My starting point is a nice feature article in the June 18, 2005 issue of Science News: Pieces of Numbers. But since the article, unfortunately, eschews any mathematical notation, it may not be as clear as it should be. (The editors may be to blame for hostility towards mathematical notation.)
I'll try to remedy this problem. Define a function p(n) on the positive integers as follows. Let p(n) be the number of different ways (not counting order) that n can be expressed as a sum of positive integers. For instance, p(3) is 3 because we can write 3 as 1+1+1, 1+2, or simply 3. The values that p(n) takes on (the "range" of the function) are called "partition numbers". We can then refer to p(n) as the nth partition number.
The remarkable Indian mathematician Srinivasa Ramanujan made the first interesting observations about partition numbers, among his very numerous other curious results. He found that starting with p(4), every value of the form p(4+5k) is divisible by 5 for nonnegative integers k. If we use another common mathematical symbolism for divisibility, we can express this fact as p(4+5k) ≡ 0 (mod 5). (In such an expression, we shall assume this means for all integers k ≥ 0.) Ramanujan also found that p(5+7k) ≡ 0 (mod 7), and p(6+11k) ≡ 0 (mod 11). But that's as far as he could go. It doesn't seem to be true that there is any prime q such that p(7+q⋅k) ≡ 0 (mod q).
However, starting in 1968, some other relationships of this sort were found, begining with p(237+17303k) ≡ 0 (mod 13). Note that the starting partition number p(237) skips over a large number of intervening partition numbers, and that the interval between partition numbers (17303) is not the same as the prime divisor. But in 2000 Ken Ono proved, remarkably, that some such relationship does exist for every prime number other than 2 and 3. A little later Ono and Scott Ahlgren proved such relationships exist for all positive powers of primes other than 2 and 3. In other words, for any prime p other than 2 or 3, and any n ≥ 1, there exist integers l and m (that depend on p and n) such that p(l+m⋅k) ≡ 0 (mod pn). There can even be different numbers l and m that work for a given prime. For instance, p(111247+157525693k) ≡ 0 (mod 13).
What is remarkable in all of this is that partition numbers represent additive properties of integers, yet they have striking multiplicative properties in terms of divisibility by prime powers.
The proofs that Ono created used sophisticated number theoretic techniques. But long before 2000 the mathematician-turned-physicist Freeman Dyson conjectured that there might be characteristic numbers of specific partitions of a given number which could be used to explain Ramanujan's results. Dyson suggested that the set of partitions of a given number could be divided into subsets of partitions that have the same characteristic number. If all subsets had the same size and the number of subsets was divisible by the prime number in questions, the result would follow.
It was shown in the 1950s that this is true for the primes 5 and 7, using a characteristic number that Dyson called the "rank" of a partition. In this case, there were 5 and 7 subsets of partitions (respectively) for 4+5k and 5+7k where the subsets were of equal size. Unfortunately, this technique didn't work for 11. Dyson guessed that there might be a different characteristic number which he called the "crank" that might do the trick, but neither he nor anyone else, including Ono, could devise a plausible candidate.
But then this year, Ono's graduate student Karl Mahlburg succeeded where others had failed. This made the proofs of all known results much simpler. However, there was a curious wrinkle. The "crank" that Mahlburg came up with divided the set of partitions of a given number into many subsets, but (except for primes 5, 7, 11), the subsets were not of equal size. Yet, surprisingly, the number of partitions in each subset were divisible be the prime or prime power involved. And so the total number of partitions, which is the sum of the sizes of all the subsets, must also be divisible by the same prime power.
Here are a couple of short articles on Mahlburg's result:
Mathematician Untangles Legendary Problem
Classic maths puzzle cracked at last
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