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

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

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...

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

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

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

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

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

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

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...

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

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

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

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...

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...

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...

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...

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...

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...

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

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...

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

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...

algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting...

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...

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

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

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...

reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic...

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...