Let $a, m, n\in\mathbb{N^{*}}$ With $m>n$. Show ${(a^{2^m}-1,a^{2^n}+1)=a^{2^n}+1}$ Let $a, m, n\in\mathbb{N^{*}}$  With $m>n$. Show $${{(a^{2^m}-1,a^{2^n}+1)=a^{2^n}+1}}$$$$$$My thoughts: If we could show that $$(a^{2^n}+1)\mid (a^{2^m}-1)$$ the property that gcd says "If  $\;a\mid b\Longrightarrow (a,b)=a$" I can not work well with two unknowns in number theory.

Hint: $(x^2-1)=(x+1)(x-1)$, so $x-1|x^2-1$ and $x+1|x^2-1$.
