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The Möbius function is a number theoretic function defined by(1)so indicates that is squarefree (Havil 2003, p. 208). The first few values of are therefore 1, , , 0, , 1, , 0, 0, 1, , 0, ... (OEIS A008683). Similarly, the first few values of for , 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).The function was introduced by Möbius (1832), and the notation was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number ] is either (when is divisible by a square), or (mod ) (when is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).The Möbius function is implemented in the WolframLanguage as MoebiusMu[n].The summatory function of the Möbius function(2)is called the Mertens function.The..

A cork plug is a three-dimensional solid that can stopper a square, triangular, or circular hole. There is an infinite family of such shapes.The shape with smallest volume has triangular cross sections.The plug with the largest volume is made using two cuts from the top diameter to the edge, as illustrated above. Such a plug has to obtain a square cross section. For a general such a plug of height and radius , the volume of the plug is

A hexagon tiling is a tiling of the planeby identical hexagons.The regular hexagon forms a regular tessellation,also called a hexagonal grid, illustrated above.There are at least three tilings of irregular hexagons,illustrated above.They are given by the following types:(1)(Gardner 1988). Note that the periodic hexagonal tessellationis a degenerate case of all three tilings with(2)and(3)Amazingly, the number of plane partitions contained in an box also gives the number of hexagon tilings by rhombi for a hexagon of side lengths , , , , , (David and Tomei 1989, Fulmek and Krattenthaler 2000). The asymptotic distribution of rhombi in a random hexagon tiling by rhombi was given by Cohn et al. (1998). A variety of enumerations for various explicit positions of rhombi are given by Fulmek and Krattenthaler (1998, 2000)...

In identical experiments, an Allais paradox occurs when the addition of an independent event influences choice behavior. Consider the choices in the following table (Kahneman and Tversky 1979).lottery1 to 333435 to 100preference018%82%0083%017%In Experiment 1, a choice of and was given, and most participants picked . In Experiment 2, a choice of and was given, and most participants picked .This observed pattern violates the independence axiom, since in both experiments, the payoff is identical if a ball is picked, while if the event is disregarded, the two experiments are identical.To see it another way, consider the event to be a black box that is always received if the random ball value is . Knowing or not knowing the contents of the black box should not influence behavior.

Assume , , and are lotteries. Denote " is preferred to " as , and indifference between them by . One version of the probability axioms are then given by the following, the last of which is the independence axiom: 1. Completeness: either or . 2. Transitivity: . 3. Continuity: a unique such that . 4. Independence: if , then for all and .

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 formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.In Hilbert's day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with well-defined meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement could be deduced.The foundations of this program were obliterated in the mid 1930s when Gödel unexpectedly proved a result now known as Gödel's second incompleteness theorem. This theorem not only showed Hilbert's goal to be impossible, but also proved to be only the first in a series of deep and counterintuitive statements about rigor and provability in mathematics.A very simple form of logic is the study of "truth..

A fraction containing each of the digits 1 through 9 is called a pandigital fraction. The following table gives the number of pandigital fractions which represent simple unit fractions. The numbers of pandigital fractions for 1/1, 1/2, 1/3, ... are 0, 12, 2, 4, 12, 3, 7, 46, 3, ... (OEIS A054383).#fractions12,2412,37,46,,,,,,,3004

The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9).The ellipse and hyperbolaare known as central conics.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under..

There are at least 15 classes of convex pentagonal tilings, as illustrated above. The first five were discovered during investigations of German mathematician Karl Reinhardt in 1918. After a gap of 50 years, R. B. Kershner found three more in 1968. Richard James subsequently discovered a ninth type of pentagonal tiling in 1975 and over the next few years, Marjorie Rice discovered another four types. Rolf Stein found a 14th tiling in 1985. The most recently discovered 15th tiling was found by Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell in 2015 using a computer to exhaustively search through a large but finite set of possibilities (Bellos 2015).It has not been proven whether these 15 cases exhaust all possible tilings, but no others are known.Note that the tile in the 14th tiling is essentially different from the others because it is unique (up to similarity), while all the others form families..

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 -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..

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..

LCF notation is a concise and convenient notation devised by Joshua Lederberg (winner of the 1958 Nobel Prize in Physiology and Medicine) for the representation of cubic Hamiltonian graphs (Lederberg 1965). The notation was subsequently modified by Frucht (1976) and Coxeter et al. (1981), and hence was dubbed "LCF notation" by Frucht (1976). Pegg (2003) used the notation to describe many of the cubic symmetric graphs. The notation only applies to Hamiltonian graphs, since it achieves its symmetry and conciseness by placing a Hamiltonian cycle in a circular embedding and then connecting specified pairs of nodes with edges.For example, the notation describes the cubical graph illustrated above. To see how this works, begin with the cycle graph . Beginning with a vertex , count three vertices clockwise () to and connect it to with an edge. Now advance to , count three vertices counterclockwise () to vertex , and connect and with an edge...

An integral graph is defined as a graph whose graph spectrum consists entirely of integers. The notion was first introduced by Harary and Schwenk (1974). The numbers of simple integral graphs on , 2, ... nodes are 0, 2, 3, 6, 10, 20, 33, 71, ... (OEIS A077027), illustrated above for small .The numbers of connected simple integral graphs on , 2, ... nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, ... (OEIS A064731), illustrated above for small .The following table lists common graph classes and the their members which are integral.graphintegral for of the formantiprism graph3complete graph allcycle graph 2, 3, 4, 6empty graphallprism graph3, 4, 6star graph wheel graph 4The following table lists some special named graphs that are integral and gives their spectra.graphgraph spectrum16-cell24-cellClebsch graphcubical graphcuboctahedral graphDesargues graphHall-Janko graphHoffman graphHoffman-Singleton graphLevi graphM22 graphMcLaughlin..

