The theory of partitions

n = n 1 + n 2 + … + n i 7 = =

p(n) = the number of partitions of n p(1) = 11 p(2) = 22, 1+1 p(3) = 33, 2+1, p(4) = 54, 3+1, 2+2, 2+1+1, p(5) = 75, 4+1, 3+2, 3+1+1, 2+2+1, ,

p(10) = 42 p(13) = 101 p(22) = 1002 p(33) = p(100) = ≈ 1.9 x 10 8 p(500) = ≈ 2.3 x 10 21

How big is p(n)?

(1+x 1 +x 1+1 +x …) (1+x 2 +x 2+2 +x …) (1+x 3 +x 3+3 +x …) (1+x 4 +x 4+4 +x …) …

p(15) = p(14) + p(13) – p(10) – p(8) + p(3) + p(0) = – 42 – = 176

as

graph Value of asymptotic formulaValue of p(n)

where and

p(200) =

Congruence properties of p(n)

p(1)1p(11)56p(21)792 p(2)2p(12)77p(22)1002 p(3)3p(13)101p(23)1255 p(4)5p(14)135p(24)1575 p(5)7p(15)176p(25)1958 p(6)11p(16)231p(26)2436 p(7)15p(17)297p(27)3010 p(8)22p(18)385p(28)3718 p(9)30p(19)490p(29)4565 p(10)42p(20)627p(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( k ) ≡ 0 (mod17)

If and then

What is the parity of p(n)? Are there infinitely many integers n for which p(n) is prime?