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 7 of 7: Check the solutions
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(p,n)=(2,2),(2,4),or (p,n)=(p,p) for any odd prime p.(p,n)=(2,2),(2,4),\quad\text{or }(p,n)=(p,p)\text{ for any odd prime }p.
Detailed analysis

For n=p, the quotient is 1. Together with the two solutions from the p=2 check, these are exactly all pairs.