Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917116359213056 |
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| author | Prunescu, Mihai |
| author_facet | Prunescu, Mihai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs Prunescu, Mihai Number Theory 11N05 (primary), 11D61, 03D20 (secondary) 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. |
| title | Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs |
| topic | Number Theory 11N05 (primary), 11D61, 03D20 (secondary) |
| url | https://arxiv.org/abs/2512.01680 |