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}.
Step 5 of 5: State the remaining solutions
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In plain words

Check the final small primes.

(n,p)=(2,2),(3,3)(n,p)=(2,2),(3,3)
Detailed analysis

The cases p=2 p=2 and p=3 p=3 with n=p n=p both satisfy the divisibility condition. Together with the immediate family, all solutions are (1,p)(1,p) for prime p p , (2,2)(2,2), and (3,3)(3,3).