MathLabs

Problem 4

Determine all pairs (n,p)(n,p) of positive integers such that p p is a prime, n n not exceeded 2p2p , and (p−1)n+1(p-1)^n+1 is divisible by np−1 n^{p-1}.
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In plain words

Separate the obvious cases.

(1,p) works for every prime p(1,p)\text{ works for every prime }p
Detailed analysis

If n=1 n=1, the divisor is 11, so (1,p)(1,p) is a solution for every prime p p . Also (2,2)(2,2) can be checked directly. Assume henceforth n>1 n>1.