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wiki_12200_chunk_0 | Extravagant number | In number theory, an extravagant number (also known as a wasteful number) is a natural number in a given number base that has fewer digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12201_chunk_0 | Quadratic residues | In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: x 2 ≡ q ( mod n ) . {\displaystyle x^{2}\equiv q{\pmod {n}}.} Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12202_chunk_0 | Knödel number | In number theory, an n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies i m − n ≡ 1 ( mod m ) {\displaystyle i^{m-n}\equiv 1{\pmod {m}}} . The concept is named after Walter Knödel.The set of all n-Knödel numbers is denoted Kn. The special case... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12203_chunk_0 | Smooth integer | In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term see... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12204_chunk_0 | Octahedral number | In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The n {\displaystyle n} th octahedral number O n {\displaystyle O_{n}} can be obtained by the formula: O n = n ( 2 n 2 + 1 ) 3 . {\displaystyle O_{n}={n(2n^{2}+1) \over 3}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12205_chunk_0 | Euler–Jacobi pseudoprime | In number theory, an odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ( a n ) ( mod n ) {\displaystyle a^{(n-1)/2}\equiv \left({\frac {a}{n}}\right){\pmod {n}}} where ( a n ) {\displaystyle \left({\frac {a}{n}}\r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12206_chunk_0 | Odious number | In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers. In computer science, an odious number is said to have odd parity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12207_chunk_0 | Unusual number | In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than n {\displaystyle {\sqrt {n}}} . A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non- n {\displaystyle {\sqrt {n}}} -smooth. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12208_chunk_0 | Special functions | In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of the monstrous moonshine theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12209_chunk_0 | Cousin prime | In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences OEIS: A023200 and OEIS: A046132 in OEIS) below 1000 are: (3, 7), (7, 11), (13, 17), (1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12210_chunk_0 | Cuspidal representation | In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L 2 {\displaystyle L^{2}} spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12211_chunk_0 | Friendly number | In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple. Being mutually friendly is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12212_chunk_0 | Friendly number | The "abundancy" index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that σ(m) / m = σ(n) / n. "Abundancy" is not the same as abundance, which is defined as σ(n) − 2n. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12213_chunk_0 | Friendly number | "Abundancy" may also be expressed as σ − 1 ( n ) {\displaystyle \sigma _{-1}(n)} where σ k {\displaystyle \sigma _{k}} denotes a divisor function with σ k ( n ) {\displaystyle \sigma _{k}(n)} equal to the sum of the k-th powers of the divisors of n. The numbers 1 through 5 are all solitary. The smallest "friendly numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12214_chunk_0 | Friendly number | Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers". In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12215_chunk_0 | Completely multiplicative | In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12216_chunk_0 | Multiplicative order | In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that a k ≡ 1 ( mod n ) {\textstyle a^{k}\ \equiv \ 1{\pmod {n}}} .In other words, the multiplicative order of a modulo n is the order of a in the multiplicative grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12217_chunk_0 | L-adic integers | In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12218_chunk_0 | Benjamin Peirce | In number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras. He first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12219_chunk_0 | Benjamin Peirce | Peirce's definition of mathematics was credited by his son, Charles Sanders Peirce, as helping to initiate the consequence-oriented philosophy of pragmatism. Like George Boole, Peirce believed that mathematics could be used to study logic. These ideas were further developed by his son Charles , who noted that logic als... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12220_chunk_0 | Factoring integers | In number theory, integer factorization is the decomposition, of a positive integer into a product of integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime (in this case, one has a "product" of a single fa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12221_chunk_0 | Factoring integers | The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12222_chunk_0 | Factoring integers | The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12223_chunk_0 | Triviality (mathematics) | In number theory, it is often important to find factors of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial". The homogeneous matrix equation A x = 0 {\displaystyle A\mathbf {x} =\mathbf {0} } , where A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12224_chunk_0 | Triviality (mathematics) | All other groups, which are more complicated, are called "nontrivial". In graph theory, the trivial graph is a graph which has only 1 vertex and no edge. Database theory has a concept called functional dependency, written X → Y {\displaystyle X\to Y} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12225_chunk_0 | Triviality (mathematics) | The dependence X → Y {\displaystyle X\to Y} is true if Y is a subset of X, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial". It can be shown that Riemann's zeta function has zeros at the negative even numbers −2, −4, … Though the proof is comparative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12226_chunk_0 | Integer-valued function | In number theory, many arithmetic functions are integer-valued. