Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs
Mihai Prunescu
TL;DR
The paper demonstrates that arithmetic closed forms can generate and count key prime-structured sets—Mersenne primes, Fermat primes, twin-prime pairs, and Sophie Germain primes—by combining Wilson-based generation terms with single-fold exponential Diophantine representations for primality tests (Lucas-Lehmer, Pépin) and Clement’s criterion for twins. It leverages Mazzanti’s method to convert exponential Diophantine problems into arithmetic terms and to produce counting terms, delivering explicit, though highly intricate, arithmetic expressions that count primes within intervals. The results extend Jones’ Diophantine framework and provide a unified scheme to encode primality and counting in arithmetic terms, facilitating theoretical analysis of finiteness questions and potential applications in symbolic computation and SEO-friendly metadata. The work emphasizes structural connections between Diophantine representations, C-recursive sequences, and classical primality tests, illustrating a pathway from primality criteria to computable closed forms. Overall, it contributes a rigorous, technique-driven approach to encoding and counting prime-related sequences within fixed arithmetic-term frameworks.
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.