The theory of partitions

1. The theory of partitions

2. n = n1 + n2 + … + ni 7 = 3 + 2 + 2 7 = 4 + 2 + 1

3. + 3 + 3 + 2 + 2 5 5 5 3 1 + + + + 1

4. 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

5. 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

6. How big is p(n)?

8. (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+…) …

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. Value of asymptotic formula Value of p(n) graph

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(5k + 4) ≡ 0 (mod5) p(11k + 6) ≡ 0 (mod11) p(13k + 7) ≡ 0 (mod13) ? p(13k + 7) ≡ 0 (mod13) p(7k + 5) ≡ 0 (mod7)

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?