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12227_chunk_0 | Lower numbering | In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12228_chunk_0 | Krasner's lemma | In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12229_chunk_0 | Lower asymptotic density | In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval as n grows large. Intuiti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12230_chunk_0 | Lower asymptotic density | However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12231_chunk_0 | Octic reciprocity | In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity. There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol ( x p ) k {\displaystyle \left({\f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12232_chunk_0 | Primes in arithmetic progression | In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} . According... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12233_chunk_0 | Primes in arithmetic progression | For example, it can be used about primes in an arithmetic progression of the form a n + b {\displaystyle an+b} , where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer k ≥ 3, an AP-k (also called PAP... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12234_chunk_0 | Quadratic Gauss sum | In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These obje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12235_chunk_0 | Quadratic integers | In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0with b and c (usual) integers. When algebraic integers are considered, the usual integers are often cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12236_chunk_0 | Quadratic integers | Another common example is the non-real cubic root of unity −1 + √−3/2, which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12237_chunk_0 | Reverse divisible number | In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS).For instance, 1089 × ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12238_chunk_0 | Sexy prime | In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six: sex. If p + 2 or p + 4 (where p is the lower prime) is also prime, then th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12239_chunk_0 | Markov constant | In number theory, specifically in Diophantine approximation theory, the Markov constant M ( α ) {\displaystyle M(\alpha )} of an irrational number α {\displaystyle \alpha } is the factor for which Dirichlet's approximation theorem can be improved for α {\displaystyle \alpha } . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12240_chunk_0 | Lonely runner conjecture | In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that n {\displaystyle n} runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12241_chunk_0 | Ankeny–Artin–Chowla congruence | In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is ε = t + u d 2 {\displaystyle \varepsilon ={\frac {t+u{\sqrt {d}}}{2}}} wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12242_chunk_0 | Stark–Heegner theorem | In number theory, the Baker–Heegner–Stark theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are unique factorization domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed clas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12243_chunk_0 | Stark–Heegner theorem | The Baker–Heegner–Stark theorem can then be stated as follows: If d < 0, then the class number of Q(√d) is one if and only if d ∈ { − 1 , − 2 , − 3 , − 7 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.} These are known as the Heegner numbers. By replacing d with the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12244_chunk_0 | Bateman-Horn conjecture | In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conj... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12245_chunk_0 | Calkin–Wilf tree | In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the fraction a/b has as its two children the numbers a/a + b and a + b/b. Every positive rational numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12246_chunk_0 | Calkin–Wilf tree | It is named after Neil Calkin and Herbert Wilf, but appears in other works including Kepler's Harmonices Mundi. The sequence of rational numbers in a breadth-first traversal of the Calkin–Wilf tree is known as the Calkin–Wilf sequence. Its sequence of numerators (or, offset by one, denominators) is Stern's diatomic ser... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12247_chunk_0 | Ax–Katz theorem | In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12248_chunk_0 | Chinese hypothesis | In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n—in other words, that an integer n is prime if and only if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} . It is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12249_chunk_0 | Davenport–Erdős theorem | In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.Let A = a 1 , a 2 , … {\displaystyle A=a_{1},a_{2},\dots } be a sequence of positive integers. Then the multiples of A {\displaystyle A} are another set M ( A ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12250_chunk_0 | Davenport–Erdős theorem | The sequential density, defined as the limit (as i {\displaystyle i} goes to infinity) of the densities of the sets M ( { a 1 , … a i } ) {\displaystyle M(\{a_{1},\dots a_{i}\})} of multiples of the first i {\displaystyle i} elements of A {\displaystyle A} . As these sets can be decomposed into finitely many disjoint a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12251_chunk_0 | Dedekind psi function | In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n ∏ p | n ( 1 + 1 p ) , {\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right),} where the product is taken over all primes p {\displaystyle p} dividing n . {\displaystyle n.} (By convent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12252_chunk_0 | Dedekind psi function | The value of ψ ( n ) {\displaystyle \psi (n)} for the first few integers n {\displaystyle n} is: 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the OEIS).The function ψ ( n ) {\displaystyle \psi (n)} is greater than n {\displaystyle n} for all n {\displaystyle n} greater than 1, and is even for all ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12253_chunk_0 | Dedekind psi function | This also leads to a proof of the generating function in terms of the Riemann zeta function, which is ∑ ψ ( n ) n s = ζ ( s ) ζ ( s − 1 ) ζ ( 2 s ) . {\displaystyle \sum {\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}.