Hybrid bounds for prime divisors

TL;DR

This paper establishes new lower bounds for integers based on the number of prime factors of $x^n-1$, with implications for primitive elements in finite fields.

Contribution

It introduces hybrid bounds for prime divisors of $x^n-1$, refining previous results by linking divisibility properties to the prime factor count.

Findings

Derived a non-trivial lower bound for $x$ based on $ ext{ω}_n$

Refined bounds by analyzing divisibility of $ ext{φ}$ dividing $x^n-1$

Applications demonstrated in primitive element existence in finite fields

Abstract

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