Twin, cousin, and sexy prime counting function. Explicit formulas

Krzysztof Kowitz

doi:10.3842/umzh.v76i8.7567

Krzysztof Kowitz Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Poland

DOI: https://doi.org/10.3842/umzh.v76i8.7567

Keywords: prime numbers, twin prime numbers, cousin prime numbers, sexy prime numbers, counting function.

Abstract

UDC 511

We give explicit formulas for the twin-prime and cousin-prime counting functions. We propose a formula that computes the number of primes less than or equal to $n$ whose difference is $m\geqslant 6.$ We also present a characterization specifying when two numbers whose difference is $n\geqslant 2$ are primes.

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