} This is also a consequence of the fact that we can write as a Dirichlet convolu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12254_chunk_0 | Dedekind psi function | There is an additive definition of the psi function as well. Quoting from Dickson, R. Dedekind proved that, if n {\displaystyle n} is decomposed in every way into a product a b {\displaystyle ab} and if e {\displaystyle e} is the g.c.d. of a , b {\displaystyle a,b} then ∑ a ( a / e ) φ ( e ) = n ∏ p | n ( 1 + 1 p ) {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12255_chunk_0 | Dirichlet hyperbola method | In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum ∑ n ≤ x f ( n ) {\displaystyle \sum _{n\leq x}f(n)} where f , g , h {\displaystyle f,g,h} are multiplicative functions with f = g ∗ h {\displaystyle f=g*h} , where ∗ {\displaystyle *} is the Dirichlet convolution. It uses the fact that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12256_chunk_0 | Eichler–Shimura congruence relation | In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was introduced by Eichler (1954) and generalized by Shimura (1958). Roughly speaking, it says that the correspondence on the modular curve inducing ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12257_chunk_0 | Elkies trinomial curves | In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particular Galois groups. One curve, C168, gives Galois group PSL(2,7) from a polynomial of deg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12258_chunk_0 | Elkies trinomial curves | The curve has genus two, and so by Faltings theorem there are only a finite number of rational points on it. These rational points were proven by Nils Bruin using the computer program Kash to be the only ones on C168, and they give only four distinct trinomial polynomials with Galois group PSL(2,7): x7-7x+3 (the Trinks... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12259_chunk_0 | Elliott–Halberstam conjecture | In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968.Stating the conjecture requires some notation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12260_chunk_0 | Elliott–Halberstam conjecture | Let π ( x ) {\displaystyle \pi (x)} , the prime-counting function, denote the number of primes less than or equal to x {\displaystyle x} . If q {\displaystyle q} is a positive integer and a {\displaystyle a} is coprime to q {\displaystyle q} , we let π ( x ; q , a ) {\displaystyle \pi (x;q,a)} denote the number of prim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12261_chunk_0 | Elliott–Halberstam conjecture | If we then define the error function E ( x ; q ) = max gcd ( a , q ) = 1 | π ( x ; q , a ) − π ( x ) φ ( q ) | {\displaystyle E(x;q)=\max _{{\text{gcd}}(a,q)=1}\left|\pi (x;q,a)-{\frac {\pi (x)}{\varphi (q)}}\right|} where the max is taken over all a {\displaystyle a} coprime to q {\displaystyle q} , then the Elliott–H... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12262_chunk_0 | Erdős arcsine law | In number theory, the Erdős arcsine law, named after Paul Erdős in 1969, states that the prime divisors of a number have a distribution related to the arcsine distribution. Specifically, say that the jth prime factor p of a given number n (in the sorted sequence of distinct prime factors) is "small" when log log p < j.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12263_chunk_0 | Erdős–Kac theorem | In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of ω ( n ) − log log n log log n {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12264_chunk_0 | Erdős–Moser equation | In number theory, the Erdős–Moser equation is 1 k + 2 k + ⋯ + m k = ( m + 1 ) k , {\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k},} where m {\displaystyle m} and k {\displaystyle k} are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12265_chunk_0 | Fermat pseudoprime | In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12266_chunk_0 | Fermat quotient | In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},} or δ p ( a ) = a − a p p {\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}} .This article is about the former; for the latter see p-derivation.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12267_chunk_0 | Fermat–Catalan conjecture | In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12268_chunk_0 | Gaussian moat | In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. More colorfully, if one imagines the Gaussian primes to be stepping stones in a sea of complex numbers,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12269_chunk_0 | Green–Tao theorem | In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemeréd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12270_chunk_0 | Hasse norm theorem | In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with N L / K ( l ) = k {\displaystyle \math... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12271_chunk_0 | Hasse norm theorem | Serre and Tate showed that another counterexample is given by the field Q ( 13 , 17 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {13}},{\sqrt {17}})/{\mathbf {Q} }} where every rational square is a local norm everywhere but 5 2 {\displaystyle 5^{2}} is not a global norm. This is an example of a theorem stating a local-gl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12272_chunk_0 | Hasse norm theorem | The full theorem is due to Hasse (1931). The special case when the degree n of the extension is 2 was proved by Hilbert (1897), and the special case when n is prime was proved by Furtwangler in 1902.The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K) is trivial ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12273_chunk_0 | Katz–Lang finiteness theorem | In number theory, the Katz–Lang finiteness theorem, proved by Nick Katz and Serge Lang (1981), states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12274_chunk_0 | Kempner function | In number theory, the Kempner function S ( n ) {\displaystyle S(n)} is defined for a given positive integer n {\displaystyle n} to be the smallest number s {\displaystyle s} such that n {\displaystyle n} divides the factorial s ! {\displaystyle s!