Mersenne numbers and the doubling map
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913169521246208 |
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| author | Alsedà, Lluís Garijo, Antonio Jarque, Xavier |
| author_facet | Alsedà, Lluís Garijo, Antonio Jarque, Xavier |
| contents | We study the connection between the Mersenne numbers $M(n) = 2^n-1$ and the dynamics of the angle-doubling map. Within this framework, we develop an algorithm to compute divisors of Mersenne numbers without explicitly evaluating $M(n)$. Determining whether $M(n)$ is prime for a prime $n$ (and knowing if there are infinitely many of them), is a central problem, traditionally addressed with the help of the Lucas-Lehmer test. We provide an alternative approach based on dynamical methods. As an application, we prove that $M(2{,}199{,}023{,}254{,}451)$ (with approximately $6.6 \times 10^{11}$ digits) is composite by exhibiting a non-trivial divisor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29130 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mersenne numbers and the doubling map Alsedà, Lluís Garijo, Antonio Jarque, Xavier Number Theory Dynamical Systems 37E10, 11Y55 We study the connection between the Mersenne numbers $M(n) = 2^n-1$ and the dynamics of the angle-doubling map. Within this framework, we develop an algorithm to compute divisors of Mersenne numbers without explicitly evaluating $M(n)$. Determining whether $M(n)$ is prime for a prime $n$ (and knowing if there are infinitely many of them), is a central problem, traditionally addressed with the help of the Lucas-Lehmer test. We provide an alternative approach based on dynamical methods. As an application, we prove that $M(2{,}199{,}023{,}254{,}451)$ (with approximately $6.6 \times 10^{11}$ digits) is composite by exhibiting a non-trivial divisor. |
| title | Mersenne numbers and the doubling map |
| topic | Number Theory Dynamical Systems 37E10, 11Y55 |
| url | https://arxiv.org/abs/2605.29130 |