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There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.Group-theoretically, let be the modular group Gamma0, and let be the compactification (by adding cusps) of , where is the upper half-plane. Also define to be the Fricke involution defined by the block matrix . For a prime, define . Then is a supersingular prime if the genus of .The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field . They have their j-invariant in .Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group...

A (general, asymmetric) lens is a lamina formed by the intersection of two offset disks of unequal radii such that the intersection is not empty, one disk does not completely enclose the other, and the centers of curvatures are on opposite sides of the lens. If the centers of curvature are on the same side, a lune results.The area of a general asymmetric lens obtained from circles of radii and and offset can be found from the formula for circle-circle intersection, namely(1)(2)Similarly, the height of such a lens is(3)(4)A symmetric lens is lens formed by the intersection of two equal disk. The area of a symmetric lens obtained from circles with radii and offset is given by(5)and the height by(6)A special type of symmetric lens is the vesica piscis (Latin for "fish bladder"), corresponding to a disk offset which is equal to the disk radii.A lens-shaped region also arises in the study of Bessel functions, is very important in the theory of..

A fact noticed by physicist G. Gamow when he had an office on the second floor and physicist M. Stern had an office on the sixth floor of a seven-story building (Gamow and Stern 1958, Gardner 1986). Gamow noticed that about 5/6 of the time, the first elevator to stop on his floor was going down, whereas about the same fraction of time, the first elevator to stop on the sixth floor was going up. This actually makes perfect sense, since 5 of the 6 floors 1, 3, 4, 5, 6, 7 are above the second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the sixth. However, the situation takes some unexpected turns if more than one elevator is involved, as discussed by Gardner (1986). Furthermore, even worse, the analysis by Gamow and Stern (1958) turns out to be incorrect (Knuth 1969)!Main character Charles Eppes discusses the elevator paradox in the Season 4 episode "Chinese Box" of the television crime drama NUMB3RS...

A schematic diagram used in logic theory to depict collectionsof sets and represent their relationships.The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing a total of four regions, , , , and (the empty set, represented by none of the regions occupied). Here, denotes the intersection of sets and .The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled , , and consist of members which are only in one set and no others, the three regions labelled , , and consist of members which are in two sets but not the third, the region consists of members which are simultaneously in all three, and no regions occupied represents .In general, an order- Venn diagram is a collection of simple closed curves in the plane such that 1. The curves partition the plane into connected..

The quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."

1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):(K. MacMillan, pers. comm., Apr. 29, 2007).This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.1729 was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character..

The Farey sequence for any positive integer is the set of irreducible rational numbers with and arranged in increasing order. The first few are(1)(2)(3)(4)(5)(OEIS A006842 and A006843). Except for , each has an odd number of terms and the middle term is always 1/2.Let , , and be three successive terms in a Farey series. Then(6)(7)These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of terms, insert the mediant fraction between terms and when (Hardy and Wright 1979, pp. 25-26; Conway and Guy 1996; Apostol 1997). Given with , let be the mediant of and . Then , and these fractions satisfy the unimodular relations(8)(9)(Apostol 1997, p. 99).The number of terms in the Farey sequence for the integer is(10)(11)where is the totient function and is the summatory function of , giving 2, 3, 5, 7, 11, 13, 19, ... (OEIS A005728). The asymptotic limit..

The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth..

Also called Chvátal's art gallery theorem. If the walls of an art gallery are made up of straight line segments, then the entire gallery can always be supervised by watchmen placed in corners, where is the floor function. This theorem was proved by Chvátal (1975). It was conjectured that an art gallery with walls and holes requires watchmen, which has now been proven by Bjorling-Sachs and Souvaine (1991, 1995) and Hoffman et al. (1991).In the Season 2 episode "Obsession" (2006) of the television crime drama NUMB3RS, Charlie mentions the art gallery theorem while building an architectural model.

A Markov chain is collection of random variables (where the index runs through 0, 1, ...) having the property that, given the present, the future is conditionally independent of the past.In other words,If a Markov sequence of random variates take the discrete values , ..., , thenand the sequence is called a Markov chain (Papoulis 1984, p. 532).A simple random walk is an example of a Markovchain.The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.

RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..

The partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other. A Voronoi diagram is sometimes also known as a Dirichlet tessellation. The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons.Voronoi diagrams were considered as early at 1644 by René Descartes and were used by Dirichlet (1850) in the investigation of positive quadratic forms. They were also studied by Voronoi (1907), who extended the investigation of Voronoi diagrams to higher dimensions. They find widespread applications in areas such as computer graphics, epidemiology, geophysics, and meteorology. A particularly notable use of a Voronoi diagram was the analysis of the 1854 cholera epidemic in London, in which physician John Snow determined a strong correlation of deaths with proximity to a particular..

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.

Percolation theory deals with fluid flow (or any other similar process) in random media.If the medium is a set of regular lattice points, then there are two main types of percolation: A site percolation considers the lattice vertices as the relevant entities; a bond percolation considers the lattice edges as the relevant entities. These two models are examples of discrete percolation theory, an umbrella term used to describe any percolation model which takes place on a regular point lattice or any other discrete set, and while they're most certainly the most-studied of the discrete models, others such as AB percolation and mixed percolation do exist and are reasonably well-studied in their own right.Contrarily, one may also talk about continuum percolation models, i.e.,models which attempt to define analogous tools and results with respect to continuous, uncountable domains. In particular, continuum percolation theory involves notions..

A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..

A Nash equilibrium of a strategic game is a profile of strategies , where ( is the strategy set of player ), such that for each player , , , where and .Another way to state the Nash equilibrium condition is that solves for each . In words, in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players.The Season 1 episode "Dirty Bomb" (2005) of the television crime drama NUMB3RS mentions Nash equilibrium.

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve..

The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann in 1928.Formally, let and be mixed strategies for players A and B. Let be the payoff matrix. Thenwhere is called the value of the game and and are called the solutions. It also turns out that if there is more than one optimal mixed strategy, there are infinitely many.In the Season 4 opening episode "Trust Metric" (2007) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that he used the minimax theorem in an attempt to derive an equation describing friendship.

For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..

A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).It is implemented in the Wolfram Languageas PolyhedronData["Cuboctahedron"].It is shown above in a number of symmetric projections.The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron..

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 hypotrochoid generated by a fixed point on a circle rolling inside a fixed circle. The curves above correspond to values of , 0.2, ..., 1.0.Additional attractive designs such as the one above can also be made by superposing individual spirographs.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features spirographs when discussing Guilloché patterns.

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an chessboard? Since this is a geometric series, the answer for squares isa Mersenne number. Plugging in then gives .

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.

Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.Mathematicians sometimes use the term "combinatorics" to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as "enumeration."The Season 1 episode "Noisy Edge" (2005) of the television crime drama NUMB3RS mentions combinatorics.

Combinatorial matrix theory is a rich branch of mathematics that combines combinatorics, graph theory, and linear algebra. It includes the theory of matrices with prescribed combinatorial properties, including permanents and Latin squares. It also comprises combinatorial proof of classical algebraic theorems such as Cayley-Hamilton theorem.As mentioned in Season 4 episodes 407 "Primacy" and 412 "Power" of the television crime drama NUMB3RS, professor Amita Ramanujan's primary teaching interest is combinatorial matrix theory.

Various handshaking problems are in circulation, the most common one being the following. In a room of people, how many different handshakes are possible?The answer is . To see this, enumerate the people present, and consider one person at a time. The first person may shake hands with other people. The next person may shake hands with other people, not counting the first person again. Continuing like this gives us a total number ofhandshakes, which is exactly the answer given above.Another popular handshake problem starts out similarly with people at a party. Not being able to shake hands with yourself, and not counting multiple handshakes with the same person, the problem is to show that there will always be two people at the party, who have shaken hands the same number of times.The solution to this problem uses Dirichlet's box principle. If there exists a person at the party, who has shaken hands zero times, then every person at the party has shaken..

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..

