sequence of moves by a..

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..

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 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.

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..

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)...

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 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.

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.

A map is called bijective if it is both injective and surjective. A bijective map is also called a bijection. A function admits an inverse (i.e., " is invertible") iff it is bijective.Two sets and are called bijective if there is a bijective map from to . In this sense, "bijective" is a synonym for "equipollent" (or "equipotent"). Bijectivity is an equivalence relation on the class of sets.

A point is said to lie between points and (where , , and are distinct collinear points) if . A number of Euclid's proofs depend on the idea of betweenness without explicit mentioning it.All points on a line segment excluding the endpointslie between the endpoints.Let be a partially ordered set, and let . If , then is said to be between and . If in and there is no that is between and , then covers . Conversely, if covers , then no is between and

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)...

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).

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)

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..

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

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..

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..

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,..

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..

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..

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 .

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,..

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.

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.

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..

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..

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..

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)..

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)...

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 ...

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 .

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 (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 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).

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..

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

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 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).

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).

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,..

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,..

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 .

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..

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 .

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 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).

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.

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

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..

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).

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).

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.

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 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..

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 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)...

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.

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.

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 .

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).

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.

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

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.

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)..

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).

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.

A snark is a connected bridgeless cubic graph (i.e., a biconnected cubic graph) with edge chromatic number of four. (By Vizing's theorem, the edge chromatic number of every cubic graph is either three or four, so a snark corresponds to the special case of four.) Snarks are therefore class 2 graphs.In order to avoid trivial cases, snarks are commonly restricted to be connected (so that the graph union of two Petersen graphs is excluded), have girth 5 or more and not to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial (Read and Wilson 1998, p. 263).Snarks that are trivial in the above senses are sometimes called "reducible" snarks. A number of reducible snarks are illustrated above.The Petersen graph is the smallest snark, and Tutte conjectured that all snarks have Petersen graph graph minors. This conjecture was proven in 2001 by Robertson, Sanders, Seymour, and Thomas,..

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,..

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 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..

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.

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

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 (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)

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).

Suppose that is a pseudograph, is the edge set of , and is the family of edge sets of graph cycles of . Then obeys the axioms for the circuits of a matroid, and hence is a matroid. Any matroid that can be obtained in this way is a graphic matroid.

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..

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..

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.

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)..

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).

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..

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..

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..

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).

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...

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..

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.

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).

Building on work of Huntington (1933ab), Robbins conjectured 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).

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.The matrix direct sum is defined by(1)(2)(Ayres 1962, pp. 13-14).The direct sum of two subspaces and is the sum of subspaces in which and have only the zero vector in common (Rosen 2000, p. 357).The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum). This general definition gives as a consequence the definition of the direct sum of Abelian groups and (since they are -modules, i.e., modules over the integers) and the direct sum of vector spaces (since they are modules over a field). Note that the direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.Note that direct products and direct sums differ for infinite indices. An element of the direct sum is..

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).

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..

The Curry triangle, also sometimes called the missing square puzzle, is a dissection fallacy created by American neuropsychiatrist L. Vosburgh Lions as an example of a phenomenon discovered by Paul Curry. The figure apparently shows that a triangle of area 60, a triangle of area 58 containing a rectangular hole, and a broken rectangle of area 59 can all be formed out of the same set of 6 polygonal pieces. The explanation for this lies in the inaccuracy of the initial subdivision. In the diagrams, the small and large right triangles are similar, hence they cannot have perpendicular sides of lengths and , respectively, as apparently shown in the drawing.

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...

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).

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.

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)...

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.

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...

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,..

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.

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...

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..

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.

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..

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)...

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.

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.

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 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..

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, 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).

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..

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, 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).

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).

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..

Let be a group and n permutation of . Then is an orthomorphism of if the self-mapping of defined by is also an permutation of .

The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements in a neighborhood set of . 1. There corresponds to each point at least one neighborhood , and each neighborhood contains the point . 2. If and are two neighborhoods of the same point , there must exist a neighborhood that is a subset of both. 3. If the point lies in , there must exist a neighborhood that is a subset of . 4. For two different points and , there are two corresponding neighborhoods and with no points in common.

Given an event in a sample space which is either finite with elements or countably infinite with elements, then we can writeand a quantity , called the probability of event , is defined such that1. . 2. . 3. Additivity: , where and are mutually exclusive. 4. Countable additivity: for , 2, ..., where , , ... are mutually exclusive (i.e., ).

One of the Eilenberg-Steenrod axioms which states that, if is homotopic to , then their induced maps and are the same.

Let be a set. An operation on is a function from a power of into . More precisely, given an ordinal number , a function from into is an -ary operation on . If is a finite ordinal, then the -ary operation is a finitary operation on .

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..

Consider the forms for which the generic characters are equal to some preassigned array of signs or ,subject to . There are possible arrays, where is the number of distinct prime divisors of a field discriminant , and the set of forms corresponding to each array is called a genus of forms. The forms for which all are called the principal genus of forms, and each genus is also a collection of proper equivalence classes (Cohn 1980, pp. 223-224).

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).

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..

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 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).

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..

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..

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..

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..

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 ...

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 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.

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..

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..

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).

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).

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..

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

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.

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..

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..

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).

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).

Given a hereditary representation of a number in base , let be the nonnegative integer which results if we syntactically replace each by (i.e., is a base change operator that 'bumps the base' from up to ). The hereditary representation of 266 in base 2 is(1)(2)so bumping the base from 2 to 3 yields(3)Now repeatedly bump the base and subtract 1,(4)(5)(6)(7)(8)(9)(10)(11)(12)etc.Starting this procedure at an integer gives the Goodstein sequence . Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's theorem states that is 0 for any and any sufficiently large . Even more amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).

Martin Gardner (1975) played an April Fool's joke by asserting that the map of 110 regions illustrated above (left figure) required five colors and constitutes a counterexample to the four-color theorem (cf. Wilson 2004, pp. 14-15; Chartrand and Zhang, p. 23, 2008; Posamentier and Lehmann, Fig. 1.13, 2013). However, because the four-color theorem is true (though not proved until 1976), the map must be (and is) four-colorable (right figure above), as demonstrated by the explicitly coloring due Wagon (1998; 1999, pp. 535-536), obtained algorithmically using the Wolfram Language.As stated by Gardner, "As a public service, I shall comment briefly on six major discoveries of 1974 that for one reason or another were inadequately reported to both the scientific community and the public at large. The most sensational of last year's discoveries in pure mathematics was surely the finding of a counterexample to..

The Kittell graph is a planar graph on 23 nodes and 63 edges that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails.It is also an identity graph.The Fritsch graph and Soifergraph provide smaller (and in fact the smallest possible) counterexamples.

Let be a planar graph whose vertices have been properly colored and suppose is colored . Define the -Kempe chain containing to be the maximal connected component of that 1. Contains , and 2. Contains only vertices that are colored with elements from (Gethner and Springer 2003).The illustration above shows a number of graphs that tangle the chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails.

The Soifer graph is a planar graph on 9 nodes that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails. As proved by Gethner and Springer, the Soifer graph is the smallest such counterexample (and is smaller than the Kittell graph and Errera graph).It is implemented in the Wolfram Languageas GraphData["SoiferGraph"].

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.

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).

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..

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.

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).

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.

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).

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).

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..

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..

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 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)...

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..

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..

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..

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..

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)...

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..

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..

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..

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..

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..

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..

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..

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'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..

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...

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),..

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).

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..

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...

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).

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

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..

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..

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