The theory of partitions

n = n1 + n2 + … + ni

7 = 3 + 2 + 2

7 = 4 + 2 + 1

5 5 113 + + + +

5

2+

3+

3+

2+

p(n) = the number of partitions of n

p(1) = 1 1p(2) = 2 2, 1+1p(3) = 3 3, 2+1, 1+1+1p(4) = 5 4, 3+1, 2+2, 2+1+1, 1+1+1+1p(5) = 7 5, 4+1, 3+2, 3+1+1, 2+2+1,

2+1+1+1, 1+1+1+1+1

p(10) = 42

p(13) = 101

p(22) = 1002

p(33) = 10143

p(100) = 190569292 ≈ 1.9 x 108

p(500) = 2300165032574323995027 ≈ 2.3 x 1021

How big is p(n)?

)1)(1)(1(

11 321 xxx

xnp n

(1+x1+x1+1+x1+1+1+…)(1+x2+x2+2+x2+2+2+…)(1+x3+x3+3+x3+3+3+…) (1+x4+x4+4+x4+4+4+…) …

2)13(1

2)13(1521 11 kknpkknpnpnpnpnp kk

p(15) = p(14) + p(13) – p(10) – p(8) + p(3) + p(0)

= 135 + 101 – 42 – 22 + 3 + 1

= 176

34

32exp

~n

n

np

n as

graph

Value of asymptotic formula Value of p(n)

nx

kk

x

xk

dxdknAnp

1

241

241

32sinh

21

1),(mod

,2exp

khkh

k khsiknihnA

1

1 21

21

,k

i khi

khi

ki

ki

khs

where

and

3 972 998 993 185.896+ 36 282.978

- 87.555+ 5.147+ 1.424+ 0.071+ 0.000

+ 0.0433 972 999 029 388.004

p(200) = 3 972 999 029 388

Congruence properties of p(n)

p(1) 1 p(11) 56 p(21) 792p(2) 2 p(12) 77 p(22) 1002p(3) 3 p(13) 101 p(23) 1255p(4) 5 p(14) 135 p(24) 1575p(5) 7 p(15) 176 p(25) 1958p(6) 11 p(16) 231 p(26) 2436p(7) 15 p(17) 297 p(27) 3010p(8) 22 p(18) 385 p(28) 3718p(9) 30 p(19) 490 p(29) 4565

p(10) 42 p(20) 627 p(30) 5604

p(5k + 4) ≡ 0 (mod5)p(7k + 5) ≡ 0 (mod7)

p(11k + 6) ≡ 0 (mod11)p(13k + 7) ≡ 0 (mod13) ?p(13k + 7) ≡ 0 (mod13)

p(48037937k + 112838) ≡ 0 (mod17)

cba 1175 mod124

cb

akp 1175mod0 22

If and

then

What is the parity of p(n)?

Are there infinitely many integers n for which p(n) is prime?