A geometric sequence is a sequence , , 1, ..., such that each term is given by a multiple of the previous one. Another equivalent definition is that a sequence is geometric iff it has a zero series bias. If the multiplier is , then the th term is given byTaking gives the simple special caseThe Season 1 episode "Identity Crisis" (2005) of the television crime drama NUMB3RS mentions geometric progressions.

The minimum spanning tree of a weighted graph is a set of edges of minimum total weight which form a spanning tree of the graph. When a graph is unweighted, any spanning tree is a minimum spanning tree.The minimum spanning tree can be found in polynomial time. Common algorithms include those due to Prim (1957) and Kruskal's algorithm (Kruskal 1956). The problem can also be formulated using matroids (Papadimitriou and Steiglitz 1982). A minimum spanning tree can be found in the Wolfram Language using the command FindSpanningTree[g].The Season 1 episodes "Vector" and "Man Hunt" (2005) and Season 2 episode "Rampage" (2006) of the television crime drama NUMB3RS feature minimal spanning trees.

A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for as many time steps as desired. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universal constructor." Cellular automata were studied in the early 1950s as a possible model for biological systems (Wolfram 2002, p. 48). Comprehensive studies of cellular automata have been performed by S. Wolfram starting in the 1980s, and Wolfram's fundamental research in the field culminated in the publication of his book A New Kind of Science (Wolfram 2002) in which Wolfram presents a gigantic collection of results concerning automata, among which are a number of groundbreaking new discoveries.The Season 2 episode..

A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..

If moves along a known curve, then describes a pursuit curve if is always directed toward and and move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.The equations of pursuit are given by(1)which specifies that the tangent vector at point is always parallel to the line connecting and , combined with(2)which specifies that the point moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives(3)The case restricting to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation..

Dijkstra's algorithm is an algorithm for finding a graph geodesic, i.e., the shortest path between two graph vertices in a graph. It functions by constructing a shortest-path tree from the initial vertex to every other vertex in the graph. The algorithm is implemented in the Wolfram Language as FineShortestPath[g, Method -> "Dijkstra"].The worst-case running time for the Dijkstra algorithm on a graph with nodes and edges is because it allows for directed cycles. It even finds the shortest paths from a source node to all other nodes in the graph. This is basically for node selection and for distance updates. While is the best possible complexity for dense graphs, the complexity can be improved significantly for sparse graphs.With slight modifications, Dijkstra's algorithm can be used as a reverse algorithm that maintains minimum spanning trees for the sink node. With further modifications, it can be extended to become bidirectional.The..

A curve on the unit sphere is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve's velocity never vanishes. A mapping forms an immersion of the circle into the sphere iff, for all ,Smale (1958) showed it is possible to turn a sphere insideout (sphere eversion) using eversion.The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions eversion.

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).The..

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..

A sequence whose terms are integers. The most complete printed references for such sequences are Sloane (1973) and its update, Sloane and Plouffe (1995). Neil Sloane maintains the sequences from both these works in a vastly expanded on-line encyclopedia known as the On-Line Encyclopedia of Integer Sequences (https://www.research.att.com/~njas/sequences/). In this listing, sequences are identified by a unique 6-digit A-number. Sequences appearing in Sloane and Plouffe (1995) are ordered lexicographically and identified with a 4-digit M-number, and those appearing in Sloane (1973) are identified with a 4-digit N-number. To look up sequences by e-mail, send a message to either mailto:[email protected] or mailto:[email protected] containing lines of the form lookup 5 14 42 132 ... (note that spaces must be used instead of commas).Integer sequences can be analyzed by a variety of techniques (Sloane and Plouffe..

If is an arbitrary integer relatively prime to and is a primitive root of , then there exists among the numbers 0, 1, 2, ..., , where is the totient function, exactly one number such thatThe number is then called the discrete logarithm of with respect to the base modulo and is denotedThe term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p. 501). In number theory, the term "index" is generally used instead (Gauss 1801; Nagell 1951, p. 112).For example, the number 7 is a positive primitive root of (in fact, the set of primitive roots of 41 is given by 6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35), and since , the number 15 has multiplicative order 3 with respect to base 7 (modulo 41) (Nagell 1951, p. 112). The generalized multiplicative order is implemented in the Wolfram Language..

