# Partitions

This page : Definition(s) | Records | References | Related Pages |
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

rankprime digitswhowhencomment
1p(221444161) 16569 c77 Apr 2017 Partitions, ECPP
2p(131328565) 12758 c77 Mar 2017 Partitions, ECPP
3p(130249452) 12705 c85 Oct 2017 Partitions, ECPP
4p(130243561) 12705 c85 Sep 2017 Partitions, ECPP
5p(130242827) 12705 c85 Nov 2017 Partitions, ECPP
6p(130232271) 12705 c85 Aug 2017 Partitions, ECPP
7p(130201087) 12703 c85 Jul 2017 Partitions, ECPP
8p(130168020) 12701 c85 Jun 2017 Partitions, ECPP
9p(130142600) 12700 c85 Apr 2017 Partitions, ECPP
10p(130123073) 12699 c85 Mar 2017 Partitions, ECPP
11p(130086648) 12697 c85 Mar 2017 Partitions, ECPP
12p(130085878) 12697 c85 Feb 2017 Partitions, ECPP
13p(130060601) 12696 c85 Dec 2016 Partitions, ECPP
14p(130000231) 12693 c59 Feb 2016 Partitions, ECPP
15p(122110618) 12302 c77 May 2015 Partitions, ECPP
16p(120052058) 12198 c59 Dec 2012 Partitions, ECPP
17p(120037981) 12197 c59 Apr 2014 Partitions, ECPP
18p(110030755) 11677 c59 Feb 2014 Partitions, ECPP
19p(100115477) 11138 c59 Mar 2016 Partitions, ECPP
20p(100090547) 11137 c59 Nov 2014 Partitions, ECPP

### 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.