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In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. [1] Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. [2] One says also a is prime to b or a ...
In mathematics, Legendre's equation is a Diophantine equation of the form: + + = The equation is named for Adrien-Marie Legendre who proved it in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers and also not all ...
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).
This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See Coin problem.)
The first step is relatively slow but only needs to be done once. Modular multiplicative inverses are used to obtain a solution of a system of linear congruences that is guaranteed by the Chinese Remainder Theorem. For example, the system X ≡ 4 (mod 5) X ≡ 4 (mod 7) X ≡ 6 (mod 11) has common solutions since 5,7 and 11 are pairwise coprime ...
If 2 k + 1 is prime and k > 0, then k itself must be a power of 2, [1] so 2 k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023 [update] , the only known Fermat primes are F 0 = 3 , F 1 = 5 , F 2 = 17 , F 3 = 257 , and F 4 = 65537 (sequence A019434 in the OEIS ).
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For instance, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint. [6] A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise ...