NJA Sloane: There are actually three sequences in the OEIS that refer to the numbers in the article that Warren mentions: A060041 and A076912, which coincide for 9 terms but probably disagree at the 10th term, and A199878, which is a published but incorrect version. Neil On Tue, Apr 17, 2012 at 3:38 PM, Warren Smith <warren.wds at gmail.com> wrote:
http://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00875-2/S0273-0979...
describes a remarkable conjectured formula (arising from generating functions ad series reversions) for the number of rational curves of degree d on a generic "quintic threefold." The formula gives the right answers for the first 9 cases. However, it is not even clear that the numbers it outputs, are always integers, although hundreds have been computed and always came out integer.
--My point was that you don't need to know anything about superstring theory, and you don't need to know anything about algebraic geometry. All you need to know is this paper gives some generating function formulas, and you can try to prove or disprove the coefficients in that generating function are always integer. 5, 2875, 609250, 317206375, 242467530000, 229305888887625, 248249742118022000, ... If you do, you've accomplished something important. How hard can it be to prove or disprove they always are integer????!!?! If Sloane is correct that A060041 and A076912 disagree at the next (uncalculated) term, I gather from the rumor mill that that too would seem very important, it would seem to be a refutation of a major article of faith in superstring theory; but you won't be able to settle that unless you know a lot more than I do. The comparison was made between this and the "monstrous moonshine" conjectures. In the MM conjecture JH Conway (and also some other people independently) noticed some numbers arising in series expansions of elliptic functions... that the same numbers were happening as the number of dimensions needed in matrix representations of the "monster" largest sporadic simple group. This was "clearly" a ridiculously lucky coincidence, since there was no was no way in hell some simple little elliptic functions could possibly know anything about the Monster. However, upon trying a slightly harder, Conway & people found more such numbers, amounting to about 24 digits worth of lucky agreements! http://en.wikipedia.org/wiki/Monstrous_moonshine Eventually Richard Borcherds, a former student of Conway's, was able more or less to understand why monstrous moonshine was happening and got awarded the Fields medal. I personally never understood it. Explanation also has some vague connections to string theory. In the present case, "obviously" superstring theory has nothing whatever to do with counting rational curves on generic rational algebraic surfaces. But... tremendous numerical coincidences, and some vague understanding via physical intuition and so forth. At present, as far as I understand, there is some partial progress but still this all is far from fully understood. Tito Piezas has got his own interesting Moonshine writeup and observations, which by the way seem to me on shallow examination to go beyond what Conway noticed and what Borcherds explained (and they also are a lot easier to read than certain other Moonshine papers...) see http://sites.google.com/site/tpiezas/0022 http://sites.google.com/site/tpiezas/0023 http://sites.google.com/site/tpiezas/0026 http://sites.google.com/site/tpiezas/0025 http://sites.google.com/site/tpiezas/0029 http://sites.google.com/site/tpiezas/0027 http://www.oocities.org/titus_piezas/Pi_formulas1.pdf [the last observed "coincidental" connection between the "baby monster" and Weber and Ramanujan functions] and I guarantee your mind will boggle in amazement if you look at this stuff. -- Warren D. Smith http://RangeVoting.org