A problem in game theory first discussed by A. Tucker. Suppose each of two prisoners and , who are not allowed to communicate with each other, is offered to be set free if he implicates the other. If neither implicates the other, both will receive the usual sentence. However, if the prisoners implicate each other, then both are presumed guilty and granted harsh sentences.A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision. Each prisoner's best strategy would appear to be to turn the other in (since if makes the worst-case assumption that will turn him in, then will walk free and will be stuck in jail if he remains silent). However, if the prisoners turn each other in, they obtain the worst possible outcome for both.Mosteller (1987) describes a different problem he terms "the prisoner's dilemma." In this problem, three prisoners , , and with apparently equally good records..

There are many mathematical and recreational problems related to folding. Origami,the Japanese art of paper folding, is one well-known example.It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular..

Cluster analysis is a technique used for classification of data in which data elements are partitioned into groups called clusters that represent collections of data elements that are proximate based on a distance or dissimilarity function.Cluster analysis is implemented as FindClusters[data] or FindClusters[data, n].The Season 1 pilot (2005) and Season 2 episode "Dark Matter" of the television crime drama NUMB3RS feature clusters and cluster analysis. In "Dark Matter," math genius Charlie Eppes runs a cluster analysis to find connections between the students that seemed to be systematically singled out by the anomalous third shooter. In Season 4 episode"Black Swan," characters Charles Eppes and Amita Ramanujan adjust cluster radii in their attempt to do a time series analysis of overlapping Voronoi regions to track the movements of a suspect. ..

The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. More concisely, a prime number is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright..

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ."The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.Because the number of..

has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..

Game theory is a branch of mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.Game theory has two distinct branches: combinatorialgame theory and classical game theory.Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.The..

The line in the complex plane on which the Riemann hypothesis asserts that all nontrivial (complex) Riemann zeta function zeros lie. The plot above shows the first few zeros of the Riemann zeta function, with the critical line shown in red. The zeros with and that do not line on the critical line are the trivial zeros of at , , .... Although it is known that an infinite number of zeros lie on the critical line and that these comprise at least 40% of all zeros, the Riemann hypothesis is still unproven.An attractive poster plotting the Riemann zeta function zeros on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes discusses the critical line after realizing that character Ethan's daughter has been kidnapped because he is close to solving..

Actuarial science is the study of risk through the use of mathematics, probability, and statistics. A person who performs risk assessment is known as an actuary. Actuaries typically are employed in financial, insurance, pensions, and other related sectors.Actuarial science is similar to medicine in that a lot of time must be taken for schooling and taking examinations, but salaries are typically rather high.The Season 1 episode "Sacrifice" (2005) of the television crime drama NUMB3RS mentions actuarial science.

Reversion to the mean, also called regression to the mean, is the statistical phenomenon stating that the greater the deviation of a random variate from its mean, the greater the probability that the next measured variate will deviate less far. In other words, an extreme event is likely to be followed by a less extreme event.Although this phenomenon appears to violate the definition of independent events, it simply reflects the fact that the probability density function of any random variable , by definition, is nonnegative over every interval and integrates to one over the interval . Thus, as you move away from the mean, the proportion of the distribution that lies closer to the mean than you do increases continuously. Formally,for .The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions regression to the mean. ..

A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability , much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1998).Benford's law was used by the character Charlie Eppes as an analogy to help solve a series of high burglaries in the Season 2 "The Running Man" episode (2006) of the television crime drama NUMB3RS.Benford's law applies to data that are not dimensionless, so the numerical values of the data depend on the units. If there..

