Small Composite Numbers in Orbits of Linear Maps

Generalized Cunningham chains are sets of the form $\{f^n(z){\}}_{n\ge 0}$ 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 $\mathrm{\ell }(z)<z$ if $z$ is big enough, where $\mathrm{\ell }(z)$ 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.