Exp(Pi*Sqrt(n)) Page

This table lists values of Exp(Pi*Sqrt(n)), for some selected values of n up to 1000. Some of these values are very close to integers. A prize will be awarded to anyone who can either convincingly argue that this is coincidence, or who can explain why this is so in terms intelligible to an intelligent college senior. Something else that might help to lift the veil on this mystery would be a predictor: for a given value of n, is Exp(Pi*Sqrt(n)) close to an integer?

-1                                           -1.0000000000000
0                                             1.0000000000000
6                                          2197.9908695437080
17                                       422150.9976756804516
18                                       614551.9928856196354
22                                      2508951.9982574244671
25                                      6635623.9993411342332
37                                    199148647.9999780465518
43                                    884736743.9997774660349
58                                  24591257751.9999998222132
59                                  30197683486.9931822609282
67                                 147197952743.9999986624542
74                                 545518122089.9991746788535
103                              70292286279654.0019412888758
148                           39660184000219160.0009666743585
149                           45116546012289599.9918302870003
163                          262537412640768743.9999999999992
164                          296853791705948489.0026726248354
177                         1418556986635586485.9961793552497
205                        34268610654606782799.0030258870981
223                       236855705574162154847.0034451037730
226                       324394960614997599147.0065272185438
232                       604729957825300084759.9999921715268
267                     19683091854079461001445.9927370407698
268                     21667237292024856735768.0002920388424
326                   4309793301730386363005719.9960116516268
359                  70997279226412702087506048.0094309359706
386                 639355180631208421212174016.9976698325078
522            14871070263238043663567627879007.9998487264827
566           288099755064053264917867975825573.9938983115610
630         17602513749954237250474851101885772.0095551338274
638         28994858898043231996779771804797161.9923729395451
652         68925893036109279891085639286943768.0000000001637
719       3842614373539548891490294277805829192.9999872495660
790     223070667213077889794379623183838336437.9920551177281
792     249433117287892229255125388685911710805.9960973230079
928  365698321891389219219142531076638716362775.9982597470174
940  677621063891416076248230276783145121158916.0018892548309
986 6954830200814801770418837940281460320666108.9946496112506

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