The thickness (or depth) (Skiena 1990, p. 251; Beineke 1997) or (Harary 1994, p. 120) of a graph is the minimum number of planar subgraphs of needed such that the graph union (Skiena 1990, p. 251).The thickness of a planar graph is therefore , and the thickness of a nonplanar graph satisfies .A lower bound for the thickness of a graph is given by(1)where is the number of edges, is the number vertices, and is the ceiling function (Skiena 1990, p. 251). The example above shows a decomposition of the complete graph into three planar subgraphs. This decomposition is minimal, so , in agreement with the bound .The thickness of a complete graph satisfies(2)except for (Vasak 1976, Alekseev and Gonchakov 1976, Beineke 1997). For , 2, ..., the thicknesses are therefore 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... (OEIS A124156).The thickness of a complete bipartite graph is given by(3)except possibly when and are both odd, and with there..

A graph is a minor of a graph if a copy of can be obtained from via repeated edge deletion and/or edge contraction.The Kuratowski reduction theorem states that any nonplanar graph has the complete graph or the complete bipartite graph as a minor. In addition, any snark has the Petersen graph as a minor, as conjectured by Tutte (1967; West 2000, p. 304) and proved by Robertson et al. The determination of graph minors is an NP-hard problem for which no good algorithms are known, although brute-force methods such as those due to Robertson, Sanders, and Thomas exist.

Assume that numbered pancakes are stacked, and that a spatula can be used to reverse the order of the top pancakes for . Then the pancake sorting problem asks how many such "prefix reversals" are sufficient to sort an arbitrary stack (Skiena 1990, p. 48).The maximum numbers of flips needed to sort a random stack of , 2, 3, ... pancakes are 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A058986), with the number of maximal stacks for , 3, ... being 1, 1, 3, 20, 2, 35, 455, ... (OEIS A067607).The following table (OEIS A092113) gives the numbers of stacks of pancakes requiring flips. A flattened version is shown above as a binary plot.0123456781121131221413611351412354820615207919928113327163014954313571903101635For example, the three stacks of four pancakes requiring the maximum of four flips are , , and , which can be ordered using the flip sequences , , and , respectively (illustrated above). Similarly, the two stacks of six pancakes..

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.

In a boarding school there are fifteen schoolgirls who always take their daily walks in rows of threes. How can it be arranged so that each schoolgirl walks in the same row with every other schoolgirl exactly once a week? Solution of this problem is equivalent to constructing a Kirkman triple system of order . A visualization is shown above. Falcone and Pavone (2011) give a number of attractive visualizations together with a visual proof of the problem.The following table gives one of the 7 distinct (up to permutations of letters) solutions to the problem. SunABCDEFGHIJKLMNOMonADHBEKCIOFLNGJMTueAEMBHNCGKDILFJOWedAFIBLOCHJDKMEGNThuAGLBDJCFMEHOIKNFriAJNBIMCELDOGFHKSatAKOBFGCDNEIJHLM(The table of Dörrie 1965 contains four omissions in which the and entries for Wednesday and Thursday are written simply as .)..

The number of multisets of length on symbols is sometimes termed " multichoose ," denoted by analogy with the binomial coefficient . multichoose is given by the simple formulawhere is a multinomial coefficient. For example, 3 multichoose 2 is given by 6, since the possible multisets of length 2 on three elements are , , , , , and .The first few values of are given in the following table. 12345112345213610153141020354151535705162156126Multichoose problems are sometimes called "bars and stars" problems. For example, suppose a recipe called for 5 pinches of spice, out of 9 spices. Each possibility is an arrangement of 5 spices (stars) and dividers between categories (bars), where the notation indicates a choice of spices 1, 1, 5, 6, and 9 (Feller 1968, p. 36). The number of possibilities in this case is then ,..

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 cuban primes, named after differences between successive cubic numbers, have the form . The first few are 7, 19, 37, 61, 127, 271, ... (OEIS A002407), which are also the prime hex numbers. They correspond to indices , 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, ... (OEIS A002504; Cunningham 1912).The numbers of cuban primes less than 1, 10, , ... are 0, 1, 4, 11, 28, 64, 173, 438, 1200, ... (OEIS A113478), which is well-approximated byCuban primes are cyclotomic in nature, being the evaluation of the third homogeneous cyclotomic polynomial, , at values and . The form therefore can only have primitive factors of the form . Also, by construction, 2 and 3 are excluded as non-primitive factors. Therefore, this form has a slightly higher density than would arbitrary numbers of the same size (P. Carmody, pers. comm., Jan. 8, 2006). ..

A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical. Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).The number of demiregular tessellations is commonly given as 14 (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by "demiregular tessellation."A more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65)...

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)...

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..

Let there be ways for a "good" selection and ways for a "bad" selection out of a total of possibilities. Take samples and let equal 1 if selection is successful and 0 if it is not. Let be the total number of successful selections,(1)The probability of successful selections is then(2)(3)(4)The hypergeometric distribution is implemented in the Wolfram Language as HypergeometricDistribution[N, n, m+n].The problem of finding the probability of such a picking problem is sometimes called the "urn problem," since it asks for the probability that out of balls drawn are "good" from an urn that contains "good" balls and "bad" balls. It therefore also describes the probability of obtaining exactly correct balls in a pick- lottery from a reservoir of balls (of which are "good" and are "bad"). For example, for and , the probabilities of obtaining correct balls..

