1. The theory of partitions

2. n = n 1 + n 2 + … + n i 7 = 3 + 2 + 2 7 = 4 + 2 + 1

3. 55 1 13 ++++ 5 2+ 3+ 3+ 2+

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

5. p(10) = 42 p(13) = 101 p(22) = 1002 p(33) = 10143 p(100) = 190569292 ≈ 1.9 x 10 8 p(500) = 2300165032574323995027 ≈ 2.3 x 10 21

6. How big is p(n)?

8. (1+x 1 +x 1+1 +x 1+1+1 +…) (1+x 2 +x 2+2 +x 2+2+2 +…) (1+x 3 +x 3+3 +x 3+3+3 +…) (1+x 4 +x 4+4 +x 4+4+4 +…) …

9. p(15) = p(14) + p(13) – p(10) – p(8) + p(3) + p(0) = 135 + 101 – 42 – 22 + 3 + 1 = 176

10. as

11. graph Value of asymptotic formulaValue of p(n)

12. where and

13. 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

14. Congruence properties of p(n)

15. 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)

16. p(48037937k + 112838) ≡ 0 (mod17)

17. If and then

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