The Top Twenty--a Prime Page Collection

Partitions

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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.

(up) 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]).

(up) Record Primes of this Type

rankprime digitswhowhencomment
1p(120052058) 12198 c59 Dec 2012 Partitions, ECPP
2p(120037981) 12197 c59 Apr 2014 Partitions, ECPP
3p(110030755) 11677 c59 Feb 2014 Partitions, ECPP
4p(100077222) 11136 c59 Nov 2012 Partitions, ECPP
5p(100057273) 11135 c59 Jan 2014 Partitions, ECPP
6p(90048122) 10563 c59 Oct 2012 Partitions, ECPP
7p(82479677) 10109 c59 Sep 2012 Partitions, ECPP
8p(82352631) 10101 c56 Jan 2012 Partitions, ECPP
9p(80036992) 9958 c46 Nov 2011 Partitions, ECPP
10p(67230446) 9126 c56 Oct 2011 Partitions, ECPP
11p(60016427) 8622 c46 Sep 2011 Partitions, ECPP
12p(54534155) 8219 c56 Oct 2011 Partitions, ECPP
13p(51983878) 8024 c4 Apr 2014 Partitions, ECPP
14p(51975657) 8023 c4 Apr 2014 Partitions, ECPP
15p(51911300) 8018 c4 Apr 2014 Partitions, ECPP
16p(51873600) 8015 c4 Apr 2014 Partitions, ECPP
17p(51864465) 8015 c4 Apr 2014 Partitions, ECPP
18p(51831641) 8012 c56 Sep 2011 Partitions, ECPP
19p(50001890) 7869 c46 Aug 2011 Partitions, ECPP
20p(41197951) 7142 c56 Jul 2011 Partitions, ECPP

(up) Related Pages

(up) 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.
Chris K. Caldwell © 1996-2014 (all rights reserved)