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Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
Branch of algebraic geometry
curves Siegel modular variety Siegel's theorem on integral points Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF)
Arithmetic_geometry
Algorithm for fast modular multiplication
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing
Montgomery modular multiplication
Montgomery_modular_multiplication
Number divisible only by 1 and itself
for intervals near a number x {\displaystyle x} ). Modular arithmetic modifies usual arithmetic by only using the numbers { 0 , 1 , 2 , … , n − 1 }
Prime_number
Branch of pure mathematics
methods in arithmetic. Its primary subjects of study are divisibility, factorization, and primality, as well as congruences in modular arithmetic. Other topics
Number_theory
Concept in modular arithmetic
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
Modular multiplicative inverse
Modular_multiplicative_inverse
Type of arithmetic where output is limited to a fixed range of values
implement integer arithmetic operations using saturation arithmetic; instead, they use the easier-to-implement modular arithmetic, in which values exceeding
Saturation_arithmetic
Set with associative invertible operation
operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined
Group_(mathematics)
factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power
List_of_number_theory_topics
Technique for selecting hash functions
multiply-shift scheme described by Dietzfelbinger et al. in 1997. By avoiding modular arithmetic, this method is much easier to implement and also runs significantly
Universal_hashing
Computational operation
Carl F. Gauss' approach to modular arithmetic in 1801. Modulo (mathematics), general use of the term in mathematics Modular exponentiation Turn (angle)
Modulo
Multi-modular arithmetic
set of modular values. Using a residue numeral system for arithmetic operations is also called multi-modular arithmetic. Multi-modular arithmetic is widely
Residue_number_system
Polish mathematician (1931–1967)
different residue classes modulo k {\displaystyle k} . Modular arithmetic modifies usual arithmetic by only using the numbers { 0 , 1 , 2 , … , n − 1 } {\displaystyle
Stanisław_Knapowski
Exponentation in modular arithmetic
perform modular exponentiation The GNU Multiple Precision Arithmetic Library (GMP) library contains a mpz_powm() function [5] to perform modular exponentiation
Modular_exponentiation
Shorthand way of determining whether a given number is divisible by a fixed divisor
also divisible by 7. Proof of correctness This method is based on modular arithmetic. Observe that: 1000 ≡ −1 (mod 7) since 1000 leaves a remainder of
Divisibility_rule
Natural number
1088/0026-1394/31/6/013. Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt
1
Implementation of arithmetic operations
Fixed-size arithmetic "Integer arithmetic", which in practice is modular arithmetic by a power of 2. Fixed-point arithmetic Modular arithmetic Multi-modular arithmetic
Computer_arithmetic
Hand game
The game can be expanded for a larger number of players by using modular arithmetic. For n players, each player is assigned a number from zero to n−1
Morra_(game)
Branch of elementary mathematics
signals to perform calculations. There are many other types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result
Arithmetic
Chinese gambling game using tiles
the total number of pips on both tiles in a hand are added using modular arithmetic (modulo 10), equivalent to how a hand in baccarat is scored. The name
Pai_gow
One over a whole number
produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into
Unit_fraction
Amount left over after computation
processed, and no more digits can be brought down. Modular Arithmetic: Utilizing modular arithmetic concepts to calculate remainders efficiently, particularly
Remainder
Topics referred to by the same term
applicable to block and stream ciphers Modulo (mathematics) Modular arithmetic Modulo operation Modular exponentiation MOD., a science museum at the University
Mod
Algorithm to calculate the day of the week
Zeller's congruence is a modular arithmetic algorithm devised by Christian Zeller in the 19th century for calculating the day of the week for a given date
Zeller's_congruence
Unique numeric book identifier since 1970
1)\\&=0+27+0+42+24+0+24+3+10+2\\&=132=12\times 11.\end{aligned}}} Formally, using modular arithmetic, this is rendered ( 10 x 1 + 9 x 2 + 8 x 3 + 7 x 4 + 6 x 5 + 5 x 6
ISBN
Division with remainder of integers
algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting
Euclidean_division
