The Ramanujan Journal

, Volume 28, Issue 3, pp 423–434 | Cite as

On the largest prime factor of the partition function of n



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 Szemerédi’s theorem, and a classical lower bound for a nonzero homogeneous linear form in logarithms of algebraic numbers due to Matveev.


Partition function Largest prime factor 

Mathematics Subject Classification (2000)

11P99 11A05 


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autonoma de MéxicoMoreliaMéxico
  2. 2.The John Knopfmacher Centre for Applicable Analysis and Number TheoryUniversity of the WitwatersrandJohannesburgSouth Africa

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