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wiki_12000_chunk_0 | Ordered Bell number | In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n {\displaystyle n} elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from n = 0 {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12001_chunk_0 | Meertens number | In number theory and mathematical logic, a Meertens number in a given number base b {\displaystyle b} is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12002_chunk_0 | Minimum overlap problem | In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12003_chunk_0 | Moebius function | In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ ( k ) {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)} for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12004_chunk_0 | Local analysis | In number theory one may study a Diophantine equation, for example, modulo p for all primes p, looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the p-adic field. This kind of local analysis provides conditions for solution that are necessary. In cases where l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12005_chunk_0 | Local analysis | It does for quadratic forms, but certainly not in general (for example for elliptic curves). The point of view that one would like to understand what extra conditions are needed has been very influential, for example for cubic forms. Some form of local analysis underlies both the standard applications of the Hardy–Litt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12006_chunk_0 | Agoh–Giuga conjecture | In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if p B p − 1 ≡ − 1 ( mod p ) . {\displaystyle pB_{p-1}\equiv -1{\pmod {p}}.} It is named after Takashi Agoh and Giuseppe Giuga. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12007_chunk_0 | N conjecture | In number theory the n conjecture is a conjecture stated by Browkin & Brzeziński (1994) as a generalization of the abc conjecture to more than three integers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12008_chunk_0 | Almost all | In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12009_chunk_0 | Khinchin's constant | In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant. That is, for x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12010_chunk_0 | Khinchin's constant | a n ) 1 / n = K 0 {\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}} where K 0 {\displaystyle K_{0}} is Khinchin's constant K 0 = ∏ r = 1 ∞ ( 1 + 1 r ( r + 2 ) ) log 2 r ≈ 2.6854520010 … {\displaystyle K_{0}=\prod _{r=1}^{\infty }{\left(1+{1 \over r(r+2)}\right)}^{\log _{2}r}\appr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12011_chunk_0 | Artin constant | In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12012_chunk_0 | Berlekamp–Rabin algorithm | In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over a field Z p {\displaystyle \mathbb {Z} _{p}} . The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorizat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12013_chunk_0 | Infinitude of prime numbers | In number theory, Bertrand's postulate is a theorem stating that for any integer n > 1 {\displaystyle n>1} , there always exists at least one prime number such that n < p < 2 n . {\displaystyle n | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12014_chunk_0 | Bertrand's postulate | In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3 {\displaystyle n>3} , there always exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that n < p < 2 n . {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12015_chunk_0 | Bonse's inequality | In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then p n # = p 1 ⋯ p n > p n + 1 2 . {\displaystyle p_{n}\#=p_{1}\cdots p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12016_chunk_0 | Brocard's conjecture | In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12017_chunk_0 | Brun's constant | In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 (sequence A065421 in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12018_chunk_0 | Büchi's problem | In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12019_chunk_0 | Carmichael's theorem | In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12020_chunk_0 | Chebyshev's bias | In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12021_chunk_0 | Chen's theorem | In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12022_chunk_0 | Cramér conjecture | In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12023_chunk_0 | Dirichlet's theorem on diophantine approximation | In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers α {\displaystyle \alpha } and N {\displaystyle N} , with 1 ≤ N {\displaystyle 1\leq N} , there exist integers p {\displaystyle p} and q {\displaystyle q} such that 1 ≤ q ≤ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12024_chunk_0 | Dirichlet's theorem on diophantine approximation | This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality 0 < | α − p q | < 1 q 2 {\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}} is sa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12025_chunk_0 | Dirichlet's theorem on arithmetic progressions | In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12026_chunk_0 | Dixon's factorization method | In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12027_chunk_0 | Euler's criterion | In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12028_chunk_0 | Euler's