Problem 3

Find all pairs of positive integers

m,n\ge3

for which there exist infinitely many positive integers

a

such that

\frac{a^m+a-1}{a^n+a^2-1}

is itself an integer.

Step 1 of 3: Turn infinitely many values into divisibility

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In plain words

Turn infinitely many values into divisibility

x^n+x^2-1\mid x^m+x-1

Detailed analysis

Divide by the monic denominator. At infinitely many large positive integers an integral quotient has remainder smaller in absolute value than the denominator, so the remainder is zero infinitely often and hence identically zero.