Let [itex]a[/itex] be a positive integer. Find all positive integers [itex]n[/itex] such that [itex]b = a^n[/itex] satisfies the condition that [itex]a^2 + b^2[/itex] is divisible by [itex]ab + 1[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Obviously if [itex]a=1[/itex] then all [itex]n[/itex] work. Otherwise, we have [itex]a^2 + b^2 = a^2 (1+a^{2(n-1)})[/itex]. Also, [itex]a^2[/itex] and [itex]a^{n+1} + 1[/itex] are relatively prime, so we need to find all [itex]n[/itex] such that [itex]a^{n+1} + 1[/itex] divides [itex]1+a^{2(n-1)}[/itex]. Clearly [itex]n=3[/itex] works, but now I'm stuck. What do I do now?

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# Find all integers with b=a^n such that a^2 + b^2 is divisible by ab+1

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