The Hardy-Ramanujan formula is a divergent series, but has the remarkable property that if you curtail it at the right spot (something like O(sqrt n) terms) then it gives the answer within epsilon of the integer answer. It's better to use Rademacher's formula, which uses cosh in place of exp and which converges. Proofs are analytic, and not suitable for polite society. The book jacket of the first edition of Hardy's Mathematician's Apology had the numerical check of the calculation of p(200) which MacMahon had calculated using Euler's pentagonal numbers formula. As the answer came out close to an integer, there were lots of nines, in Hardy's very distinctive hand. I've used his nine ever since I first saw it, and it came in handy during WW2 (that's not Winning Ways 2nd edition) when we had to do a lot of deciphering using one-time pads -- subtract in base 10 without carrying. We soon found that it was quickest to use two people, one subtracting, the other writing. One got so fast at subtracting, that speed of writing was the limiting factor, so I devised digits that could be made with a single stroke. Most are obvious. Don't know if I can ASCII the others (start at X): /V <<<<<X >>>> / V V | \ / V V _ \ | <<<<<<<<X V/ \ <<<X/ V V / V <<< / / R. On Mon, 19 May 2003 asimovd@aol.com wrote:
Just saw a play, "Partition", performed in Berkeley by the Aurora Theatre Company, which is all about the interaction between Hardy & Ramanujan.
Thoroughly enjoyed it. I don't know if it will be performed in other cities, but its run in the Bay Area has just ended.
It's nice, the number of math-related plays recently. Alas, I haven't seen either Arcadia or Proof. Has anyone out there seen either of these, and if so, how did you like it?
ALSO: Was moved to check out the extensive MathWorld entry on partitions, at <http://mathworld.wolfram.com/PartitionFunctionP.html>, and was reminded of the mystical asymptotic formula
P(n) ~ exp(pi sqrt(2n/3)) / (4 sqrt(3) n)
discovered by Hardy & Ramanujan. Is anyone familiar enough with a proof of this -- or a heuristic argument -- to post a sketch of it to math-fun?
--Dan