Let $p$ be a fixed prime and $r$ a fixed prime power. Let $x\_{p'}$ denote the largest divisor of a positive integer $x$ such that $x\_{p'}$ is not divisible by $p$. (For example, $60\_{2'}=15$.) I would like to prove:


**Claim**: $\dfrac{(r^n-1)\_{p'}}{n}\rightarrow\infty$ as $n\rightarrow \infty$.


Sorry if this is obvious and thank you for your help.



A limit concerning prime numbers
This is true, and due to Walter Feit (On large Zsigmondy Primes, PAMS 1988). (What he shows [Theorem B in the quoted paper] is the following:


Let $N$ be a positive integer. Then, for all but finitely many pairs of integers $<a, n>$ with $a>1$ and $n>2$ there exists a Zsigmondy prime with $|a^n-1|\_p > n N +1.$ A Zsigmondy prime is one with multiplicative order of $a$ modulo $p$ being equal to $n$ (so, one that divides $a^n-1$ but not lower powers of $a$). The norm is the $p$-adic valuation (so, the largest power of $p$ dividing the number). )


A very nice proof of this fact (and lots of other cool stuff) is in the short paper:


On Zsigmondy Primes


Moshe Roitman,


PAMS 1997


