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In number theory, the gcd-sum function, [1] also called Pillai's arithmetical function, [1] is defined for every by = = (,) or equivalently ...
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c. [6] The greatest common divisor can be visualized as follows. [7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly.
Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 2 2 × 3 = 12.. The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, [1] [2] is an algorithm that computes the greatest common divisor (GCD) of two nonnegative integers.
Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is mainly used for big integers that have a representation as a string of digits relative to some chosen numeral system base , say β = 1000 or β = 2 32 .
The greatest common divisor of p and q is usually denoted "gcd(p, q)". The greatest common divisor is not unique: if d is a GCD of p and q , then the polynomial f is another GCD if and only if there is an invertible element u of F such that f = u d {\displaystyle f=ud} and d = u − 1 f . {\displaystyle d=u^{-1}f.}
In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. [2] [3] The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8.
The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition, where f + g is defined by (f + g)(n) = f(n) + g(n), and Dirichlet convolution.