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 2 of 3: Bound the exponent
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In plain words

Bound the exponent

n<m≤2nn<m\le2n
Detailed analysis

Clearly m>nm>n. If m>2nm>2n, choose 0<r<10<r<1 with rn+r2=1r^n+r^2=1. Divisibility would give rm+r−1=0r^m+r-1=0, but r>3/4r>3/4 and rm<r2n=(1−r2)2<1−rr^m<r^{2n}=(1-r^2)^2<1-r, contradiction.