On The Infinitude of the Twin Primes
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866908492220071936 |
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| author | Gensel, Berndt |
| author_facet | Gensel, Berndt |
| contents | We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the conjecture into the existence problem of such $x$.
Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x \pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions.
Due to the \emph{parity problem} we cannot solve the problem with a Selberg sieve. It requires other sieves or methods. The author is searching for them and invites all interested people to help. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_07975 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On The Infinitude of the Twin Primes Gensel, Berndt General Mathematics 2010: 11A41 We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the conjecture into the existence problem of such $x$. Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x \pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions. Due to the \emph{parity problem} we cannot solve the problem with a Selberg sieve. It requires other sieves or methods. The author is searching for them and invites all interested people to help. |
| title | On The Infinitude of the Twin Primes |
| topic | General Mathematics 2010: 11A41 |
| url | https://arxiv.org/abs/1909.07975 |