Wouter Meussen writes: << ... 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