Generalization of the Legendre symbol in number theory
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 in computational
Jacobi_symbol
Type of pseudorandom number generation algorithm
A permuted congruential generator (PCG) is a pseudorandom number generation algorithm developed in 2014 by Dr. M.E. O'Neill which applies an output permutation
Permuted congruential generator
Permuted_congruential_generator
Number
consequently dividing by 0 is generally considered to be undefined in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it
0
a^{p}\equiv a{\pmod {p}}} for every prime number p and every integer a (see modular arithmetic). Some of the proofs of Fermat's little theorem given below depend
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Orientation-preserving mapping class group of the torus
group" comes from the relation to moduli spaces, and not from modular arithmetic. The modular group Γ is the group of fractional linear transformations of
Modular_group
Modular arithmetic concept
(1965) [1801]. Untersuchungen über höhere Arithmetik [Studies of Higher Arithmetic] (in German). Translated by Maser, H. (2nd ed.). New York, NY: Chelsea
Primitive_root_modulo_n
Algorithm for computing greatest common divisors
reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic
Euclidean_algorithm
Condition under which an integer is a quadratic residue
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
Gauss's_lemma_(number_theory)
Theorem on prime numbers
is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial ( n − 1 ) ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) {\displaystyle
Wilson's_theorem
Decimal error detection code
The Verhoeff algorithm is a checksum for error detection first published by Dutch mathematician Jacobus Verhoeff in 1969. It was the first decimal check
Verhoeff_algorithm
Simple checksum formula
The Luhn algorithm or Luhn formula (creator: IBM scientist Hans Peter Luhn), also known as the "modulus 10" or "mod 10" algorithm, is a simple check digit
Luhn_algorithm
Algorithm in modular arithmetic
In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of a mod n {\displaystyle a\,{\bmod {\,}}n\,} without needing
Barrett_reduction
Group of units of the ring of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set { 0 , 1 , … , n − 1 } {\displaystyle \{0,1,\dots ,n-1\}} of n non-negative
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
value as a principle in contemporary sieve theory received recognition. Arithmetic progressions of integers are sets of the form a + nd where n is any integer
Rogers_sieving_theorem
Type of average of a collection of numbers
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection
Arithmetic_mean
About simultaneous modular congruences
rings of integers modulo the ni. This means that for doing a sequence of arithmetic operations in Z / N Z , {\displaystyle \mathbb {Z} /N\mathbb {Z} ,} one
Chinese_remainder_theorem
Symbol in number theory
In number theory, the Kronecker symbol, written as ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} or ( a | n ) {\displaystyle (a|n)} , is a generalization
Kronecker_symbol
A prime p divides a^p–a for any integer a
the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv
Fermat's_little_theorem
Concept in modular arithmetic
examples of multiplicative order in various languages Discrete logarithm Modular arithmetic Niven, Zuckerman & Montgomery 1991, Section 2.8 Definition 2.6 von
Multiplicative_order
Open problem on 3x+1 and x/2 functions
divide it by two. If the number is odd, triple it and add one. In modular arithmetic notation, define the function f as follows: f ( n ) = { n / 2 if
Collatz_conjecture
Problem of inverting exponentiation in groups
k} such that b k = a {\displaystyle b^{k}=a} . In the special case of arithmetic modulo an integer m {\displaystyle m} , the more commonly used term is
Discrete_logarithm
Theorem on polynomial roots modulo prime powers
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
Hensel's_lemma
Process of converting plaintext to ciphertext
asymmetric-key). Many complex cryptographic algorithms often use simple modular arithmetic in their implementations. In symmetric-key schemes, the encryption
Encryption
Representation of modular integers by "small" fractions
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
Thue's_lemma
Period of the Fibonacci sequence modulo an integer
MathWorld. On Arithmetical functions related to the Fibonacci numbers. Acta Arithmetica XVI (1969). Retrieved 22 September 2011. A Theorem on Modular Fibonacci
Pisano_period
Algorithm used in modular arithmetic
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
