(See the sibling comment with a "10." in it for context.) 2. Partition numbers. Suppose I give you a positive integer; say 5. How many ways are there to write

(See the sibling comment with a "10." in it for context.) 2. Partition numbers. Suppose I give you a positive integer; say 5. How many ways are there to write it as a sum of positive integers? Seven: 1+1+1+1+1; 1+1+1+2; 1+1+3; 1+2+2; 1+4; 2+3; 5. How do these numbers behave? How fast do they grow? Does anything about them repeat periodically? Back in 1918, Hardy and Ramanujan found an amazing formula, an asymptotic series that you can use to compute p(n) exactly with a manageable amount of computation. (The series doesn't converge, but if you stop adding at the right point you get the right answer!) Rademacher later tweaked it to make it actually convergent. Ramanujan also found some very startling results of the following kind: If n leaves remainder 4 when divided by 5, then p(n) is a multiple of 5. If n leaves remainder 5 when divided by 7, then p(n) is a multiple of 7. If n leaves remainder 6 when divided by 11, then p(n) is a multiple of 11. (Why 4, 5, and 6? Because 24x4 is 1 mod 5, and 24x5 is 1 mod 7, and 24x6 is 1 mod 11. No, this is not meant to be obvious.) Unfortunately, no such thing is true for 13, or in fact for any other prime number. (Ramanujan also found related results for powers of 5, 7, and 11.) But! There are similar things for other primes, and Ono proved that for any prime other than 2 and 3 there are theorems of the form p(ak+b)=0 mod [your prime]. (These differ from Ramanujan's in that we don't get to take a = the prime in question.) And then in 2011 -- this is the result Elwes is referring to -- Ono and Bruinier generalized this further by finding a sort of fractal structure in the partition function. The idea is that there are relations of the form p(q^m0.a0+b0) and p(q^m1.a1+b1) modulo q, where m1<m0, so that you can compute p(q^m0.a0+b0) mod q in terms of p(much smaller things). The number of much smaller things you need depends on q, and it turns out that (0) for q<=3 this stuff doesn't work, (1) for 5<=q<=11 you need no much smaller things so that you get p(...) = 0 mod q, (2) for 13<=q<=31 you need one much smaller thing so that you get p(...) = constant times p(smaller) mod q, etc. This is all sufficiently constructive that Ono can, e.g., exhibit the congruences whose existence he's proved, and use them to calculate partition numbers. I don't have much understanding of how this all works, but it uses an important bit of machinery called modular forms. (Closely related to the stuff I waved my hands vaguely towards when talking about the Langlands program.) These are mathematical functions with a certain kind of "almost-periodic" behaviour, which turn out to be connected to pretty much everything in number theory. Improvements in understanding these could be very valuable elsewhere.