Track / Stage 4 / 155 of 340 #415 of 1964

Problem 415

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

(French-Slovak Competition 1996) Find all strictly positive integers $x, y, p$ such that $p^{x}-y^{p}=1$ with $p$ prime.

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Let's rewrite the equation in the form $y^{p}+1^{p}=p^{x}$ . If $y=1$ , we see that $p=2$ and $x=1$ . If $p=2$ , it readily follows that necessarily $x, y=1$ . Therefore, we assume $p$ is odd so that $y+1$ divides $y^{p}+1$ .

If the conditions of Zsigmondy's theorem are satisfied, there exists a prime factor of $y^{p}+1$ that does not divide $y+1$ , so $y^{p}+1$ cannot be a power of a prime number. It remains to handle the case $(y, p)=(2,3)$ , which gives the solution $x=2$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.