Hybrid bounds for prime divisors

Abstract

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $\ddot{I}{‰}_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $\ddot{I}†\mid x^n-1$ for $\ddot{I}†\mid n$, we obtain a further refinement. This bound has applications for existence problems relating to primitive elements in finite fields.

19 pages