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

[durt]

Let a be a positive integer. Find all positive integers n such that b = a^n satisfies the condition that a^2 + b^2 is divisible by ab + 1.

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

 

[morphism]

What can you say about the polynomial x^{2k} + 1?

 

[durt]

I don't know. It has no real zeros? That doesn't mean it can't be factored. For example 2^2+1|2^6+1.

 

[morphism]

Right, but there's something special we can say about its factors. Here's another hint: What can you say about x^{2^m k} + 1 when k is odd?

 

[durt]

Let k=2j+1. Then x^{2^m (2j+1)} + 1 = (x^{2^m}+1)(x^{2^m (2j)} - x^{2^m (2j - 1)} + ... + 1). So x^{2^m}+1 | x^{2^m k} + 1. I'm still stuck .