theorem | In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function, then a raised to the power φ ( n ) {\displaystyle \varphi (n)} is congruent to 1 modulo n; tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12029_chunk_0 | Euler's theorem | The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo n {\displaystyle n} . For example, consider finding the ones place decimal digit of 7 222 {\displaystyle 7^{222}} , i.e. 7 222 ( mod 10 ) {\displaystyle 7^{222}{\pmod {10}}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12030_chunk_0 | Euler's theorem | The integers 7 and 10 are coprime, and φ ( 10 ) = 4 {\displaystyle \varphi (10)=4} . So Euler's theorem yields 7 4 ≡ 1 ( mod 10 ) {\displaystyle 7^{4}\equiv 1{\pmod {10}}} , and we get 7 222 ≡ 7 4 × 55 + 2 ≡ ( 7 4 ) 55 × 7 2 ≡ 1 55 × 7 2 ≡ 49 ≡ 9 ( mod 10 ) {\displaystyle 7^{222}\equiv 7^{4\times 55+2}\equiv (7^{4})^{5... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12031_chunk_0 | Euler totient | In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ ( n ) {\displaystyle \varphi (n)} or ϕ ( n ) {\displaystyle \phi (n)} , and may also be called Euler's phi function. In other words, it is the num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12032_chunk_0 | Euler totient | Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } ). It is also used for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12033_chunk_0 | Mathematical conjecture | In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} can satisfy the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} for any integer value of n {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12034_chunk_0 | Fermat’s Last Theorem | In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12035_chunk_0 | Fermat’s Last Theorem | Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12036_chunk_0 | Fermat’s Last Theorem | It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12037_chunk_0 | Glaisher's theorem | In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12038_chunk_0 | Goldbach's weak conjecture | In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum. )This conjecture is called "weak" ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12039_chunk_0 | Goldbach's weak conjecture | For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3). In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12040_chunk_0 | Goldbach's weak conjecture | The proof was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since; fully-refereed chapters in close to final form are being made public in the process.Some state the conjecture as Every odd number greater than 7 can be expressed as the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12041_chunk_0 | Grimm's conjecture | In number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12042_chunk_0 | Hilbert's irreducibility theorem | In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12043_chunk_0 | Hurwitz's theorem (number theory) | In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that The condition that ξ is irrational cannot be omitted. Moreover the constant 5 {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12044_chunk_0 | Schur's theorem | In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values { p ( n ) ≠ 0: n ∈ N } {\displaystyle {\begin{Bmatrix}p(n)\neq 0:n\in \mathbb {N} \end{Bmatrix}}} , then the set of primes that divide some member of S is infinite. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12045_chunk_0 | Iwasawa theory | In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generaliza... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12046_chunk_0 | Kaprekar's routine | In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12047_chunk_0 | Lagrange's theorem (number theory) | In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if p is a prime number, x ∈ Z / p Z {\displaystyle x\in \mathbb {Z} /p\mathbb {Z} } , and f ( x ) ∈ Z {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12048_chunk_0 | Lemoine's conjecture | In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12049_chunk_0 | Lochs' theorem | In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs.The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fractio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12050_chunk_0 | Lochs' theorem | A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12051_chunk_0 | Lucas' theorem | In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12052_chunk_0 | Maier's theorem | In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the prime-counting function and λ is greater than 1 then π ( x + ( log x ) λ ) − π ( x ) ( log x ) λ − 1 {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12053_chunk_0 | Mazur's control theorem | In number theory, Mazur's control theorem, introduced by Mazur (1972), describes the behavior in Zp extensions of the Selmer group of an abelian variety over a number field. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12054_chunk_0 | Meyer's theorem | In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation Q(x) = 0has a non-zero real solution, then it has a non-zero rational solution (the converse is obvio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12055_chunk_0 | Meyer's theorem | Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement: A rational quadratic form in five or more variables represents zero over the field Qp of the p-adic numbers for all p.Meyer's theorem is best possible with respect to the number of variables: there ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12056_chunk_0 | Mills' constant | In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is a prime number for all positive natural numbers n. This constant is named after William Harold Mills who proved in 19... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12057_chunk_0 | Moessner's theorem | In number theory, Moessner's theorem or Moessner's magic is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers 1 n , 2 n , 3 n , 4 n , ⋯ , {\displaystyle 1^{n},2^{n},3^{n},4^{n},\cdots ~,} with n ≥ 1 , {\displaystyle n\geq 1~,} by recursively manipulating the sequ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12058_chunk_0 | Niven's constant | In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12059_chunk_0 | Ostrowski's theorem | In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12060_chunk_0 | Poisson summation formula | In number theory, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians . Put q = e i π τ {\displaystyle q=e^{i\pi \tau }} , for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12061_chunk_0 | Proth's theorem | In number theory, Proth's theorem is a primality test for Proth numbers. It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which a p − 1 2 ≡ − 1 ( mod p ) , {\displaystyle a^{\frac {p-1}{2}}\equiv -1{\pmod {p}},} then p is prime. In this case p is cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12062_chunk_0 | Proth's theorem | In practice, however, a quadratic nonresidue of p is found via a modified Euclid's algorithm and taken as the value of a, since if a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. For such an a the Legendre symbol is ( a p ) = − 1. {\displaystyle \left({\frac {a}{p}}\righ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12063_chunk_0 | Proth's theorem | Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly. Note that if a is chosen to be a quadratic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12064_chunk_0 | Ramanujan's sum | In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},} where (a, q) = 1 means that a only takes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12065_chunk_0 | Rosser's theorem | In number theory, Rosser's theorem states that the n {\displaystyle n} th prime number is greater than n log n {\displaystyle n\log n} , where log {\displaystyle \log } is the natural logarithm function. It was published by J. Barkley Rosser in 1939.Its full statement is: Let p n {\displaystyle p_{n}} be the n {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12066_chunk_0 | Selberg's identity | In number theory, Selberg's identity is an approximate identity involving logarithms of primes named after Atle Selberg. The identity, discovered jointly by Selberg and Paul Erdős, was used in the first elementary proof for the prime number theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12067_chunk_0 | Skewes's number | In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x {\displaystyle x} for which π ( x ) > li ( x ) , {\displaystyle \pi (x)>\operatorname {li} (x),} where π is the prime-counting function and li is t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12068_chunk_0 | Sophie Germain's theorem | In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x p + y p = z p {\displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime p {\displaystyle p} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12069_chunk_0 | Størmer's theorem | In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations. It follows from the Thue–Siegel–Roth theorem that there are only a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12070_chunk_0 | Sylvester's sequence | In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).Sylvester's sequence is named after James Joseph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12071_chunk_0 | Szpiro's conjecture | In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12072_chunk_0 | Tate's thesis | In number theory, Tate's thesis is the 1950 PhD thesis of John Tate (1950) completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12073_chunk_0 | Tijdeman's theorem | In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation y m = x n + 1 , {\displaystyle y^{m}=x^{n}+1,} for exponents n and m greater than one, is finite. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12074_chunk_0 | Tunnell's theorem | In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12075_chunk_0 | Vantieghems theorem | In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n≥3 is prime if and only if ∏ 1 ≤ k ≤ n − 1 ( 2 k − 1 ) ≡ n mod ( 2 n − 1 ) . {\displaystyle \prod _{1\leq k\leq n-1}\left(2^{k}-1\right)\equiv n\mod \left(2^{n}-1\right).} Similarly, n is prime, if and only if the following... