Wieferich pair In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof[1] of Mihăilescu's theorem (formerly known as Catalan's conjecture).[2] Known Wieferich pairs[edit] There are only 7 Wieferich pairs known:[3][4] (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). Wieferich triple[edit] A Wieferich triple is a triple of prime numbers p, q and r that satisfy pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2). There are 17 known Wieferich triples: Barker sequence[edit] Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12).[5] Wieferich sequence[edit] 3, 11, 71, 47, ?
Independent set (graph theory) Unrelated vertices in graphs of vertices such that for every two vertices in A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph . and is usually denoted by Every maximum independent set also is maximal, but the converse implication does not necessarily hold. Properties[edit] Relationship to other graph parameters[edit] and the size of a minimum vertex cover is equal to the number of vertices in the graph. A vertex coloring of a graph corresponds to a partition of its vertex set into independent subsets. , is at least the quotient of the number of vertices in and the independent number Maximal independent set[edit] Finding independent sets[edit] In computer science, several computational problems related to independent sets have been studied. Maximum independent sets and maximum cliques[edit] Exact algorithms[edit] Approximation algorithms[edit] See also[edit]
Wike's law of low odd primes From Wikipedia, the free encyclopedia Wike's law of low odd primes is a methodological principle to help design sound experiments in psychology. It is: "If the number of experimental treatments is a low odd prime number, then the experimental design is unbalanced and partially confounded" (Wike, 1973, pp. 192–193). This law was stated by Edwin Wike in a humorous article in which he also admits that the association of his name with the law is an example of Stigler's law of eponymy.
Ant on a rubber rope Mathematics problem The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary,[1][2] but a typical form is as follows: An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). An ant (red dot) crawling on a stretchable rope at a constant speed of 1 cm/s. A formal statement of the problem [edit] For sake of analysis, the following is a formalized version of the puzzle. Consider an ideal elastic rope on the -axis such that at time its endpoints are at (the starting point) and (the target point) for constants and . the target point is at the position and that as varies the target point moves at constant velocity . it begins at the starting point, moving along the rope at constant velocity is 1 km, is 1 km/s, and is 1 cm/s.
SuperPrime From Wikipedia, the free encyclopedia Background information[edit] In August 1995, the calculation of Pi up to 4,294,960,000 decimal digits was achieved by using a supercomputer at the University of Tokyo. The program used to achieve this was ported to personal computers, for operating systems such as Windows NT and Windows 95 and called Super-PI. Landmarks[edit] On September 29, 2006, a milestone was broken when bachus_anonym of www.xtremesystems.org broke the 30 seconds barrier using a highly overclocked Core 2 Duo machine [1] See also[edit] Erodov.com, the 'home forum' for the SuperPrime benchmark. References[edit] External links[edit]
linear algebra - What is the difference between a point and a vector? Sphenic number From Wikipedia, the free encyclopedia Positive integer that is the product of three distinct prime numbers Definition[edit] A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. Examples[edit] The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. As of 2020[ref] the largest known sphenic number is It is the product of the three largest known primes. Divisors[edit] All sphenic numbers have exactly eight divisors. , where p, q, and r are distinct primes, then the set of divisors of n will be: The converse does not hold. Properties[edit] All sphenic numbers are by definition squarefree, because the prime factors must be distinct. The Möbius function of any sphenic number is −1. The cyclotomic polynomials , taken over all sphenic numbers n, may contain arbitrarily large coefficients[1] (for n a product of two primes the coefficients are or 0). Any multiple of a sphenic number (except by 1) isn't a sphenic number. See also[edit]
1 + 2 + 3 + 4 + ⋯ The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is −1/12. The sum of all natural numbers 1 + 2 + 3 + 4 + · · · is a divergent series. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory. In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".[2] Partial sums[edit] The first six triangular numbers The partial sums of the series 1 + 2 + 3 + 4 + 5 + ⋯ are 1, 3, 6, 10, 15, etc. This equation was known to the Pythagoreans as early as the sixth century B.C.E.[3] Numbers of this form are called triangular numbers, because they can be arranged in a triangle. Summability[edit] Heuristics[edit] Dividing both sides by −3, one gets c = −1/12. . . .
Smarandache–Wellin number From Wikipedia, the free encyclopedia In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin. The first decimal Smarandache–Wellin numbers are: Smarandache–Wellin prime A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The primes at the end of the concatenation in the Smarandache–Wellin primes are 2, 3, 7, 719, 1033, 2297, 3037, 11927, ... The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are: 1, 2, 4, 128, 174, 342, 435, 1429, ... The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. See also References External links
Sierpiński number is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. If the form is instead , then k is a Riesel number. Known Sierpiński numbers[edit] The sequence of currently known Sierpiński numbers begins with: 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ... The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form 78557⋅2n + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. However, in 1995 A. Sierpiński problem[edit] Unsolved problem in mathematics: Is 78,557 the smallest Sierpiński number? The Sierpiński problem asks for the value of the smallest Sierpiński number. k = 21181, 22699, 24737, 55459, and 67607. having been eliminated.[6] Prime Sierpiński problem[edit] . .
Semiprime Product of two prime numbers Examples and variations[edit] The semiprimes less than 100 are: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95 (sequence A001358 in the OEIS) Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, ... The semiprimes are the case of the -almost primes, numbers with exactly prime factors. 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, ... Formula for number of semiprimes[edit] A semiprime counting formula was discovered by E. denote the number of semiprimes less than or equal to n. where is the prime-counting function and denotes the kth prime.[3] Properties[edit] For a squarefree semiprime (with ) the value of Euler's totient function Applications[edit]
Selberg's identity From Wikipedia, the free encyclopedia Approximate identity involving logarithms of primes In number theory, Selberg's identity is an approximate identity involving logarithms of primes named after Atle Selberg. Statement[edit] There are several different but equivalent forms of Selberg's identity. where the sums are over primes p and q. Explanation[edit] The strange-looking expression on the left side of Selberg's identity is (up to smaller terms) the sum where the numbers are the coefficients of the Dirichlet series Another variation of the identity[edit] Selberg's identity sometimes also refers to the following divisor sum identity involving the von Mangoldt function and the Möbius function when This variant of Selberg's identity is proved using the concept of taking derivatives of arithmetic functions defined by in Section 2.18 of Apostol's book (see also this link). References[edit]
Ruth–Aaron pair From Wikipedia, the free encyclopedia Two consecutive integers for which the sums of the prime factors of each are equal In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: and There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization. The name was given by Carl Pomerance for Babe Ruth and Hank Aaron, as Ruth's career regular-season home run total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Examples[edit] If only distinct prime factors are counted, the first few Ruth–Aaron pairs are: (The lesser of each pair is listed in OEIS: A006145). Counting repeated prime factors (e.g., 8 = 2×2×2 and 9 = 3×3 with 2+2+2 = 3+3), the first few Ruth–Aaron pairs are: (The lesser of each pair is listed in OEIS: A039752). The intersection of the two lists begins: Density[edit]