Find all pairs of positive integers for which there exist infinitely many positive integers such that is itself an integer.
Laurentiu Panaitopol, Romania
%V0
Find all pairs of positive integers $m,n\geq3$ 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.
Laurentiu Panaitopol, Romania
Izvor: Međunarodna matematička olimpijada, shortlist 2002
Školjka 
for which there exist infinitely many positive integers 
such that 
is itself an integer.