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 1 of 7: Start with the size condition
pn+1≤np+1⟹pn≤np.p^n+1\le n^p+1\Longrightarrow p^n\le n^p.
Detailed analysis

If the quotient is a positive integer, its denominator cannot exceed its numerator. Thus p^n≤n^p.