Small Composite Numbers in Orbits of Linear Maps

TL;DR

This paper extends the concept of Cunningham chains to include non-prime starting points, establishing bounds on chain length based solely on the size of the initial number, independent of its prime factors.

Contribution

It generalizes Cunningham chains to non-prime starts and proves a size-based bound on chain length, removing previous prime factorization constraints.

Findings

Bound of chain length is less than the initial number for large enough z.

Chain length depends only on the size of z, not its prime factors.

Generalization broadens the scope of Cunningham chain analysis.

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.