Partitions

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

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(120052058) 12198 c59 Dec 2012 Partitions, ECPP
2 p(100077222) 11136 c59 Nov 2012 Partitions, ECPP
3 p(90048122) 10563 c59 Oct 2012 Partitions, ECPP
4 p(82479677) 10109 c59 Sep 2012 Partitions, ECPP
5 p(82352631) 10101 c56 Jan 2012 Partitions, ECPP
6 p(80036992) 9958 c46 Nov 2011 Partitions, ECPP
7 p(67230446) 9126 c56 Oct 2011 Partitions, ECPP
8 p(60016427) 8622 c46 Sep 2011 Partitions, ECPP
9 p(54534155) 8219 c56 Oct 2011 Partitions, ECPP
10 p(51831641) 8012 c56 Sep 2011 Partitions, ECPP
11 p(50001890) 7869 c46 Aug 2011 Partitions, ECPP
12 p(41197951) 7142 c56 Jul 2011 Partitions, ECPP
13 p(40100918) 7047 c46 Jun 2011 Partitions, ECPP
14 p(39576498) 7000 c56 May 2011 Partitions, ECPP
15 p(30248445) 6119 c46 Apr 2011 Partitions, ECPP
16 p(30245335) 6119 c46 Apr 2011 Partitions, ECPP
17 p(30244992) 6119 c46 Apr 2011 Partitions, ECPP
18 p(30191251) 6113 c46 Apr 2011 Partitions, ECPP
19 p(30158067) 6110 c46 Apr 2011 Partitions, ECPP
20 p(30147428) 6109 c46 Apr 2011 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.