# Partitions

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.

### 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(1289844341) 40000 c84 Feb 2020 Partitions, ECPP
2p(1000007396) 35219 E4 Aug 2022 Partitions, ECPP
3p(398256632) 22223 E1 Aug 2022 Partitions, ECPP
4p(355646102) 21000 E1 Jul 2022 Partitions, ECPP
5p(350199893) 20838 E7 Jul 2022 Partitions, ECPP
6p(322610098) 20000 E1 Jun 2022 Partitions, ECPP
7p(221444161) 16569 c77 Apr 2017 Partitions, ECPP
8p(158375386) 14011 E1 Jul 2022 Partitions, ECPP
9p(158295265) 14007 E1 Jul 2022 Partitions, ECPP
10p(158221457) 14004 E1 Jul 2022 Partitions, ECPP
11p(141528106) 13244 E6 Jul 2022 Partitions, ECPP
12p(141513546) 13244 E6 Jul 2022 Partitions, ECPP
13p(141512238) 13244 E6 Jul 2022 Partitions, ECPP
14p(141255053) 13232 E6 Jul 2022 Partitions, ECPP
15p(141150528) 13227 E6 Jul 2022 Partitions, ECPP
16p(141112026) 13225 E6 Jul 2022 Partitions, ECPP
17p(141111278) 13225 E6 Jul 2022 Partitions, ECPP
18p(140859260) 13213 E6 Jul 2022 Partitions, ECPP
19p(140807155) 13211 E6 Jul 2022 Partitions, ECPP
20p(140791396) 13210 E6 Jul 2022 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.