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 $n\to \infty$ ?

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The settling result is recorded on the claim page Stewart 2013.

Er65b: Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
La21: L. Lai, On the largest prime divisor of $n!+1$ . arXiv:2103.14894 (2021).
MuWo02: Murty, Ram and Wong, Siman, The $ABC$ conjecture and prime divisors of the Lucas and Lehmer sequences. (2002), 43-54.
Sc62: Schinzel, A., On primitive prime factors of $a\sp{n}-b\sp{n}$ . Proc. Cambridge Philos. Soc. (1962), 555-562.
St13: Stewart, Cameron L., On divisors of Lucas and Lehmer numbers. Acta Math. (2013), 291-314.
St74b: Stewart, C. L., The greatest prime factor of $a\sp{n}-b\sp{n}$ . Acta Arith. (1974/75), 427-433.