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 4 of 7: Extract a congruence
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p≥3⟹n is odd,p+1∣pn+1 and np+1,n2p≡1(modp+1).p\ge3\Longrightarrow n\text{ is odd},\quad p+1\mid p^n+1\text{ and }n^p+1,\quad n^{2p}\equiv1\pmod{p+1}.
Detailed analysis

Both numerator expressions in the divisibility condition are even, so n is odd. Since an odd exponent makes x to that exponent plus 1 divisible by x+1, p+1 divides both numerators. Therefore the p-th power of n is congruent to −1 modulo p+1, and the 2p-th power is congruent to 1.