Let $f(n)$ be the greatest prime factor of $2^n+1$
Is it true that for any $c>0$ ,there is an integer $n>c$ such that $f(n+1)<f(n)$ ?
This is true if there are infinitely many Fermat primes:
If $n>1$ and $2^{2^n}+1=x^4+1$ is prime,then
$2^{2^n+2}+1=4x^4+1=(1-2x+2x^2)(1+2x+2x^2),\\ f(2^n+2)<1+2x+2x^2<x^4+1=f(2^n).$