MathLabs

Problem 3

Find all pairs of positive integers m,n≥3m,n\ge3 for which there exist infinitely many positive integers aa such that am+a−1an+a2−1\frac{a^m+a-1}{a^n+a^2-1} is itself an integer.
Step 3 of 3: Reduce the remaining cases
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In plain words

Reduce the remaining cases

m=5, n=3m=5,\ n=3
Detailed analysis

Put k=m−nk=m-n. Modulo the denominator, the numerator reduces to xk−xk+2+x−1x^k-x^{k+2}+x-1. Checking k≤n−2k\le n-2, k=n−1k=n-1, and k=nk=n leaves only n=3,k=2n=3,k=2. Finally x5+x−1=(x2−x+1)(x3+x2−1)x^5+x-1=(x^2-x+1)(x^3+x^2-1), so (5,3)(5,3) works.