Gardner showed how to dissect a square into eight and nine acute scalene triangles.W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, illustrated above (pers. comm. to Ed Pegg, Jr., Oct 25, 2002). The coordinates can be found from solving the four simultaneous equations (1)(2)(3)(4)for the four unknowns and picking the solutions for which . The solutions are roots of 12th order polynomials with numerical values given approximately by(5)(6)(7)(8)Pegg has constructed a dissection of a square into 22 acute isosceles triangles.Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).

The number of different triangles which have integer side lengths and perimeter is(1)(2)(3)where is the partition function giving the number of ways of writing as a sum of exactly terms, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly complicated closed form is given by(4)The values of for , 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.The generating function for is given by(5)(6)(7) also satisfies(8)It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational..

Given a pick-7 lottery with 23 numbers that pays a prize to anyone matching at least 4 of the 7 numbers, there is a set of 253 tickets that guarantees a win. This set corresponds to the Witt design.More formally, the Witt design on 23 points is a 4-(23,7,1) block design (Witt 1938). It is one of the most remarkable structures in all of combinatorics (Godsil and Royle 2001). It can be constructed by factoring over GF(2), into , whereThe 2048 powers , , , ..., are then computed, mod . This set of vectors happens to be the [23,12,7] Golay code with 253 weight-7 vectors, 1288 weight-11 vectors, and 506 weight-15 vectors. For example, is a weight-7 vector.The Witt design is the set of 253 weight-7 vectors acting on 23 points.Consider as vertices the 253 vectors () and 23 points (). Set edges such that are adjacent if , and are adjacent if they share a single term. Select an arbitrary vertex. For all 176 neighbors of that vertex, change edges to non-edges, and non-edges..

The Golay code is a perfect linear error-correcting code. There are two essentially distinct versions of the Golay code: a binary version and a ternary version.The binary version is a binary linear code consisting of codewords of length 23 and minimum distance 7. The ternary version is a ternary linear code, consisting of codewords of length 11 with minimum distance 5.A parity check matrix for the binary Golay code is given by the matrix , where is the identity matrix and is the matrixBy adding a parity check bit to each codeword in , the extended Golay code , which is a nearly perfect binary linear code, is obtained. The automorphism group of is the Mathieu group .A second generator is the adjacency matrix for the icosahedron, with appended, where is a unit matrix and is an identity matrix.A third generator begins a list with the 24-bit 0 word (000...000) and repeatedly appends first 24-bit word that has eight or more differences from all words in the list...

An amphichiral knot is a knot that is capable of being continuously deformed into its own mirror image. More formally, a knot is amphichiral (also called achiral or amphicheiral) if there exists an orientation-reversing homeomorphism of mapping to itself (Hoste et al. 1998). (If the words "orientation-reversing" are omitted, all knots are equivalent to their mirror images.)Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are amphichiral using the command KnotData[knot, "Amphichiral"].There are 20 amphichiral knots having ten or fewer crossings, namely (the figure eight knot), , , , , , , , , , , , , , , , , , , and (Jones 1985), the first few of which are illustrated above.The following table gives the total number of prime amphichiral knots, number of amphichiral noninvertible prime knots, amphichiral noninvertible prime knots, and fully amphichiral invertible knots prime knots () with..

Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..

A semiprime, also called a 2-almost prime, biprime (Conway et al. 2008), or -number, is a composite number that is the product of two (possibly equal) primes. The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (OEIS A001358). The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, 10, 14, 15, 21, 22, 26, 33, 34, ... (OEIS A006881).The square of any prime number is by definition a semiprime. The largest known semiprime is therefore the square of the largest known prime.A formula for the number of semiprimes less than or equal to is given by(1)where is the prime counting function and is the th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006).The numbers of semiprimes less than for , 2, ... are 3, 34, 299, 2625, 23378, 210035, ... (OEIS A066265).For with and distinct, the following..

The golden ratio can be written in terms of a nested radical in the beautiful form(1)which can be written recursively as(2)for , with .Paris (1987) proved approaches at a constant rate, namely(3)as , where(4)(OEIS A105415) is the Paris constant.A product formula for is given by(5)(Finch 2003, p. 8).Another formula is given by letting be the analytic solution to the functional equation(6)for , subject to initial conditions and . Then(7)(Finch 2003, p. 8).A close approximation is , which is good to 4 decimal places (M. Stark, pers. comm.).

The irrational constant(1)(2)(OEIS A060295), which is very close to an integer. Numbers such as the Ramanujan constant can be found using the theory of modular functions. In fact, the nine Heegner numbers (which include 163) share a deep number theoretic property related to some amazing properties of the j-function that leads to this sort of near-identity.Although Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such ), he did not actually mention the particular near-identity given above. In fact, Hermite (1859) observed this property of 163 long before Ramanujan's work. The name "Ramanujan's constant" was coined by Simon Plouffe and derives from an April Fool's joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American. In his column, Gardner claimed that was exactly an integer, and that Ramanujan had conjectured this in his 1914 paper. Gardner admitted his hoax a..

The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..

The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...

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).

