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arXiv · 2508.18305

Small Composite Numbers in Orbits of Linear Maps

Jose Reyes
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Abstract

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.

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Jose Reyes. 2025-08-23. Small Composite Numbers in Orbits of Linear Maps. https://arxiv.org/abs/2508.18305

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