Partitions

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The Prime Pages

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The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

Definitions and Notes

The number of (unrestrict) partiton of n, denoted p(n), s the number of ways of writing the integer n as a sum of positive integers. For example,

5 = 5
  = 4 + 1
  = 3 + 2
  = 3 + 1 + 1
  = 2 + 2 + 1
  = 2 + 1 + 1 + 1
  = 1 + 1 + 1 + 1 + 1

so p(5) = 7. The value of p(n) for n = 1, 2, ..., is 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...

How often is p(n) prime? Weisstein states that Leibniz noticed that p(n) is prime for n = 2, 3, 4, 5, 6, but not 7. p(n) is prime for 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ...

Kolberg [Kolberg1959] proved that there are infinitely many even and odd values of p(n), so it is composite infintely often, and congruence properties of p(n) have been very repeatedly studied (e.g., [Ramanujan1919], [Ramanujan1921], [Ono2000] and [Ahlgren2001]).

Record Primes of this Type

rank prime digits who when comment

1 p(1289844341) 40000 c84 Feb 2020 Partitions, ECPP
2 p(221444161) 16569 c77 Apr 2017 Partitions, ECPP
3 p(131328565) 12758 c77 Mar 2017 Partitions, ECPP
4 p(130249452) 12705 c85 Oct 2017 Partitions, ECPP
5 p(130243561) 12705 c85 Sep 2017 Partitions, ECPP
6 p(130242827) 12705 c85 Nov 2017 Partitions, ECPP
7 p(130232271) 12705 c85 Aug 2017 Partitions, ECPP
8 p(130201087) 12703 c85 Jul 2017 Partitions, ECPP
9 p(130168020) 12701 c85 Jun 2017 Partitions, ECPP
10 p(130142600) 12700 c85 Apr 2017 Partitions, ECPP
11 p(130123073) 12699 c85 Mar 2017 Partitions, ECPP
12 p(130086648) 12697 c85 Mar 2017 Partitions, ECPP
13 p(130085878) 12697 c85 Feb 2017 Partitions, ECPP
14 p(130060601) 12696 c85 Dec 2016 Partitions, ECPP
15 p(130000231) 12693 c59 Feb 2016 Partitions, ECPP
16 p(122110618) 12302 c77 May 2015 Partitions, ECPP
17 p(120052058) 12198 c59 Dec 2012 Partitions, ECPP
18 p(120037981) 12197 c59 Apr 2014 Partitions, ECPP
19 p(110030755) 11677 c59 Feb 2014 Partitions, ECPP
20 p(100115477) 11138 c59 Mar 2016 Partitions, ECPP

MathWorld's Partition Function P Congruences
Sloane's Sequence A046063 prime partition indices
Sloane's Sequence A049575 prime partition indices

References

AB2003
S. Ahlgren and M. Boylan, "Arithmetic properties of the partition function," Invent. Math., 153:3 (2003) 487--502. MR2000466
Ahlgren2000
S. Ahlgren, "Distribution of the partition function modulo composite integers M," Math. Ann., 318:4 (2000) 795--803. MR1802511
HW79
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford University Press, 1979. ISBN 0198531702. MR 81i:10002 (Annotation available)
Kolberg1959
O. Kolberg, "Note on the parity of the partition function," Math. Scand., 7 (1959) 377--378. MR0117213
Ono2000
K. Ono, "Distribution of the partition function modulo m," Ann. of Math. (2), 151:1 (2000) 293--307. MR1745012
Ramanujan1919
S. Ramanujan, "Congruence properties of partitions," Proc. London Math. Soc., 19 (1919) 207--210.
Ramanujan1921
S. Ramanujan, "Congruence properties of partitions," Math. Z., 9 (1921) 147--153.