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 3 of 5: Use the size bound
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In plain words

There is no room for another prime factor.

n=por(n,p)=(4,2)n=p\quad\text{or}\quad(n,p)=(4,2)
Detailed analysis

Since p p is the smallest prime divisor of n n and n≤2p n\le2p , either n=p n=p or p=2,n=4 p=2,n=4. The latter fails by direct substitution, so only n=p n=p remains.