Small Composite Numbers in Orbits of Linear Maps
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912554469556224 |
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| author | Reyes, Jose |
| author_facet | Reyes, Jose |
| contents | Generalized Cunningham chains are sets of the form $\{f^n(z)\}_{n\ge0}$ where all its elements are prime numbers and $f$ is a linear polynomial with integer coefficients. We generalize this definition further to include starting terms that are not prime, and we obtain the bound of $\ell(z)< z$ if $z$ is big enough, where $\ell(z)$ is the size of the generalized Cunningham chain. Unlike a direct generalization of previous results, which require $z$ to have a prime factor that does not divide the leading term of $f$, this result is only dependent on the size of $z$ and not on its prime factorization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small Composite Numbers in Orbits of Linear Maps Reyes, Jose Number Theory 11B83 Generalized Cunningham chains are sets of the form $\{f^n(z)\}_{n\ge0}$ where all its elements are prime numbers and $f$ is a linear polynomial with integer coefficients. We generalize this definition further to include starting terms that are not prime, and we obtain the bound of $\ell(z)< z$ if $z$ is big enough, where $\ell(z)$ is the size of the generalized Cunningham chain. Unlike a direct generalization of previous results, which require $z$ to have a prime factor that does not divide the leading term of $f$, this result is only dependent on the size of $z$ and not on its prime factorization. |
| title | Small Composite Numbers in Orbits of Linear Maps |
| topic | Number Theory 11B83 |
| url | https://arxiv.org/abs/2508.18305 |