Now, these expression *seem* to huddle uncomfortably close to integers:
2,806133
3,074844
6,995495
20,986486
79,000346
357,009124
1879,002190
11276,988463
75966,991041
567381,021008
4652071,037121
41534492,955918
401057934,821915
4164175845,053300
46260731383,985200
but, the loss of accuracy towards the end troubles me.
...
>>
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...
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.
Dan Asimov