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wiki_12100_chunk_0 | Poincaré series (modular form) | In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they generalize classical Eisenstein series. They are named after Henri Poincaré. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12101_chunk_0 | Poincaré series (modular form) | If Γ is a finite group acting on a domain D and H(z) is any meromorphic function on D, then one obtains an automorphic function by averaging over Γ: ∑ γ ∈ Γ H ( γ ( z ) ) . {\displaystyle \sum _{\gamma \in \Gamma }H(\gamma (z)).} However, if Γ is a discrete group, then additional factors must be introduced in order to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12102_chunk_0 | Shimura variety | In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12103_chunk_0 | Sidon sequence | In number theory, a Sidon sequence is a sequence A = { a 0 , a 1 , a 2 , … } {\displaystyle A=\{a_{0},a_{1},a_{2},\dots \}} of natural numbers in which all pairwise sums a i + a j {\displaystyle a_{i}+a_{j}} (for i ≤ j {\displaystyle i\leq j} ) are different. Sidon sequences are also called Sidon sets; they are named a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12104_chunk_0 | Sierpinski problem | In number theory, a Sierpiński number is an odd natural number k such that k × 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all memb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12105_chunk_0 | Smith number | In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12106_chunk_0 | 321 prime | In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 ⋅ 2 n − 1 {\displaystyle 3\cdot 2^{n}-1} for a non-negative integer n. The first few Thabit numbers are: 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 157286... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12107_chunk_0 | Wagstaff prime | In number theory, a Wagstaff prime is a prime number of the form 2 p + 1 3 {\displaystyle {{2^{p}+1} \over 3}} where p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr. ; the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12108_chunk_0 | Wall–Sun–Sun prime | In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12109_chunk_0 | Generalized Wieferich prime | In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12110_chunk_0 | Wilson prime | In number theory, a Wilson prime is a prime number p {\displaystyle p} such that p 2 {\displaystyle p^{2}} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle p} divides ( p − ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12111_chunk_0 | Wilson prime | + 1 {\displaystyle (p-1)!+1} . Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12112_chunk_0 | Wolstenholme prime | In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12113_chunk_0 | Wolstenholme prime | Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. The only two known Wolstenholme ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12114_chunk_0 | Woodall number | In number theory, a Woodall number (Wn) is any natural number of the form W n = n ⋅ 2 n − 1 {\displaystyle W_{n}=n\cdot 2^{n}-1} for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12115_chunk_0 | Balanced prime | In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number p n {\displaystyle p_{n}} , where n is its index in the ordered set of prime numbers, p ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12116_chunk_0 | Bi-twin chain | In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers n − 1 , n + 1 , 2 n − 1 , 2 n + 1 , … , 2 k n − 1 , 2 k n + 1 {\displaystyle n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,} in which every number is prime.The numbers n − 1 , 2 n − 1 , … , 2 k n − 1 {\displaystyle n-1,2n-1,\dots ,2^{k}n-1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12117_chunk_0 | Dickson's conjecture | In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by Dickson (1904) that for a finite set of linear forms a1 + b1n, a2 + b2n, ..., ak + bkn with bi ≥ 1, there are infinitely many positive integers n for which they are all prime, unless there is a congruence condition preventing th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12118_chunk_0 | Ramanujan's ternary quadratic form | In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z. Srinivasa Ramanujan considered this expression in a footnote in a paper published in 1916 and briefly discussed the representability of integers in this form. Aft... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12119_chunk_0 | Hilbert number | In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... (sequence A016813 in the OEIS)) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12120_chunk_0 | Mirimanoff's congruence | In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12121_chunk_0 | Cusp form | In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12122_chunk_0 | Highly cototient number | In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k} than any other integer below k {\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12123_chunk_0 | Reduced totient | In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12124_chunk_0 | Special number field sieve | In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form re ± s, where r and s are small (for instance Mersenne nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12125_chunk_0 | Cluster prime | In number theory, a cluster prime is a prime number p such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding p (OEIS: A038134). For example, the number 23 is a cluster prime because 23 − 3 = 20, and every even integer from 2 to 20, inclusive, is the diff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12126_chunk_0 | Compatible system of ℓ-adic representations | In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility properties for almost all ℓ. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12127_chunk_0 | Congruence of squares | In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12128_chunk_0 | Congruent number | In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.The sequence of (integer) congruent numbers starts with 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12129_chunk_0 | Congruum | In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , and z 2 {\displaystyle z^{2}} (for integers x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12130_chunk_0 | Congruum | Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle. Congrua are also closely connected with congruent numbers: every congruu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12131_chunk_0 | P-adic cyclotomic character | In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ: G → AutR(R(1)) ≈ GL(1, R)). