Partitions

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The Prime 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

rank prime digits who when comment

1 p(29099391) 6002 c39 Feb 2010 Partitions, ECPP
2 p(25235715) 5588 c46 Sep 2009 Partitions, ECPP
3 p(25102542) 5574 c39 May 2009 Partitions, ECPP
4 p(24512858) 5508 c42 Aug 2007 Partitions, ECPP
5 p(24503300) 5507 c42 May 2007 Partitions, ECPP
6 p(23028252) 5338 c42 Dec 2008 Partitions, ECPP
7 p(23010067) 5336 c42 Oct 2007 Partitions, ECPP
8 p(22857207) 5318 c46 Aug 2009 Partitions, ECPP
9 p(22810361) 5313 c46 Oct 2009 Partitions, ECPP
10 p(22312025) 5254 c39 Mar 2007 Partitions, ECPP
11 p(20186952) 4998 c46 Aug 2009 Partitions, ECPP
12 p(17819598) 4695 c46 Aug 2009 Partitions, ECPP
13 p(17120312) 4602 c39 May 2007 Partitions, ECPP
14 p(17120303) 4602 c39 Jun 2007 Partitions, ECPP
15 p(16102957) 4463 c46 Jul 2009 Partitions, ECPP
16 p(16026516) 4452 c39 Oct 2006 Partitions, ECPP
17 p(15502228) 4379 c46 Jul 2009 Partitions, ECPP
18 p(15446832) 4371 c8 Sep 2006 Partitions, ECPP
19 p(15432340) 4369 c8 Oct 2006 Partitions, ECPP
20 p(15421217) 4367 c8 Nov 2006 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.