} . For example, the number 8 {\displaystyle 8} does not divide 1 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12275_chunk_0 | Kempner function | {\displaystyle 1!} , 2 ! {\displaystyle 2!} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12276_chunk_0 | Kempner function | , or 3 ! {\displaystyle 3!} , but does divide 4 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12277_chunk_0 | Kempner function | {\displaystyle 4!} , so S ( 8 ) = 4 {\displaystyle S(8)=4} . This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12278_chunk_0 | Arithmetic derivative | In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. There are many versions of "arithmetic derivatives", including the one disc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12279_chunk_0 | Mertens function | In number theory, the Mertens function is defined for all positive integers n as M ( n ) = ∑ k = 1 n μ ( k ) , {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k),} where μ ( k ) {\displaystyle \mu (k)} is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12280_chunk_0 | Mertens function | {\displaystyle M(x)=M(\lfloor x\rfloor ).} Less formally, M ( x ) {\displaystyle M(x)} is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. The first 143 M(n) values are (sequence A002321 in the OEIS) The Mertens function slowly grows ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12281_chunk_0 | Mertens function | This implies, for θ = 0 {\displaystyle \theta =0} that M ( x ) = O ( x log h x ) . {\displaystyle M(x)=O\left({\frac {x}{\log ^{h}x}}\right)\ .} The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12282_chunk_0 | Mertens function | However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely M(x) = O(x1/2 + ε). Since high values for M(x) grow at least as fast as x {\displaystyle {\sqrt {x}}} , this puts a rather tight bound on its rate of growth. Here, O refers to big O notation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12283_chunk_0 | Mertens function | The true rate of growth of M(x) is not known. An unpublished conjecture of Steve Gonek states that 0 < lim sup x → ∞ | M ( x ) | x ( log log log x ) 5 / 4 < ∞ . {\displaystyle 0<\limsup _{x\to \infty }{\frac {|M(x)|}{{\sqrt {x}}(\log \log \log x)^{5/4}}}<\infty .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12284_chunk_0 | Mertens function | Probabilistic evidence towards this conjecture is given by Nathan Ng. In particular, Ng gives a conditional proof that the function e − y / 2 M ( e y ) {\displaystyle e^{-y/2}M(e^{y})} has a limiting distribution ν {\displaystyle \nu } on R {\displaystyle \mathbb {R} } . That is, for all bounded Lipschitz continuous fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12285_chunk_0 | Moser–de Bruijn sequence | In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4. Equivalently, they are the numbers whose binary representations are nonzero only in even positions. The Moser–de Bruijn numbers in this sequence grow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12286_chunk_0 | Néron–Tate height | In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12287_chunk_0 | Padovan sequence | In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle P(0)=P(1)=P(2)=1,} and the recurrence relation P ( n ) = P ( n − 2 ) + P ( n − 3 ) . {\displaystyle P(n)=P(n-2)+P(n-3).} The first few values of P(n) are 1, 1, 1, 2, 2, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12288_chunk_0 | Poussin proof | In number theory, the Poussin proof is the proof of an identity related to the fractional part of a ratio. In 1838, Peter Gustav Lejeune Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to n: ∑ k = 1 n d ( k ) n ≈ ln n + 2 γ − 1 , {\displaystyle {\frac {\sum _{k=1}^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12289_chunk_0 | Pólya conjecture | In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12290_chunk_0 | Selberg sieve | In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12291_chunk_0 | Stark conjectures | In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjecture... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12292_chunk_0 | Stern–Brocot tree | In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree. The Stern–Brocot tree was introduced independently by Moritz Stern (1858) and Achille Brocot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12293_chunk_0 | Teichmüller character | In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where q = p {\displaystyle q=p} if p {\displaystyle p} is odd and q = 4 {\displaystyle q=4} if p = 2 {\displaystyle p=2} , taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identify... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12294_chunk_0 | Turán sieve | In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12295_chunk_0 | Restricted divisor function | In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is, s ( n ) = ∑ d | n , d ≠ n d . {\displaystyle s(n)=\sum \nolimits _{d|n,\ d\neq n}d.} It can be used to characterize the prime numbers, perfect numbers, sociable... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12296_chunk_0 | Class number formula | In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12297_chunk_0 | Classical modular curve | In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0,such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant. The curve is sometimes called X0(n), though often that notation is used for the abstract algebraic curve fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12298_chunk_0 | Classical modular curve | A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x). It is important to note that the classical modular curves are part of the larger theory of modular curves. In particular it has another expression as a compactified quotient of the complex upper half-plane H. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12299_chunk_0 | Continued fraction factorization | In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer n, not depending on special form or properties. It was described by D. H. Lehmer and R. E. Powers in 1931, and developed a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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