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.

This page is about one of those forms.

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(1000007396) 35219 E4 Aug 2022 Partitions, ECPP 3 p(398256632) 22223 E1 Aug 2022 Partitions, ECPP 4 p(355646102) 21000 E1 Jul 2022 Partitions, ECPP 5 p(350199893) 20838 E7 Jul 2022 Partitions, ECPP 6 p(322610098) 20000 E1 Jun 2022 Partitions, ECPP 7 p(221444161) 16569 c77 Apr 2017 Partitions, ECPP 8 p(158931035) 14036 E1 May 2026 Partitions, ECPP 9 p(158917403) 14035 E1 May 2026 Partitions, ECPP 10 p(158898550) 14034 E1 May 2026 Partitions, ECPP 11 p(158788742) 14029 E1 May 2026 Partitions, ECPP 12 p(158635927) 14022 E1 May 2026 Partitions, ECPP 13 p(158569348) 14020 E1 May 2026 Partitions, ECPP 14 p(158507386) 14017 E1 May 2026 Partitions, ECPP 15 p(158447043) 14014 E1 May 2026 Partitions, ECPP 16 p(158430002) 14013 E1 May 2026 Partitions, ECPP 17 p(158381443) 14011 E1 May 2026 Partitions, ECPP 18 p(158375386) 14011 E1 Jul 2022 Partitions, ECPP 19 p(158295265) 14007 E1 Jul 2022 Partitions, ECPP 20 p(158221457) 14004 E1 Jul 2022 Partitions, ECPP

Related Pages

• 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.