Erdős Problem #977

If \(P(m)\) is the greatest prime divisor of \(m\), then is it true that \[\frac{P(2^n-1)}{n}\to \infty\] as \(n\to \infty\)? Schinzel [Sc62] proved that \(P(2^n-1)>2n\) for \(n>12\). In [Er65b] Erdős also asks about \(P(n!+1)\). Stewart [St74b] proved that this conjecture is true if we restrict \(n\) to those integers with \(<\frac{1}{\log 2}\log\log n\) many prime factors. This was proved in the affirmative by Stewart [St13], who proved that \(P(2^n-1)\gg n^{1+\frac{1}{104\log\log n}}\) for all large \(n\).