Tonelli–Shanks_algorithm
Count of the possible partitions of a set
doi:10.1017/S1757748900002334. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2):
Bell_number
Formula concerning prime numbers
and 22 = 4. We can do these calculations faster by using various modular arithmetic and Legendre symbol properties. If we keep calculating the values
Euler's_criterion
Topics referred to by the same term
value of a real or complex number ( |c| ) Modulus (modular arithmetic), base of modular arithmetic Similarly, the modulus of a Dirichlet character Moduli
Modulus
Word with multiple distinct meanings
factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many
Modulo_(mathematics)
Composite number in number theory
Carmichael number is a composite number n {\displaystyle n} which in modular arithmetic satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle
Carmichael_number
Group obtained by aggregating similar elements of a larger group
\mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } ) SL(2, Z {\displaystyle \mathbb {Z} } ) Arithmetic group Lattice Hyperbolic
Quotient_group
Function in number theory
manipulation. Since no efficient factorization algorithm is known, but efficient modular exponentiation algorithms are, in general it is more efficient to use Legendre's
Legendre_symbol
Multiplication table in Indian mathematics
1 is dark and the digital root of (base-1) is light. Latin square Modular arithmetic Monoid Lin, Chia-Yu (2016). "Digital Root Patterns of Three-Dimensional
Vedic_square
Theorem on modular exponentiation
arithmetica nova methodo demonstrata" (Proof of a new method in the theory of arithmetic), Novi Commentarii academiae scientiarum Petropolitanae, 8 : 74–104. Euler's
Euler's_theorem
Type of machine learning model
by an LLM. For instance, the authors trained small transformers on modular arithmetic addition. The resulting models were reverse-engineered, and it turned
Large_language_model
Symbols used to write numbers
tallies. A great convenience of modular arithmetic is that it is easy to multiply. This makes use of modular arithmetic for provisions especially attractive
Numerical_digit
Equivalence relation in algebra
corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo n {\displaystyle
Congruence_relation
Arithmetic operation
denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The
Addition
Type of linear congruential generator with no additive constant
on the Cray XD1. Cray User Group 2009. The die is determined using modular arithmetic, e.g., lrand48() % 6 + 1, ... The CRAY RANF function only rolls three
Lehmer random number generator
Lehmer_random_number_generator
Number of integers coprime to and less than n
relatively prime to p k {\displaystyle p^{k}} . The fundamental theorem of arithmetic states that if n > 1 there is a unique expression n = p 1 k 1 p 2 k 2
Euler's_totient_function
Congruence used in integer factorization algorithms
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. Given a positive integer n, Fermat's factorization
Congruence_of_squares
Analytic function on the upper half-plane with a certain behavior under the modular group
\Gamma <{\text{SL}}_{2}(\mathbb {Z} )} of finite index (called an arithmetic group), a modular form of level Γ {\displaystyle \Gamma } and weight k {\displaystyle
Modular_form
Algebraic structure with addition, multiplication, and division
prime order, are most directly accessible using modular arithmetic. For a fixed positive integer n, arithmetic "modulo n" means to work with the numbers Z/nZ
Field_(mathematics)
Theory in number theory
geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related
Anabelian_geometry
Algorithm for generating pseudo-randomized numbers
implemented and fast, especially on computer hardware which can provide modular arithmetic by storage-bit truncation. The generator is defined by the recurrence
Linear_congruential_generator
Mathematics of varieties with integer coordinates
these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems of fundamental importance in Diophantine geometry
Diophantine_geometry
information is. The Feige–Fiat–Shamir identification scheme, however, uses modular arithmetic and a parallel verification process that limits the number of communications
Feige–Fiat–Shamir identification scheme
Feige–Fiat–Shamir_identification_scheme
Integer that is a perfect square modulo some integer
exhibits some striking regularities. Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder
Quadratic_residue
Gives conditions for the solvability of quadratic equations modulo prime numbers
number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo
Quadratic_reciprocity
Probabilistic primality test
(a^{2}r)+un} is divisible by a 2 {\displaystyle a^{2}} in ordinary integer arithmetic, then r = ( lift ( a 2 r ) + u n ) / a 2 {\displaystyle r=(\operatorname
Fermat_primality_test
Chinese mathematician
remarkable combination of sophisticated techniques, in which the arithmetic and geometry of modular curves and of Shimura varieties play a central role, and have
Yunqing_Tang
Set of residue classes modulo n, relatively prime to n
Congruence relation Euler's totient function Greatest common divisor Modular arithmetic Number theory Residue number system Long (1972, p. 85) Pettofrezzo
Reduced_residue_system
On unit fractions adding to 4/n
integer solution to the equation. Nevertheless, modular arithmetic, and identities based on modular arithmetic, have proven a very important tool in the study
Erdős–Straus_conjecture
variables occurring in it. 2. In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer. 3. May denote a logical
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
for x, then x is called a primitive kth root of unity modulo n. See modular arithmetic for notation and terminology. The roots of unity modulo n are exactly
Root_of_unity_modulo_n
Integer side lengths of a right triangle
Euler brick Heronian triangle Hilbert's theorem 90 Integer triangle Modular arithmetic Nonhypotenuse number Plimpton 322 Pythagorean prime Pythagorean quadruple
Pythagorean_triple
Function in mathematical number theory
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 (
Carmichael_function
Puzzle of reconstructing equations that have been enciphered into words
possible and with 2+8=10+U, U=0. The use of modular arithmetic often helps. For example, use of mod-10 arithmetic allows the columns of an addition problem
Verbal_arithmetic
1955 random number compilation
some statistical biases; this was resolved by scrambling through modular arithmetic. The book was released to acclaim by mathematicians and statisticians
A Million Random Digits with 100,000 Normal Deviates
A_Million_Random_Digits_with_100,000_Normal_Deviates
Type of mathematical expression
integers modulo some prime number as the coefficient ring R (see modular arithmetic). If R is commutative, then one can associate with every polynomial
Polynomial
Algorithm for fast exponentiation
exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation
Exponentiation_by_squaring
this above computation, remembering that something close to complex modular arithmetic is going on here) As such: ( 2 + 2 2 − 10 ) 13 2 ⋅ 7 mod 13 3 ≡ 1540
Cipolla's_algorithm
Coprime number less than a given integer
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
Totative
Result in number theory showing congruences involving Bernoulli numbers
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer. Kubota & Leopoldt (1964) used Kummer's
Kummer's_congruence
Cryptographic algorithm created by Adi Shamir
"wrapping around" behavior of modular arithmetic prevents the leakage of "S is even", unlike the example with integer arithmetic above. For purposes of keeping
Shamir's_secret_sharing
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
Mirimanoff's_congruence
Algebraic variety
structure and for this reason they play an important role in arithmetic geometry. The level N modular curve X(N) is the moduli space for elliptic curves with
Modular_curve
Topics referred to by the same term
to perform QR decomposition Quadratic reciprocity, a theorem from modular arithmetic Quasireversibility, a property of some queues Reaction quotient (Qr)
QR
Arithmetical function
primes is a cyclotomic polynomial of p − k {\displaystyle p^{-k}} ), the arithmetic functions defined by J k ( n ) J 1 ( n ) {\displaystyle {\frac {J_{k}(n)}{J_{1}(n)}}}
Jordan's_totient_function
Arithmetical puzzle game
large number of cages is to add up the cages using 'clock' arithmetic (formally, Modular Arithmetic modulo 10), in which all digits other than the last in
Killer_sudoku
Number of partitions of an integer
discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the
Partition function (number theory)
Partition_function_(number_theory)
Pseudo-random number generator algorithm
A combined linear congruential generator (CLCG) is a pseudo-random number generator algorithm based on combining two or more linear congruential generators
Combined linear congruential generator
Combined_linear_congruential_generator
(signed)((i1 << 12) - (i2 << 12)) Date windowing Lollipop sequence numbering Modular arithmetic RFC 1323: "TCP Extensions for High Performance", section 4.2. RFC 2182
Serial_number_arithmetic
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