Bayesian analysis is a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a "prior distribution" which may be based on anything, including an assessment of the relative likelihoods of parameters or the results of non-Bayesian observations. In practice, it is common to assume a uniform distribution over the appropriate range of values for the prior distribution.Given the prior distribution, collect data to obtain the observed distribution. Then calculate the likelihood of the observed distribution as a function of parameter values, multiply this likelihood function by the prior distribution, and normalize to obtain a unit probability over all possible values. This is called the posterior distribution. The mode of the distribution is then the parameter estimate, and "probability intervals" (the Bayesian analog of confidence..

Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in the Wolfram Language by L. Zamiatina.To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming..

In the early 1950s, Ernst Straus asked 1. Is every region illuminable from every point in the region? 2. Is every region illuminable from at least one point in the region? Here, illuminable means that there is a path from every point to every other by repeated reflections.In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle. Penrose's room, illustrated above, consists of two half-ellipses at the top and bottom and two mushroom-shaped protuberances (which are in turn built up from straight line segments and smaller half-ellipses) on the left and right sides. The ellipses and mushrooms are strategically placed as shown, with the red points being the foci of the half-ellipses. There are essentially three possible configurations of illumination. In this figure, lit regions are indicated in white, unilluminated..

The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation(1)with . As a result of the definition (1), it is conventional to define .The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .Fibonacci numbers are implemented in the WolframLanguage as Fibonacci[n].The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation).The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43,..

An SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time. The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people , number of people infected , and number of people who have recovered . One of the simplest SIR models is the Kermack-McKendrick model.The Season 1 episode "Vector" (2005) of the television crime drama NUMB3RS features SIR models.

The traveling salesman problem is a problem in graph theory requiring the most efficient (i.e., least total distance) Hamiltonian cycle a salesman can take through each of cities. No general method of solution is known, and the problem is NP-hard.The Wolfram Language command FindShortestTour[g] attempts to find a shortest tour, which is a Hamiltonian cycle (with initial vertex repeated at the end) for a Hamiltonian graph if it returns a list with first element equal to the vertex count of .The traveling salesman problem is mentioned by the character Larry Fleinhardt in the Season 2 episode "Rampage" (2006) of the television crime drama NUMB3RS.

Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Simplistically, linear programming is the optimization of an outcome based on some set of constraints using a linear mathematical model.Linear programming is implemented in the Wolfram Language as LinearProgramming[c, m, b], which finds a vector which minimizes the quantity subject to the constraints and for .Linear programming theory falls within convex optimization theory and is also considered to be an important part of operations research. Linear programming is extensively used in business and economics, but may also be used to solve certain engineering problems.Examples from economics include Leontief's input-output model, the determination of shadow prices, etc., an example of a business application would be maximizing..

Set covering deployment (sometimes written "set-covering deployment" and abbreviated SCDP for "set covering deployment problem") seeks an optimal stationing of troops in a set of regions so that a relatively small number of troop units can control a large geographic region. ReVelle and Rosing (2000) first described this in a study of Emperor Constantine the Great's mobile field army placements to secure the Roman Empire. Set covering deployment can be mathematically formulated as a (0,1)-integer programming problem.To formulate the Roman domination problem, consider the eight provinces of the Constantinian Roman Empire illustrated above. Each region is represented as a white disk, and the red lines indicate region connections. Call a region secured if one or more field armies are stationed in that region, and call a region securable if a field army can be deployed to that area from an adjacent area. In addition,..

Wavelets are a class of a functions used to localize a given function in both space and scaling. A family of wavelets can be constructed from a function , sometimes known as a "mother wavelet," which is confined in a finite interval. "Daughter wavelets" are then formed by translation () and contraction (). Wavelets are especially useful for compressing image data, since a wavelet transform has properties which are in some ways superior to a conventional Fourier transform.An individual wavelet can be defined by(1)Then(2)and Calderón's formula gives(3)A common type of wavelet is defined using Haar functions.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features wavelets.

A Gröbner basis for a system of polynomials is an equivalence system that possesses useful properties, for example, that another polynomial is a combination of those in iff the remainder of with respect to is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination.The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems (Trott 2006, p. 37).Gröbner bases are pervasive in the construction of symbolic algebra algorithms, and Gröbner bases with respect to lexicographic order are very useful for solving equations and for elimination..

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