Abstract
We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo m. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pó lya, we obtain a heuristic formula for the “probability” that a pair of consecutive prime numbers of size approximately x will be congruent to (a, a+d) modulo m. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for x in various ranges.
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