OK, I decided to acquiesce to Mathematica's ridiculous demands. (Why should you have to apply AccountingForm[] just to get a number in standard decimal format?) But gripes aside, appended are the numbers N (other than 1237, 1249, 1256, which it seemed to choke on), 1 <= N <= 2000, for which exp(pi*sqrt(N)) is within .01 of an integer. Besides the mysterious number theory explanations for why some are so very close, it's also puzzling why so many more of these are just above an integer than just below one. --Dan ------------------------------------------------------------------------------ In[52]:= g[n_] := Exp[Pi*Sqrt[n]] In[53]:= K[max_] := For[n=1,n<=max,n=n+1,If[N[Abs[FractionalPart[g[n]]-1/2],3] > .49, Print[n," -> ", AccountingForm[N[g[n],70]]], Null]] In[54]:= K[2000] 17 -> 422150.9976756804516223118279935465573938998932276160355506282041005362 18 -> 614551.9928856196354139298276299092637721589143090195364477936836190191 22 -> 2508951.998257424467165529194121687162476410734868206435729660830423741 25 -> 6635623.999341134233266264067099104921836106934150987203247166936354241 37 -> 199148647.9999780465518567665009238753359004336658664318323947056203771 43 -> 884736743.9997774660349066619374620785853768473991271391609175146278345 58 -> 24591257751.99999982221324146957619235526581222761017107146978074727952 59 -> 30197683486.99318226092820317456691819659501730074805447587158906382489 67 -> 147197952743.9999986624542245068292613125786285081833125038167126333713 74 -> 545518122089.9991746788535498566430173362368690907009069646240495704728 103 -> 70292286279654.00194128887588078323607339804118079323350802547781482432 148 -> 39660184000219160.00096667435857524642577260262167893454080881851324518 149 -> 45116546012289599.99183028700036243820106838306889700225231435049418180 163 -> 262537412640768743.9999999999992500725971981856888793538563373369908627 164 -> 296853791705948489.0026726248354647230571999609920977505604998845455156 177 -> 1418556986635586485.996179355249780456304054322609687324822525165865396 205 -> 34268610654606782799.00302588709819877957417711509991955739178349255586 223 -> 236855705574162154847.0034451037730456731334687474109472878887472801644 226 -> 324394960614997599147.0065272185438674490961889863263111309144075035267 232 -> 604729957825300084759.9999921715268564302785946808125512858845316413892 267 -> 19683091854079461001445.99273704076983901658449709887562598444136513736 268 -> 21667237292024856735768.00029203884241295945428349129791564877323635540 326 -> 4309793301730386363005719.996011651626851524869273743200857493473875733 386 -> 639355180631208421212174016.9976698325078231117287843106768301634122745 522 -> 14871070263238043663567627879007.99984872648279481477388379523706422285 566 -> 288099755064053264917867975825573.9938983115610667783462631133127007347 638 -> 28994858898043231996779771804797161.99237293954516048470864023572454327 652 -> 68925893036109279891085639286943768.00000000016373864420923460757232906 719 -> 3842614373539548891490294277805829192.999987249566012187563270183657068 790 -> 223070667213077889794379623183838336437.9920551177281967668330417762103 792 -> 249433117287892229255125388685911710805.9960973230079997367491007072130 928 -> 365698321891389219219142531076638716362775.9982597470174543148220891454 940 -> 677621063891416076248230276783145121158916.0018892548309602322004898534 986 -> 6954830200814801770418837940281460320666108.994649611250605331131122063 1005 -> 17910081680940580833529050259595126076784743.00652422098176402344384390 1169 -> 44555719382988281777368496770130045948309444044.99996080286386846150243 1194 -> 139661526073504116557581973059759277212070858620.0003900603188035532686 1213 -> 330144200785402970319166028643329915082011865217.0064534054200532543561 N::meprec: Internal precision limit $MaxExtraPrecision = 50. reached while evaluating -( 1938766283555706196415301648109962894754207498417 Sqrt[1237] Pi -------------------------------------------------) + E . 2 1245 -> 1384892132296689864688150528414254481739817624464.999423621295089731039 N::meprec: Internal precision limit $MaxExtraPrecision = 50. reached while evaluating -( 3309173472789647250136240543502288628041647543181 Sqrt[1249] Pi -------------------------------------------------) + E . 2 N::meprec: Internal precision limit $MaxExtraPrecision = 50. reached while evaluating -( 4514931544489018513609201179433753739448459339035 2 Sqrt[314] Pi -------------------------------------------------) + E . 2 General::stop: Further output of N::meprec will be suppressed during this calculation. 1257 -> 2359752187588332572926453870820027357203549890449.993034752664193580926 1293 -> 11499177736948672237714068560631910541735301233989.99496136788442126388 1326 -> 48171098188871186483096169726268818132470483648300.99134484794080091307 1332 -> 62382700950473563258623210672143849649476315193453.00092102414610595950 1467 -> 18095625621654510801615355531263454706630064771074975.99999999012369367 1556 -> 659805499174190199819286657618770540166630302112791359.9959424231045979 1850 -> 48310987197300327887464627364483701432184761367510399635913.99985950299 1872 -> 107633344087088750110483164611005972787304802294303445731718.9992468149 1960 -> 2532305471868198233465298022928017175581855191972722719356926.993687927