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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).
A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial () that satisfies (1)–(3), () is prime for at least one positive integer : but then, since the translated polynomial (+) still satisfies (1)–(3), in view of the weaker statement () is prime for at least one positive integer >, so ...
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 ...
The Golomb topology, [2] or relatively prime integer topology, [6] on the set > of positive integers is obtained by taking as a base the collection of all + with , > and and relatively prime. [2] Equivalently, [ 7 ] the subcollection of such sets with the extra condition b < a {\displaystyle b<a} also forms a base for the topology. [ 6 ]
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.
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 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.