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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 developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.
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 and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
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 remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
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 modulo n.
The modular inverse of aR mod N is REDC((aR mod N) −1 (R 3 mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.
In that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic functions (geometrically, the n 2 -fold covering map from a 2- torus to itself given by the mapping x → n · x on the underlying group) expressed in terms of complex analysis .
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 coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
Two important examples are the partitions restricted to only odd integer parts or only even integer parts, with the corresponding partition functions often denoted () and (). A theorem from Euler shows that the number of strict partitions is equal to the number of partitions with only odd parts: for all n , q ( n ) = p o ( n ) {\displaystyle q ...
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