Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs

TL;DR

This paper develops arithmetic closed-form formulas to generate and count special prime sets like Mersenne, Fermat, twin primes, and Sophie Germain primes, using classical number theory tests and operations.

Contribution

It introduces explicit arithmetic closed forms for generating and counting these prime sets, combining classical theorems with novel formulas.

Findings

Closed forms generate all Mersenne and Fermat primes with repetitions.

Counting formulas use primality tests like Lucas-Lehmer and Pepin.

Formulas count primes within specific numerical ranges.

Abstract

We construct closed forms that generate with repetitions all Mersenne primes, respectively all Fermat primes, all twin-prime pairs and all Sophie Germain primes. Also, we construct closed forms that count all Mersenne primes between $0$ and $2^{n+2}-1$, respectively all Fermat primes between $0$ and $6n+5$ and all twin-prime pairs between $0$ and $n$. Every closed form is an arithmetic term, i. e. a fixed finite composition of the following arithmetic operations: addition, subtraction, multiplication, division with remainder and the exponentiation $2^n$. While for generating these sets with repetitions, only Wilson's Theorem is applied, for the counting forms we use more specific tests, i.e. Lucas-Lehmer, respectively Pepin, and we apply to some extent Jones' work (see Acta Arithmetica XXXV, pg. 210 - 221, 1979). To count twin primes we apply Clement's Theorem, which is closely related to Wilson's. A closed form to count the Sophie Germain primes can be constructed similarly.