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 5 of 7: Determine the order
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e=ord⁡p+1(n)∣2p,e∈{1,2,p,2p},e=2.e=\operatorname{ord}_{p+1}(n)\mid2p,\quad e\in\{1,2,p,2p\},\quad e=2.
Detailed analysis

Let e be the least positive exponent with n^e≡1 modulo p+1. Division of 2p by e and minimality give e|2p. The cases e=1 or p contradict n^p≡−1. Euler's theorem and φ(p+1)<2p exclude e=2p, leaving e=2.