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If every pair in a set of integers is coprime, then the set is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. For example, the ...
If p, q, and r are pairwise relatively prime positive integers then the link of the singularity x p + y q + z r = 0 (in other words, the intersection of a small 3-sphere around 0 with this complex surface) is a Brieskorn manifold that is a homology 3-sphere, called a Brieskorn 3-sphere Σ(p, q, r).
The Beal conjecture, a generalization of Fermat's Last Theorem proposing that if A, B, C, x, y, and z are positive integers with A x + B y = C z and x, y, z > 2, then A, B, and C have a common prime factor. The abc conjecture would imply that there are only finitely many counterexamples.
More generally, a positive integer c is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of c is congruent to 1 modulo 4; that is, each prime factor has the form 4n + 1. In this case, the number of primitive Pythagorean triples (a, b, c) with a < b is 2 k−1, where k is the number of distinct prime factors of c ...
Assume that is an odd prime, that + + = for pairwise relatively prime integers (i.e. in ) ,, and that . This is the first case of Fermat's Last Theorem. (The second case is when .
The prime number theorem implies that a random integer in a suitable interval around N is prime with probability 1 / ln N. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that F 5 , ..., F 32 are composite, then the expected number of Fermat primes beyond F 4 (or ...
But in such a large, complex market, items frequently fall through the cracks, so if you do discover a counterfeit item in your Amazon Prime box, Dimyan suggests two actions. "Utilize Amazon's ...
The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements.