mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers...

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...

Arithmetic dynamics Arithmetic of abelian varieties Birch and Swinnerton-Dyer conjecture Moduli of algebraic curves Siegel modular variety Siegel's theorem...

In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing...

for intervals near a number ⁠ x {\displaystyle x} ⁠). Modular arithmetic modifies usual arithmetic by only using the numbers ⁠ { 0 , 1 , 2 , … , n − 1 }...

implement integer arithmetic operations using saturation arithmetic; instead, they use the easier-to-implement modular arithmetic, in which values exceeding...

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...

methods in arithmetic. Its primary subjects of study are divisibility, factorization, and primality, as well as congruences in modular arithmetic. Other topics...

perform modular exponentiation The GNU Multiple Precision Arithmetic Library (GMP) library contains a mpz_powm() function [5] to perform modular exponentiation...

operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined...

Carl F. Gauss' approach to modular arithmetic in 1801. Modulo (mathematics), general use of the term in mathematics Modular exponentiation Turn (angle)...

factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power...

set of modular values. Using a residue numeral system for arithmetic operations is also called multi-modular arithmetic. Multi-modular arithmetic is widely...

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...

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...

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 Floating-point arithmetic Interval arithmetic Arbitrary-precision arithmetic Modular arithmetic Multi-modular arithmetic p-adic arithmetic...

means 10 ≡ 1 ( mod 3 ) {\displaystyle 10\equiv 1{\pmod {3}}} (see modular arithmetic). The same for all the higher powers of 10: 10 n ≡ 1 n ≡ 1 ( mod 3...

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...

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...

\mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } ) SL(2, Z {\displaystyle \mathbb {Z} } ) Arithmetic group Lattice Hyperbolic...

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...

produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into...

integer k {\displaystyle k} such that b k = a {\displaystyle b^{k}=a} . In arithmetic modulo an integer m {\displaystyle m} , the more commonly used term is...

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...

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...

Carmichael number is a composite number ⁠ n {\displaystyle n} ⁠ which in modular arithmetic satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle...

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...

In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of a mod n {\displaystyle a\,{\bmod {\,}}n\,} without needing...

arithmetica nova methodo demonstrata" (Proof of a new method in the theory of arithmetic), Novi Commentarii academiae scientiarum Petropolitanae, 8 : 74–104. Euler's...