Vizing's theorem states that a graph can be edge-colored in either or colors, where is the maximum vertex degree of the graph. A graph with edge chromatic number equal to is known as a class 2 graph.Class 2 graphs include the Petersen graph, complete graphs for , 5, 7, ..., and the snarks.All regular graphs with an odd number of nodes are class 2 by parity. Such graphs automatically have an even number of edges per vertex.A graph is trivially class 2 if the maximum independent edge sets are not large enough to cover all edges. In particular, a graph is trivially class 2 ifwhere is the matching number, the maximum vertex degree, and the edge count of .The following table summarizes some named class 2 graphs.graph triangle graph3pentatope graph5Petersen graph10first Blanuša snark18second Blanuša snark18Robertson graph19flower snark2025-Grünbaum graph25Doyle graph27double star snark30Szekeres snark50McLaughlin graph275The..

Vizing's theorem states that a graph can be edge-colored in either or colors, where is the maximum vertex degree of the graph. A graph with edge chromatic number equal to is known as a class 1 graph.König's line coloring theorem states that all bipartite graphs are class 1. All cubic Hamiltonian graphs are class 1, as are planar graphs with maximum vertex degree (Cole and Kowalik 2008).Class 1 graphs have both edge chromatic number and fractional edge chromatic number equal to .Families of non-bipartite graphs that appear to be class 1 (or at least whose smallest members are all class 1) include king, Lindgren-Sousselier, and windmill graphs (S. Wagon, pers. comm., Oct. 27-28, 2011). Keller graphs are class 1 (Jarnicki et al. 2015). Aubert and Schneider (1982) showed that rook graphs admit Hamiltonian decomposition, meaning they are class 1 when they have even vertex count and class 2 when they have odd vertex count (because..

Convergents of the pi continued fractions are the simplest approximants to . The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.Two approximations follow from the near-identity function evaluated at and , giving(1)(2)which are good to 2 and 3 digits, respectively.Kochanski's approximation is the rootof(3)given by(4)which is good to 4 digits.Another curious fact is the almost integer(5)which can also be written as(6)(7)Here, is Gelfond's constant. Applying cosine a few more times gives(8)Another approximation involving is given by(9)which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).An apparently interesting near-identity is given by(10)which becomes less surprising when it is noted that 555555 is a repdigit,so the above is..

An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by(1)found by R. Sabey in 2004 (Friedman 2004).Castellanos (1988ab) gives several curious approximations to ,(2)(3)(4)(5)(6)(7)which are good to 6, 7, 9, 10, 12, and 15 digits respectively.E. Pegg Jr. (pers. comm., Mar. 2, 2002), found(8)which is good to 7 digits.J. Lafont (pers. comm., MAy 16, 2008) found(9)where is a harmonic number, which is good to 7 digits.N. Davidson (pers. comm., Sept. 7, 2004) found(10)which is good to 6 digits.D. Barron noticed the curious approximation(11)where is Catalan's constant and is the Euler-Mascheroni constant, which however, is only good to 3 digits.

An almost integer is a number that is very close to an integer.Surprising examples are given by(1)which equals to within 5 digits and(2)which equals to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004). The first of these comes from the half-angle formula identity(3)where 22 is the numerator of the convergent 22/7 to , so . It therefore follows that any pi approximation gives a near-identity of the form .Another surprising example involving both e andpi is(4)which can also be written as(5)(6)Here, is Gelfond's constant. Applying cosine a few more times gives(7)This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" is true has yet been discovered.Another nested cosine almost integer is given by(8)(P. Rolli, pers. comm., Feb. 19, 2004).An..

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..

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..

A cubic nonplanar graph is a graph that is both cubicand nonplanar.The following table summarizes some named nonplanar cubic graphs.graph utility graph6Petersen graph10Franklin graph12Heawood graph14Möbius-Kantor graph16first Blanusa snark18second Blanusa snark18Pappus graph18Desargues graph20flower snark20McGee graph24Coxeter graph28double star snark30Levi graph30Dyck graph32Szekeres snark50Gray graph54Balaban 10-cage70Foster graph90Biggs-Smith graph102Balaban 11-cage112Tutte 12-cage126The largest cubic nonplanar graphs having diameters 3 and 4 are illustrated above. They have 20 and 38 vertices, respectively.

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..

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..

A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume. (Note that the volume of a tetrahedron can be computed using the Cayley-Menger determinant.)The integer Heronian tetrahedron having smallest maximum side length is the one with edge lengths 51, 52, 53, 80, 84, 117; faces (117, 80, 53), (117, 84, 51), (80, 84, 52), (53, 51, 52); face areas 1170, 1800, 1890, 2016; and volume 18144 (Buchholz 1992; Guy 1994, p. 191). This is the only integer Heronian triangle with all side lengths less than 157.The integer Heronian tetrahedron with smallest possible surface area and volume has edges 25, 39, 56, 120, 153, and 160; areas 420, 1404, 1872, and 2688 (for a total surface area of 6384); and volume 8064 (Buchholz 1992, Peterson..

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..

"SOHCAHTOA" is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent,(1)(2)(3)Other mnemonics include 1. "Tommy On A Ship Of His Caught A Herring" (probably more common in Great Britain than the United States). 2. "Oscar Has A Hold On Angie." 3. "Oscar Had A Heap of Apples." 4. "The Old Army Colonel And His Son Often Hiccup" (which gives the functions in the order tangent, cosine, sine). 5. "Studying Our Homework Can Always Help To Obtain Achievement." 6. "Some Old Hippy Caught Another Hippy Tripping On Acid."

