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 1

p(2) = 2 2, 1+1

p(3) = 3 3, 2+1, 1+1+1

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

p(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 kk

npkk

npnpnpnpnp kk

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

= 135 + 101 – 42 – 22 + 3 + 1

= 176

34

3

2exp

~n

n

np

n as

graph

Value of asymptotic formula Value of p(n)

nx

kk

x

xk

dx

dknAnp

1

24

1

24

1

3

2sinh

2

1

1),(

mod

,2

exp

kh

khk khsi

k

nihnA

1

1 2

1

2

1,

k

i k

hi

k

hi

k

i

k

ikhs

where

and

3 972 998 993 185.896

+ 36 282.978

- 87.555

+ 5.147

+ 1.424

+ 0.071

+ 0.000

+ 0.043

3 972 999 029 388.004

p(200) = 3 972 999 029 388

Congruence properties of p(n)

p(1) 1 p(11) 56 p(21) 792

p(2) 2 p(12) 77 p(22) 1002

p(3) 3 p(13) 101 p(23) 1255

p(4) 5 p(14) 135 p(24) 1575

p(5) 7 p(15) 176 p(25) 1958

p(6) 11 p(16) 231 p(26) 2436

p(7) 15 p(17) 297 p(27) 3010

p(8) 22 p(18) 385 p(28) 3718

p(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 2

2

If and

then

What is the parity of p(n)?

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