On the largest prime factor of the partition function of n

Abstract

Let p(n) be the function that counts the number of partitions of n. For a positive integer m, let P(m) be the largest prime factor of m. Here, we show that P(p(n)) tends to infinity when n tends to infinity through some set of asymptotic density 1. In fact, we show that the inequality P(p(n))>loglogloglogloglogn holds for almost all positive integers n. Features of the proof are the first term in Rademacher’s formula for p(n), Gowers’ effective version of Szemeredi’s theorem, and a classical lower bound for a nonzero homogeneous linear form in logarithms of algebraic numbers due to Matveev.