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12076_chunk_0 | Vieta jumping | In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine equation from k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12077_chunk_0 | Vinogradov's theorem | In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12078_chunk_0 | Vinogradov's theorem | The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 101346 is sufficiently large. Additionally numbers up to 1020 ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12079_chunk_0 | Waring's prime number conjecture | In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring. It states that every odd number exceeding 3 is either a prime number or the sum of three prime numbers. It follows from the generalized Riemann hypothesis, and (trivia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12080_chunk_0 | Hilbert–Waring theorem | In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12081_chunk_0 | Weyl's inequality (number theory) | In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies | c − a / q | ≤ t q − 2 , {\displaystyle |c-a/q|\leq tq^{-2},} for some t greater than or equal to 1, then for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12082_chunk_0 | Znám's problem | In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had conside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12083_chunk_0 | Zolotarev's lemma | In number theory, Zolotarev's lemma states that the Legendre symbol ( a p ) {\displaystyle \left({\frac {a}{p}}\right)} for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: ( a p ) = ε ( π a ) {\displaystyle \left({\frac {a}{p}}\right)=\varepsilon (\pi ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12084_chunk_0 | Zsigmondy's theorem | In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 {\displaystyle a>b>0} are coprime integers, then for any integer n ≥ 1 {\displaystyle n\geq 1} , there is a prime number p (called a primitive prime divisor) that divides a n − b n {\displaystyle a^{n}-b^{n}} and does not divide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12085_chunk_0 | Behrend sequence | In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12086_chunk_0 | Carmichael number | In number theory, a Carmichael number is a composite number n {\displaystyle n} , which in modular arithmetic satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle b^{n}\equiv b{\pmod {n}}} for all integers b {\displaystyle b} . The relation may also be expressed in the form: b n − 1 ≡ 1 ( mod n ) {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12087_chunk_0 | Carmichael number | They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.The Carmichael numbers form the subset K1 of the Knödel numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12088_chunk_0 | Dudeney number | In number theory, a Dudeney number in a given number base b {\displaystyle b} is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his pu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12089_chunk_0 | Durfee square | In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts with values ≥ s. An equivalent, but more visual, definition is that the Durfee square is the largest square that is conta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12090_chunk_0 | Fermi–Dirac prime | In number theory, a Fermi–Dirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to Fermi–Dirac statistics in physics based on the fact that each integer has a unique representation as a product of Fermi–Dirac primes without repetition. Each element of the sequence of Fer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12091_chunk_0 | Frobenius pseudoprime | In number theory, a Frobenius pseudoprime is a pseudoprime, whose definition was inspired by the quadratic Frobenius test described by Jon Grantham in a 1998 preprint and published in 2000. Frobenius pseudoprimes can be defined with respect to polynomials of degree at least 2, but they have been most extensively studie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12092_chunk_0 | Gaussian integers | In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z {\displaystyle \mathbf {Z} } or Z . {\displaystyle \mathbb {Z} .} Gaussian integ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12093_chunk_0 | Algebraic Hecke character | In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12094_chunk_0 | Ramanujan constant | In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field Q {\displaystyle \mathbb {Q} \left} has class number 1. Equivalently, the ring of algebraic integers of Q {\displaystyle \mathbb {Q} \left} has unique factorization.The determin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12095_chunk_0 | Leyland number | In number theory, a Leyland number is a number of the form x y + y x {\displaystyle x^{y}+y^{x}} where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 (sequence A076980 in the OEIS... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12096_chunk_0 | Irrationality measure | In number theory, a Liouville number is a real number x {\displaystyle x} with the property that, for every positive integer n {\displaystyle n} , there exists a pair of integers ( p , q ) {\displaystyle (p,q)} with q > 1 {\displaystyle q>1} such that Liouville numbers are "almost rational", and can thus be approximate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12097_chunk_0 | Parshin chain | In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by Parshin (1978) in order to define an analogue of the idele class group for 2-dimensional schemes. A Parshin chain of dimension s on a scheme is a finite sequence of points p0, p1, ..., ps ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12098_chunk_0 | Pierpont prime | In number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic section... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12099_chunk_0 | Pillai prime | In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, n ! ≡ − 1 mod p {\displaystyle n!\equiv -1\mod p} but p ≢ 1 mod n {\displaystyle p\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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