WireWorld is a two-dimensional four-color cellular automaton introduced by Brian Silverman in 1987. The rule for the automaton uses the cell's old value together with the number of its eight neighbors that are set to 1 according to a system that roughly models the flow of currents in wires according to the following rules. 0. The color 0 is considered background, and always stays as background. 1. The color 1 is considered an electron head, and always turns into an electron tail. 2. The color 2 is an electron tail, and always turns into wire. 3. The color 3 is wire, which remains wire unless is 1 or 2, in which case it becomes an electron head. With these rules, digital logic circuits can be constructed, as illustrated above for OR, XOR, and AND gates.In 2002, Nick Gardner demonstrated how two 8-bit binarynumbers could be multiplied with a WireWorld construction using the network above...

Let a tree be a subgraph of a cubic graph . The graph excision is the graph resulting from removing the tree, then merging the edges. For example, if in the Levi graph (left figure) the tree formed by the 6 interior points (middle figure) is excised, the McGee graph (right figure) results. Similarly, excising the Heawood graph gives the Petersen graph, and excising the generalized hexagon (i.e., the unique 12-cage graph) gives the Balaban 11-cage (Biggs 1998).The reverse of excision is insertion. Both operations are used in the analysis ofcages.The following table gives some cubic symmetricgraphs with named edge-excised graphs, illustrated above.graphedge-excised graphutility graphtetrahedral graphcubical graph3-prism graphPetersen graph4-Möbius ladderHeawood graph12-cubic graph 84Möbius-Kantor graph14-cubic graph 503dodecahedral graphcubic polyhedral graph Cp34..

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

A directed strongly regular graph is a simple directed graph with adjacency matrix such that the span of , the identity matrix , and the unit matrix is closed under matrix multiplication.

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

A state diagram is a labeled directed graph together with state information that can be used to indicate that certain paths on in a system may be traversed only in a certain way. State diagrams are also known as problem space models (Atallah 1998, p. 36-2). For example, in the left figure above (due to R. Abbott), a car must traverse a town while obeying all traffic laws, and making no U-turns. Initially, the car is at position 4, travelling east, and has a choice of moving to position 1, travelling east, or position 5, travelling east. The State diagram corresponding to the maze is illustrated in the above right figure.Another example created by R. Abbott is illustrated above. While the maze is very similar to the first one, the state diagram is not. This maze therefore illustrates that a minor change to the rules of a system can have a large impact on its state diagram.Other puzzles, problems, and programs use state diagrams as an analysis..

Taking a connected graph or network with a high graph diameter and adding a very small number of edges randomly, the diameter tends to drop drastically. This is known as the small world phenomenon. It is sometimes also known as "six degrees of separation" since, in the social network of the world, any person turns out to be linked to any other person by roughly six connections.Short-term memory uses small world networks between neurons to remember this sentence.In modern mathematics, the center of the network of coauthorship is considered to be P. Erdős, resulting in the so-called Erdős number. In movies, Kevin Bacon is often mentioned as the center of the movie universe, but a recent study (Reynolds) has shown Christopher Lee to be the actual center. Both actors have co-starred with Julius LeFlore, so the Lee-Bacon distance is two...

Let be a set of elements together with a set of 3-subset (triples) of such that every 2-subset of occurs in exactly one triple of . Then is called a Steiner triple system and is a special case of a Steiner system with and . A Steiner triple system of order exists iff (Kirkman 1847). In addition, if Steiner triple systems and of orders and exist, then so does a Steiner triple system of order (Ryser 1963, p. 101).Examples of Steiner triple systems of small orders are(1)(2)(3)The Steiner triple system is illustrated above.The numbers of nonisomorphic Steiner triple systems of orders , 9, 13, 15, 19, ... (i.e., ) are 1, 1, 2, 80, 11084874829, ... (Stinson and Ferch 1985; Colbourn and Dinitz 1996, pp. 14-15; Kaski and Östergård 2004; OEIS A030129). is the same as the finite projective plane of order 2. is a finite affine plane which can be constructed from the array(4)One of the two s is a finite hyperbolic plane. The 80 Steiner triple systems..

The real projective plane is the closed topological manifold, denoted , that is obtained by projecting the points of a plane from a fixed point (not on the plane), with the addition of the line at infinity. It can be described by connecting the sides of a square in the orientations illustrated above (Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324).There is then a one-to-one correspondence between points in and lines through not parallel to . Lines through that are parallel to have a one-to-one correspondence with points on the line at infinity. Since each line through intersects the sphere centered at and tangent to in two antipodal points, can be described as a quotient space of by identifying any two such points. The real projective plane is a nonorientable surface. The equator of (which, in the quotient space, is itself a projective line) corresponds to the line at infinity.The complete graph on 6 vertices can be drawn in the projective..

Given the above figure (without looking at the figure below!), determine how to disengage the two slotted cube halves without cutting, breaking, or distorting.One possible solution is shown above; the slots are not perpendicular to one another but instead consist of parallel slotted tracks. Other solutions are also possible. For example, another construction involves two circular arcs sharing the same center (Gardner 2001, p. 117).

Sudoku (literally, "single number"), sometimes also is a pencil-and-paper logic puzzle whose goal is to complete a grid satisfying various constraints. In the "classic" Sudoku, a square is divided into "regions", with various squares filled with "givens." Valid solutions use each of the numbers 1-9 exactly once within each row, column and region. This kind of sudoku is therefore a particular case of a Latin square.Under the U.S.-only trademarked name "Number Place," Sudoku was first published anonymously by Garns (1979) for Dell Pencil Puzzles. In 1984, the puzzle was used by Nikoli with the Japan-only trademarked name Sudoku (Su = number, Doku = single). Due to the trademark issues, in Japan, the puzzle became well-known as nanpure, or Number Place, often using the English name. Outside Japan, the Japanese name predominates.The puzzle received a large amount of attention in the..

