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?
Top Related