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12132_chunk_0 | Cyclotomic fields | In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12133_chunk_0 | Deficient number | In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12134_chunk_0 | Diophantine quintuple | In number theory, a diophantine m-tuple is a set of m positive integers { a 1 , a 2 , a 3 , a 4 , … , a m } {\displaystyle \{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\}} such that a i a j + 1 {\displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m . {\displaystyle 1\leq i | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12135_chunk_0 | Factorion | In number theory, a factorion in a given number base b {\displaystyle b} is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12136_chunk_0 | Formula for primes | In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12137_chunk_0 | Frugal number | In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers (sequence A046759 in the OEIS). Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12138_chunk_0 | Full reptend prime | In number theory, a full reptend prime, full repetend prime, proper prime: 166 or long prime in base b is an odd prime number p such that the Fermat quotient q p ( b ) = b p − 1 − 1 p {\displaystyle q_{p}(b)={\frac {b^{p-1}-1}{p}}} (where p does not divide b) gives a cyclic number. Therefore, the base b expansion of 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12139_chunk_0 | Genus character | In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the abs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12140_chunk_0 | Happy number | In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1 2 + 3 2 = 10 {\displaystyle 1^{2}+3^{2}=10} , and 1 2 + 0 2 = 1 {\displaystyle 1^{2}+0^{2}=1} . On the other hand, 4 is not a happy number because th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12141_chunk_0 | Happy number | More generally, a b {\displaystyle b} -happy number is a natural number in a given number base b {\displaystyle b} that eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} .The origin of happy numbers is not clear. Happy numbers were brought to the attention of R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12142_chunk_0 | Hemiperfect number | In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the divisor function, the sum of all positive divisors of n. The first few hemiperfect numbers are: 2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12143_chunk_0 | Juggler sequence | In number theory, a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12144_chunk_0 | Root of unity modulo n | In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) x k ≡ 1 ( mod n ) {\displaystyle x^{k}\equiv 1{\pmod {n}}} . If k is the smallest such exponent for x, then x is called a primitive kt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12145_chunk_0 | Right-truncatable prime | In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12146_chunk_0 | Right-truncatable prime | 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime. A left-and-right-truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12147_chunk_0 | Lucky number | In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12148_chunk_0 | Lucky number | Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if Ln... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12149_chunk_0 | Multiplicative functions | In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12150_chunk_0 | Multiplicative partition | In number theory, a multiplicative partition or unordered factorization of an integer n {\displaystyle n} is a way of writing n {\displaystyle n} as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n {\displaystyle n} is itself cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12151_chunk_0 | Narcissistic number | In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base b {\displaystyle b} is a number that is the sum of its own digits each raised to the power of the number of digits. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12152_chunk_0 | Almost prime | In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents): Ω ( n ) := ∑ a i if n = ∏ p i a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12153_chunk_0 | Almost prime | , {\displaystyle \pi _{k}(n)\sim \left({\frac {n}{\log n}}\right){\frac {(\log \log n)^{k-1}}{(k-1)! }},} a result of Landau. See also the Hardy–Ramanujan theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12154_chunk_0 | Norm group | In number theory, a norm group is a group of the form N L / K ( L × ) {\displaystyle N_{L/K}(L^{\times })} where L / K {\displaystyle L/K} is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in K × {\displaystyle K^{\times }} are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12155_chunk_0 | Normal order of an arithmetic function | In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let f be a function on the natural numbers. We say that g is a normal order of f if for every ε > 0, the inequalities ( 1 − ε ) g ( n ) ≤ f ( n ) ≤ ( 1 +... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12156_chunk_0 | Totally real number field | In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers. Equivalent conditions are that F is generated over Q by one root of an integer polynomial P, all of the roots of P being real; or that the tensor product algebra of F wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12157_chunk_0 | Totally real number field | The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two. Any number field that is Galois over the rationals must be either totally real or totally imaginary. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12158_chunk_0 | Perfect digit-to-digit invariant | In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle b} that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because 3435 = 3 3 + 4 4 + 3 3 + 5 5 {\displaystyle 3435... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12159_chunk_0 | Perfect digital invariant | In number theory, a perfect digital invariant (PDI) is a number in a given number base ( b {\displaystyle b} ) that is the sum of its own digits each raised to a given power ( p {\displaystyle p} ). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12160_chunk_0 | Odd perfect number | In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12161_chunk_0 | Odd perfect number | Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12162_chunk_0 | Perfect totient number | In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12163_chunk_0 | Pernicious number | In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12164_chunk_0 | Polite number | In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two. P... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12165_chunk_0 | Erdős–Woods number | In number theory, a positive integer k is said to be an Erdős–Woods number if it has the following property: there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, k is an E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12166_chunk_0 | Practical number | In number theory, a practical number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented as sums of distinct divisors of n {\displaystyle n} . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12167_chunk_0 | Prime constellation | In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, …), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, …) are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12168_chunk_0 | Safe and Sophie Germain primes | In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician So... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12169_chunk_0 | Safe and Sophie Germain primes | One attempt by Germain to prove Fermat’s Last Theorem was to let p be a prime number of the form 8k + 7 and to let n = p – 1. In this case, x n + y n = z n {\displaystyle x^{n}+y^{n}=z^{n}} is unsolvable. Germain’s proof, however, remained unfinished. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12170_chunk_0 | Safe and Sophie Germain primes | Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if p is an odd prime and 2p + 1 is also prime, then p must divide x, y, or z. Otherwise, x n + y n ≠ z n {\textstyle x^{n}+y^{n}\neq z^{n}} . This case where p does not divide x, y, or z is c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12171_chunk_0 | Prime sextuplet | In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form { p , p + 2 , p + 6 , p + 8 } . {\displaystyle \{p,\ p+2,\ p+6,\ p+8\}.} This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12172_chunk_0 | Prime triplet | In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12173_chunk_0 | Probable prime | In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called ps... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12174_chunk_0 | Probable prime | Calculate an − 1 modulo n. If the result is not 1, then n is composite. If the result is 1, then n is likely to be prime; n is then called a probable prime to base a. A weak probable prime to base a is an integer that is a probable prime to base a, but which is not a strong probable prime to base a (see below). For a f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12175_chunk_0 | Provable prime | In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography. Contrast with probable prime, which is likely (but not certain) to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12176_chunk_0 | Elliptic pseudoprime | In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in Q ( − d ) {\displaystyle \mathbb {Q} {\big (}{\sqrt {-d}}{\big )}} , having equation y2 = x3 + ax + b with a, b integers, P bei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12177_chunk_0 | Rational reciprocity law | In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (x|p)k to be +1 if x is a k-th power modulo the prime p an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12178_chunk_0 | Regular prime | In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: 3, 5, 7, 11, 13, 17, 19, 23, 29,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12179_chunk_0 | Self prime | In number theory, a self number or Devlali number in a given number base b {\displaystyle b} is a natural number that cannot be written as the sum of any other natural number n {\displaystyle n} and the individual digits of n {\displaystyle n} . 20 is a self number (in base 10), because no such combination can be found... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12180_chunk_0 | Pseudoperfect number | In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ...... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12181_chunk_0 | Sphenic number | In number theory, a sphenic number (from Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12182_chunk_0 | Strong prime | In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p1, p2, p3, ...) = (2, 3, 5, ...), pn is a st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12183_chunk_0 | Sublime number | In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12184_chunk_0 | Totative | In number theory, a totative of a given positive integer n is an integer k such that 0 < k ≤ n and k is coprime to n. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12185_chunk_0 | Vampire number | In number theory, a vampire number (or true vampire number) is a composite natural number with an even number of digits, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors contain precisely all the digits of the original number, in any order, counti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12186_chunk_0 | Weird number | In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12187_chunk_0 | Power of three | In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. A conjecture of Paul Erdős states that this sequence contains no powers of two other... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12188_chunk_0 | Erdős–Nicolas number | In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number n is an Erdős–Nicolas number when there exists another number m such that ∑ d ∣ n , d ≤ m d = n . {\displaystyle \sum _{d\mid n,\ d\leq m}d=n.} The first ten Erdős–Nicola... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12189_chunk_0 | Euler product | In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12190_chunk_0 | Abundant number | In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12191_chunk_0 | Additive arithmetic function | In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12192_chunk_0 | Arithmetic number | In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 1 + 2 + 3 + 6 4 = 3 , {\displaystyle {\frac {1+2+3+6}{4}}=3,} which is also an integer. However, 2 is not an arithmetic n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12193_chunk_0 | Arithmetic functions | In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12194_chunk_0 | Aurifeuillian factorization | In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12195_chunk_0 | Average order of an arithmetic function | In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function. We say that an average order of f {\displaystyle f} is g {\displaystyle g} if as x {\displaystyle x} tends to infinity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12196_chunk_0 | Average order of an arithmetic function | It is conventional to choose an approximating function g {\displaystyle g} that is continuous and monotone. But even so an average order is of course not unique. In cases where the limit exists, it is said that f {\displaystyle f} has a mean value (average value) c {\displaystyle c} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12197_chunk_0 | Eigencurve | In number theory, an eigencurve is a rigid analytic curve that parametrizes certain p-adic families of modular forms, and an eigenvariety is a higher-dimensional generalization of this. Eigencurves were introduced by Coleman and Mazur (1998), and the term "eigenvariety" seems to have been introduced around 2001 by Kevi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12198_chunk_0 | Equidigital number | In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1. For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigita... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12199_chunk_0 | Evil numbers | In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set. Non-negative integers that are not evil are called odious n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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