TITLE: Is $7^4-1=2^5 \times 3 \times 5^2$ the only counterexample?
QUESTION [9 upvotes]: My friend once told me his conjecture:

For every prime $p$ and every integer $n$ which is greater than $2$, $p^n-1$ must have prime factor greater than $p$.

A few minutes later I found a counterexample:

$7^4-1=2^5 \times 3 \times 5^2$

However, is there any other counterexample? Could you help me?

REPLY [5 votes]: Searching for the smallest counterexample for each power gives the following:
$$\begin{gather*}
67^3-1=300762=2\times3\times7^2\times11\times31\\
7^4-1=2400=2^5\times3\times5^2\\
7307^5-1=20830300574702353306=2\times11\times13\times151\times191\times281\times911\times1481\times6661\\
263^6-1=330928743953808=2^4\times3^2\times7^2\times11\times13\times103\times109\times131\times223\\
493397^7-1=7128392726072389152475777731457041578312=2^2\times23\times29^2\times31\times127\times173\times1163\times2129\times4229\times26041\times50177\times71359\times138349\\
10181^8-1=115431276825545828982660756380640=2^5\times3\times5\times17\times37\times509\times677\times1657\times1697\times2069\times4657\times5113\times8009\\
734197^{10}-1=45512014174960703207280356067658088285424423446151729273048=2^3\times3\times11\times17\times19\times41\times59\times61\times71\times131\times139\times191\times761\times2161\times9871\times10301\times12391\times14341\times27901\times181711\times699511\\
\end{gather*}$$