# Category: Modular arithmetic

Automorphic number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base whose square "ends" in the same digits as the number itself.
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of th
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 u
Zeller's congruence
Zeller's congruence is an algorithm devised by Christian Zeller in the 19th century to calculate the day of the week for any Julian or Gregorian calendar date. It can be considered to be based on the
Primitive root modulo n
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a copr
Gauss's lemma (number theory)
Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of
Permuted congruential generator
A permuted congruential generator (PCG) is a pseudorandom number generation algorithm developed in 2014 which applies an output permutation function to improve the statistical properties of a modulo-2
Luhn algorithm
The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple checksum formula used to validate a variet
Vedic square
In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings i.e. the rem
Pisano period
In number theory, the nth Pisano period, written as π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better know
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: Otherwise, q is called a quadratic
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, the number is an integer multiple of p. In the notation of modular arithmetic, this is expressed as For example, if
Cipolla's algorithm
In computational number theory, Cipolla's algorithm is a technique for solving a congruence of the form where , so n is the square of x, and where is an odd prime. Here denotes the finite field with e
In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
Proofs of Fermat's little theorem
This article collects together a variety of proofs of Fermat's little theorem, which states that for every prime number p and every integer a (see modular arithmetic).
Discrete logarithm records
Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation given elements g and h of a fini
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is
Kronecker's congruence
In mathematics, Kronecker's congruence, introduced by Kronecker, states that where p is a prime and Φp(x,y) is the modular polynomial of order p, given by for j the elliptic modular function and τ run
Solovay–Strassen primality test
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic test to determine if a number is composite or probably prime. The idea behind the te
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 Euler
Luhn mod N algorithm
The Luhn mod N algorithm is an extension to the Luhn algorithm (also known as mod 10 algorithm) that allows it to work with sequences of values in any even-numbered base. This can be useful when a che
Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in
Euler's totient function
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 or , and may also be called
Method of successive substitution
In modular arithmetic, the method of successive substitution is a method of solving problems of simultaneous congruences by using the definition of the congruence equation. It is commonly applied in c
Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modu
Residue number system
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remai
Kummer's congruence
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer. used Kummer's congruences to define the p-adic zeta function.
Table of congruences
In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences.
Modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exc
Congruence of squares
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
Carmichael number
In number theory, a Carmichael number is a composite number , which in modular arithmetic satisfies the congruence relation: for all integers which are relatively prime to . The relation may also be e
Montgomery modular multiplication
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it
Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciproc
Tonelli–Shanks algorithm
The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r2 ≡ n (mod p), where p is a prime: that is, to fi
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 . In other words, the multiplicative ord
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number p: its value at a (nonzero) quadratic residue mod p is
Linear congruential generator
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the ol
Combined linear congruential generator
A combined linear congruential generator (CLCG) is a pseudo-random number generator algorithm based on combining two or more linear congruential generators (LCG). A traditional LCG has a period which
Mod n cryptanalysis
In cryptography, mod n cryptanalysis is an attack applicable to block and stream ciphers. It is a form of partitioning cryptanalysis that exploits unevenness in how the cipher operates over equivalenc
Vantieghems theorem
In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n(n≥3) is prime if and only if Similarly, n is prime, if and only if the following congruence for polyno
Kochanski multiplication
Kochanski multiplication is an algorithm that allows modular arithmetic (multiplication or operations based on it, such as exponentiation) to be performed efficiently when the modulus is large (typica
Thue's lemma
In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater th
Modulo operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation). Given two positive numbers a a
Discrete logarithm
In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarith
Canon arithmeticus
In mathematics, the Canon arithmeticus is a table of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by Carl Gustav Jacob Jacobi. The tables we
Cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from t
Barrett reduction
In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett. A naive way of computing would be to use a fast division algorithm. Barrett reduction is an algori
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n
Polydivisible number
In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties: 1. * Its first digit a is not 0. 2. * The number fo
Kronecker symbol
In number theory, the Kronecker symbol, written as or , is a generalization of the Jacobi symbol to all integers . It was introduced by Leopold Kronecker .
Reduced residue system
In mathematics, a subset R of the integers is called a reduced residue system modulo n if: 1. * gcd(r, n) = 1 for each r in R, 2. * R contains φ(n) elements, 3. * no two elements of R are congruent
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 is Euler's totient function, then a rai
Fermat primality test
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime.
Verhoeff algorithm
The Verhoeff algorithm is a checksum formula for error detection developed by the Dutch mathematician Jacobus Verhoeff and was first published in 1969. It was the first decimal check digit algorithm w
Carmichael function
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 holds for every integer a coprime to n. In algebraic terms,
Hensel's lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a pri
Jordan's totient function
Let be a positive integer. In number theory, the Jordan's totient function of a positive integer equals the number of -tuples of positive integers that are less than or equal to and that together with
Root of unity modulo n
In mathematics, namely ring 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 congruen
Pocklington's algorithm
Pocklington's algorithm is a technique for solving a congruence of the form where x and a are integers and a is a quadratic residue. The algorithm is one of the first efficient methods to solve such a
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was
Modular multiplicative inverse
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In th
Lehmer random number generator
The Lehmer random number generator (named after D. H. Lehmer), sometimes also referred to as the Park–Miller random number generator (after Stephen K. Park and Keith W. Miller), is a type of linear co