Let be a group, and let be a set of group elements such that the identity element . The Cayley graph associated with is then defined as the directed graph having one vertex associated with each group element and directed edges whenever . The Cayley graph may depend on the choice of a generating set, and is connected iff generates (i.e., the set are group generators of ).Care is needed since the term "Cayley graph" is also used when is implicitly understood to be a set of generators for the group, in which case the graph is always connected (but in general, still dependent on the choice of generators). This sort of Cayley graph of a group may be computed in the Wolfram Language using CayleyGraph[G], where the generators used are those returned by GroupGenerators[G].To complicate matters further, undirected versions of proper directed Cayley graphs are also known as Cayley graphs.An undirected Cayley graph of a particular generating set of..

The braced square problem asks: given a hinged square composed of four equal rods (indicated by the thick lines above), how many more hinged rods must be added in the same plane (with no two rods crossing) so that the original square is rigid in the plane. The best solution known (left figure above), uses a total of 27 rods, where , , and are collinear (Gardner 1964; Gardner 1984; Wells 1991). If rods are allowed to cross, the best known solution requires 19 rods (right figure above).Friedman has also considered the minimum number of rods needed to construct rigid regular -gons. In 1963, T. H. O'Beirne found solutions for the pentagon (64 rods), octagon (105 rods), and dodecagon (45 rods). His 64-rod solution for the pentagon is illustrated above (Frederickson 2002, p. 71). The best solutions known with overlapping permitted for polygons with , 4, ... sides require 3, 19, 31, 11, 79, 31, 51, ... rods (A. Khodulyov; cited in Friedman..

Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where(OEIS A093825; Steinhaus 1999, p. 202;Wells 1986, p. 29; Wells 1991, p. 237).In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.While the Kepler conjecture is intuitively obvious, the proof remained..

is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated octahedron. It has conjugacy classes 1, , , , , , , , , and (Cotton 1990). Its multiplication table is illustrated above. The octahedral group is implemented in the Wolfram Language as FiniteGroupData["Octahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Oh", "PermutationGroupRepresentation"].The great rhombicuboctahedron can be generated using the matrix representation of using the basis vector .The octahedral group has a pure rotation subgroup denoted that is isomorphic to the tetrahedral group . is of order 24 and has conjugacy classes 1, , , , and (Cotton 1990, pp. 50 and 434). Its multiplication table is illustrated above. The pure..

The tetrahedral group is the point group of symmetries of the tetrahedron including the inversion operation. It is one of the 12 non-Abelian groups of order 24. The tetrahedral group has conjugacy classes 1, , , , and (Cotton 1990, pp. 47 and 434). Its multiplication table is illustrated above. The tetrahedral group is implemented in the Wolfram Language as FiniteGroupData["Tetrahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Td", "PermutationGroupRepresentation"]. has a pure rotational subgroup of order 12 denoted (Cotton 1990, pp. 50 and 433). It is isomorphic to the alternating group and has conjugacy classes 1, , , and . It has 10 subgroups: one of length 1, three of length 2, 4 of length 3, one of length 4, and one of length 12. Of these, only the trivial subgroup, subgroup of order 4, and complete..

The icosahedral group is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product of the alternating group and cyclic group . The icosahedral group consists of the conjugacy classes 1, , , , , , , , , and (Cotton 1990, pp. 49 and 436). Its multiplication table is illustrated above. The icosahedral group is a subgroup of the special orthogonal group . The icosahedal group is implemented in the Wolfram Language as FiniteGroupData["Icosahedral", "PermutationGroupRepresentation"].Icosahedral symmetry is possible as a rotational group but is not compatible with translational symmetry. As a result, there are no crystals with this symmetry and so, unlike the octahedral group and tetrahedral group , is not one of the 32 point groups.The great rhombicosidodecahedron can be generated using the matrix representation of using the basis vector , where is the golden..

Turmites, also called turning machines, are 2-dimensional Turing machines in which the "tape" consists of a grid of spaces that can be written and erased by an active ("head") element that turns at each iteration on the basis of the state of its current grid square. The "head" of the system is usually called a "vant," "ant," or "turmite" on square grids, and a "bee," "worm," or "turtle" on hexagonal grids. (The term "turtle" is named after Seymour Papert's turtle geometry). The turmite tracks its position, direction, and current state.Amazingly, the turmite with rule illustrated above mimics binary counting. In this turmite, the pattern of bands above the red line corresponds to incrementing binary digits. After each cycle constructing the upper pattern, the same pattern is produced (mirrored left-to-right) below the red..

A 4-state two-dimensional Turing machine invented in the 1980s. The ant starts out on a grid containing black and white cells, and then follows the following set of rules. 1. If the ant is on a black square, it turns right and moves forward one unit. 2. If the ant is on a white square, it turns left and moves forward one unit. 3. When the ant leaves a square, it inverts the color. When the ant is started on an empty grid, it eventually builds a "highway" that is a series of 104 steps that repeat indefinitely, each time displacing the ant two pixels vertically and horizontally. The plots above show the ant starting from a completely white grid after 386 (left figure) and (right figure) steps. In the right figure, the highway is being constructed towards the lower right-hand corner. The fact that the ant's path is unbounded is guaranteed by the Cohen-Kung theorem. It is believed that no matter what initial pattern the ant is started on, it will eventually..

