MathLabs

Problem 3

Determine all pairs (p,n), where p is a prime number and n is a positive integer, for which (n^p+1)/(p^n+1) is an integer.
Step 3 of 7: Bound n for odd p
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p≥3, n>p⟹np<pn, hence n≤p.p\ge3,\ n>p\Longrightarrow n^p<p^n,\text{ hence }n\le p.
Detailed analysis

For an integer s≥p, the binomial estimate in the official solution gives the corresponding strict growth inequality from s to s+1 whenever it holds at s. Induction, starting at s=p, yields the reverse growth comparison for every n>p. This contradicts the size condition, so n≤p.