Vtyjkyuk

This is a question from my discrete maths homework. I have done parts i) and ii) but need help with the conclusion.

i) If $m>n$, Prove that $(a^2^n + 1) | (a^2^m − 1)$

ii) Show that if $a$, $m$, $n$ are positive integers with
$m ne n$, then prove that
$$ textgcd(a^2^m+1,a^2^n+1)=begincases
1 & textif atext is even; \
2 & textif atext is odd. \
endcases
$$gcd is greatest common divisor

iii) Deduce ( using the above parts ) that there are infinitely many primes.

I was able to do the first two parts $-$ By factorisation in i), and ii) using i). But I can't think of any approach to prove the third part.

For each $ninmathbb N$, let $p_n$ be a prime factor of $2^2^n+1$. It follows from what you did that $mneq nimplies p_mneq p_n$. Since the function $nmapsto p_n$ is injective, there are infinitely many primes.

Prove for fixed even $a$ no two members of the sequence $a^2^n+1$ have a common factor $>1$, and hence no common prime factor. But each is at least $2$ and hence has some prime factor, and the first $n$ terms have at least $n$ distinct prime factors between them.