Define a Bouniakowsky polynomial as an irreducible polynomial with integer coefficients, degree , and . The Bouniakowsky conjecture states that is prime for an infinite number of integers (Bouniakowsky 1857). As an example of the greatest common divisor caveat, the polynomial is irreducible, but always divisible by 2.Irreducible degree 1 polynomials () always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that is prime for an infinite number of integers .Various prime-generating polynomialsare known, but none of these always generates a prime (Legendre).Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, produces no primes until , 764400, 933660, ... (OEIS A122131)...

Let a chess piece make a tour on an chessboard whose squares are numbered from 1 to along the path of the chess piece. Then the tour is called a magic tour if the resulting arrangement of numbers is a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. If the first and last squares traversed are connected by a move, the tour is said to be closed (or "re-entrant"); otherwise it is open. (Note some care with terminology is necessary. For example, Jelliss terms a semimagic tour a "magic tour" and a magic tour a "diagonally magic tour.")Magic knight graph tours are not possible on boards for odd. However, as had long been known, they are possible for all boards of size for . However, the () remained open even since it was first investigated by authors such as Beverley (1848). It was not resolved until an exhaustive computer enumeration of all possibilities was completed on August 5,..

The Penrose triangle, also called the tribar (Cerf), tri-bar (Ernst 1987), impossible tribar (Pappas 1989, p. 13), or impossible triangle, is an impossible figure published by Penrose and Penrose (1958). Penrose triangles appear prominently in the works of Escher, who not only inspired creation of this object (Escher 1954, Penrose and Penrose 1958), but also subsequently publicized it.The Penrose triangle can be extended to -gonal barred objects (Cerf, Elber), including the so-called tribox.The figure was drawn earlier by artist Oscar Reutersvärd in 1934 during a "long lecture." For this, he was honored with a stamp by the government of Sweden in 1982 (Miller).The Penrose triangle appears on the cover of Raghavachary (2004).Henderson (2006) offers an impossible triangle net...

The factorization of a number into its constituent primes, also called prime decomposition. Given a positive integer , the prime factorization is writtenwhere the s are the prime factors, each of order . Each factor is called a primary. Prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.Through his invention of the Pratt certificate, Pratt (1975) became the first to establish that prime factorization lies in the complexity class NP.The following Wolfram Language code can be used to give a nicely typeset form of a number : FactorForm[n_?NumberQ, fac_:Automatic] := Times @@ (HoldForm[Power[##]]& @@@ FactorInteger[n, fac])The first few prime factorizations (the number 1, by definition, has a prime factorization of "1") are given in the following table.prime factorizationprime factorization11111122123313134145515616771717818919191020The..

According to Hardy and Wright (1979), the 44-digit Ferrier's primedetermined to be prime using only a mechanical calculator, is the largest prime found before the days of electronic computers. The Wolfram Language can verify primality of this number in a (small) fraction of a second, showing how far the art of numerical computation has advanced in the intervening years. It can be shown to be a probable prime almost instantaneously In[1]:= FerrierPrime = (2^148 + 1)/17; In[2]:= PrimeQ[FerrierPrime] // Timing Out[2]= {0.01 Second, True}and verified to be an actual prime complete with primalitycertificate almost as quickly In[3]:= <<PrimalityProving` In[4]:= ProvablePrimeQ[FerrierPrime, "Certificate" -> True] // Timing Out[4]= {0.04 Second,{True, {20988936657440586486151264256610222593863921,17, {2,{3,2,{2}},{5,2,{2}},{7,3,{2,{3,2,{2}}}}, {13,2,{2,{3,2,{2}}}},{19, 2,{2,{3,2,{2}}}},{37,2,{2,{3,2,{2}}}},{73,5,{..

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function that, roughly speaking, any nonvanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the zeta function in the critical strip.More precisely, let and suppose that is a nonvanishing continuous function on the disk that is analytic in the interior. Then for any , there exists a positive real number such thatMoreover, the set of these has positive lower density, i.e.,Garunkštis (2003) obtained explicit estimates for the first approximating and the positive lower density, provided that is sufficiently small and sufficiently smooth. The condition that have no zeros for is necessary.The Riemann hypothesis is known to be true iff can approximate itself uniformly in the sense of Voronin's theorem (Bohr 1922, Bagchi 1987). It is also known that there exists a rich zoo of Dirichlet series having this or some similar..

An attracting set that has zero measure in the embedding phase space and has fractal dimension. Trajectories within a strange attractor appear to skip around randomly.A selection of strange attractors for a general quadraticmap(1)(2)are illustrated above, where the letters to stand for coefficients of the quadratic from to 1.2 in steps of 0.1 (Sprott 1993c). These represent a small selection of the approximately 1.6% of all possible such maps that are chaotic (Sprott 1993bc).

A topological space, also called an abstract topological space, is a set together with a collection of open subsets that satisfies the four conditions: 1. The empty set is in . 2. is in . 3. The intersection of a finite number of sets in is also in . 4. The union of an arbitrary number of sets in is also in . Alternatively, may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become: 3. The intersection of an arbitrary number of sets in is also in . 4. The union of a finite number of sets in is also in . These axioms are designed so that the traditional definitions of open and closed intervals of the real line continue to be true. For example, the restriction in (3) can be seen to be necessary by considering , where an infinite intersection of open intervals is a closed set.In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff axioms (which..

