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Let $p(n)$ denote the number of integer partitions of $n$ for $n\in\mathbb{N}$.

Is it possible to list the cases where $p(n)$ is prime? Are such natural numbers finite (if so how to compute a bound) or can you prove that there are infinitely many $n\in\mathbb{N}$ with $p(n)$ is prime?

I was writing a program computing $p(n)$ when $n$ is given as an input and I just wanted to check the cases where $p(n)$ is prime. For $n\leq 400$ there are $17$ cases : $2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, 188, 212, 216, 302, 366$.

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  • $\begingroup$ I don't know, but there is an explicit formula you might want to start from. See aimath.org/news/partition $\endgroup$
    – AnalysisStudent0414
    Commented Sep 26, 2016 at 19:03
  • $\begingroup$ It seems quite hard for me to understand. Can you explain the relation between the paper and the question? $\endgroup$
    – Levent
    Commented Sep 26, 2016 at 19:06
  • $\begingroup$ If you know an explicit formula for $p(n)$ you might be able to find conditions for $p(n)$ to be prime. Before this breakthrough, the behaviour of the function was often described as random and chaotic, so I doubt there is any simple result about when $p(n)$ is a prime number (and I'm not even sure it means something about $n$). Why are you interested in this problem, by the way? $\endgroup$
    – AnalysisStudent0414
    Commented Sep 26, 2016 at 19:08
  • $\begingroup$ I see. I don't have a particular reason, as I explained I thought about it while I was listing writing a code calculating $p(n)$. $\endgroup$
    – Levent
    Commented Sep 26, 2016 at 19:14
  • $\begingroup$ I see. Well good luck! $\endgroup$
    – AnalysisStudent0414
    Commented Sep 26, 2016 at 19:14

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