At 01:30 PM 12/5/2004, Daniel Asimov wrote:
Fwiw, a related pattern also occurs for the first few numbers exp(sqrt(d)*pi) when the class number of Z[sqrt(-d)] is 1 (i.e., the ring Z[sqrt(-d)] has unique factorization):
exp(sqrt(163)*pi) = 262537412640768743.99999999999925... exp(sqrt( 67)*pi) = 147197952743.9999986... exp(sqrt( 43)*pi) = 884736743.99977... exp(sqrt( 19)*pi) = 88549.77...
Typo there: that last should be 885479.77...
and after this there's no pattern of near-integers at all.
HOLY COW! I just noticed that the first 3 of these numbers end with 743 before the decimal point! This must be a coincidence (right?). But the chance of such a coincidence is only one out of a million.
Some other equations to consider: 640320^3 + 744 = 262537412640768744. 5280^3 + 744 = 147197952744, 960^3 + 744 = 884736744, 96^3 + 744 = 885480, None of these are coincidence. The 744 comes from the q-expansion of the modular function j: q^-1 + 744 + 196884q + 21493760q^2 + 864299970q^3 + ... When q = -exp(-sqrt(19)*pi), j is exactly -96^3; similarly for the other cases. The coefficients of the j function figure in another famous "coincidence"--they are closely related to the degrees of irreducible representations of the Monster group. This surprising fact is what Conway called "Monstrous moonshine". 744 is also the number on my office desk key. Now that just may be a coincidence. -- Fred W. Helenius <fredh@ix.netcom.com> "There are no coincidences in mathematics. None; not even two equals two." -- Professor Gian-Carlo Rota