The margin of error is an estimate of a confidenceinterval for a given measurement, result, etc. and is frequently cited in statistics.While phrases such as, "The poll has a margin of error of plus or minus 3.1 percentage points" are commonly heard, an additional qualification such as "at a 95 percent confidence level" is also needed in order to precisely indicate what the error refers to.For a given confidence interval , standard deviation , and sample size , the margin of error (for a normal distribution) iswhere is the inverse erf function.

Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation to denote the greatest common divisor, two integers and are relatively prime if . Relatively prime integers are sometimes also called strangers or coprime and are denoted . The plot above plots and along the two axes and colors a square black if and white otherwise (left figure) and simply colored according to (right figure).Two numbers can be tested to see if they are relatively prime in the Wolfram Language using CoprimeQ[m, n].Two distinct primes and are always relatively prime, , as are any positive integer powers of distinct primes and , .Relative primality is not transitive. For example, and , but .The probability that two integers and picked at random are relatively prime is(1)(OEIS A059956; Cesàro and Sylvester 1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986, p. 28; Borwein and Bailey 2003, p. 139;..

A line graph (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or -obrazom graph) of a simple graph is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of have a vertex in common (Gross and Yellen 2006, p. 20).The line graph of a directed graph is the directed graph whose vertex set corresponds to the arc set of and having an arc directed from an edge to an edge if in , the head of meets the tail of (Gross and Yellen 2006, p. 265).Line graphs are implemented in the Wolfram Language as LineGraph[g]. Precomputed line graph identifications of many named graphs can be obtained in the Wolfram Language using GraphData[graph, "LineGraphName"].The numbers of simple line graphs on , 2, ... vertices are 1, 2, 4, 10, 24, 63, 166, 471, 1408, ... (OEIS A132220), and the numbers of connected..

Min Max Min Max Re Im Given a number , the cube root of , denoted or ( to the 1/3 power), is a number such that . The cube root is therefore an nth root with . Every real number has a unique real cube root, and every nonzero complex number has three distinct cube roots.The schoolbook definition of the cube root of a negative number is . However, extension of the cube root into the complex plane gives a branch cut along the negative real axis for the principal value of the cube root as illustrated above. By convention, "the" (principal) cube root is therefore a complex number with positive imaginary part. As a result, the Wolfram Language and other symbolic algebra languages and programs that return results valid over the entire complex plane therefore return complex results for . For example, in the Wolfram Language, ComplexExpand[(-1)^(1/3)] gives the result .When considering a positive real number , the Wolfram Language function CubeRoot[x],..

The generalized Petersen graph , also denoted (Biggs 1993, p. 119; Pemmaraju and Skiena 2003, p. 215), for and is a connected cubic graph consisting of an inner star polygon (circulant graph ) and an outer regular polygon (cycle graph ) with corresponding vertices in the inner and outer polygons connected with edges. These graphs were introduced by Coxeter (1950) and named by Watkins (1969).Since the generalized Petersen graph is cubic, , where is the edge count and is the vertex count. More specifically, has nodes and edges. Generalized Petersen graphs are implemented in the Wolfram Language as PetersenGraph[k, n] and their properties are available using GraphData["GeneralizedPetersen", k, n].Generalized Petersen graphs may be further generalized to Igraphs.For odd, is isomorphic to . So, for example, , , , , and so on. The numbers of nonisomorphic generalized Petersen graphs on , 8, ... nodes are 1, 1, 2, 2, 2, 3,..

"The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular graph with intersection array (Brouwer et al. 1989, §13.1.2; Brouwer and Haemers 1993). It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge, then deleting the 14 vertices within distance 2 of that edge.It has graph diameter 3, girth 5, graph radius 3, is Hamiltonian, and nonplanar. It has chromatic number 4, edge connectivity 5, vertex connectivity 5, and edge chromatic number 5.It is an integral graph and has graph spectrum (Brouwer and Haemers 1993).The Sylvester graph of a configuration is the set of ordinarypoints and ordinary lines.

The Delannoy numbers are the number of lattice paths from to in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e., , , and ). They are given by the recurrence relation(1)with . The are also given by the sums(2)(3)(4)where is a hypergeometric function.A table for values for the Delannoy numbers is given by(5)(OEIS A008288) for , 1, ... increasing from left to right and , 1, ... increasing from top to bottom.They have the generating function(6)(Comtet 1974, p. 81).Taking gives the central Delannoy numbers , which are the number of "king walks" from the corner of an square to the upper right corner . These are given by(7)where is a Legendre polynomial (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression is(8)(9)(10)where is a binomial coefficient and is a hypergeometric function. These numbers have a surprising connection with the Cantor set (E. W. Weisstein, Apr. 9,..

A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity: There exists an element such that for all , , 4. Additive inverse: For every there exists such that , 5. Left and right distributivity: For all , and , 6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring). Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).Rings may also satisfy various optional conditions: 7. Multiplicative commutativity:..

An ellipse is a curve that is the locus of all points in the plane the sum of whose distances and from two fixed points and (the foci) separated by a distance of is a given positive constant (Hilbert and Cohn-Vossen 1999, p. 2). This results in the two-center bipolar coordinate equation(1)where is the semimajor axis and the origin of the coordinate system is at one of the foci. The corresponding parameter is known as the semiminor axis.The ellipse is a conic section and a Lissajouscurve.An ellipse can be specified in the Wolfram Language using Circle[x, y, a, b].If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse. This is known as the trammel construction of an ellipse (Eves 1965, p. 177).It is possible to construct elliptical gears that rotate smoothly against one another (Brown 1871, pp. 14-15; Reuleaux and Kennedy 1876,..

A parabola (plural "parabolas"; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas and . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that..

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