Counting prime numbers in arithmetic sequences

Abstract

This paper presents the theoretical elements that support the calculation of the prime number counting function, π(x), based on properties of the sequences (6n-1) and (6n+1), n ≥ 1. As a result, Sufficient primality criteria are exposed for the terms of both sequences that support a deterministic computational algorithm that reduces the number of operations in calculating the function π(x) by exonerating all multiples of 3 from the analysis. In analyzing the primality of a particular term, the divisions by factor 3 are excluded. Such an algorithm can be applied to the search for prime numbers in each sequence separately, a possibility that allows approximately half the time of number search in practical applications.