Dear Simon Plouffe, I have made a new e computation (with verification) : 1,250,000,000 digits That's nearly all I could do since I had only 5 G. of disk space free and I needed 4 G. of disk space for the computation. (Notice that a previous computation of e to 1,7 billion digits was made several months ago by Patrick Demichel, but he never published the result because no verification was made. It appears that its first 1.25 billion digits are OK.) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Computation of e : ================= Number of digits computed (and verified) : 1,250,000,000 Who : Xavier Gourdon When : 1999, Nov, 14 (verification : 1999, Nov 21) Timing : The computation took 39h and 52 minutes on a IBM ThinkPad (PII 350 Mhz, 320 Mo of memory). The verification took 40 hours 25 minutes on the same machine. Technique used : ---------------- The classical exponential series was used e = sum_{n>=0} 1/n! together with a binary splitting process (see http://xavier.gourdon.free.fr/Constants/Algorithms/splitting.html). The multiplication was done with a FFT approach, and a NTT modulo several primes for huge multiplications, with disk memory use. The corresponding library is the same as the one used by the program PiFast (see http://xavier.gourdon.free.fr/Constants/PiProgram/pifast.html). The verification was made with the formula e = 1/[ sum_{n>=0} (-1)^n/n! ]. Again, a binary splitting method was used. Information about the computation : --------------------------------- The physical memory used during the computation was 102 Mo. The system used the other physical memory as disk cache (the machine has 320 Mo of physical memory). 40 % of the time was spent in disk memory access. 4 Go of disk memory was necessary during the computation. Remark: ====== A previous computation, several months ago, was done by Patrick Demichel. He computed 1.7 billions digits, but no verification of his digits was done. His first 1.250 billions digits agree with my computation and verification. Digits : ======= The following data contains 100 digits every 10 million digits : 7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 : 100 1648408993 2201833912 4927792742 7019674074 7544061929 7700299155 1759476095 7317367144 9657243259 9234884647 : 10000100 7870549821 1624155516 7539648061 8596466899 5036858932 2734928783 2609451953 9591546510 5816698812 9809991957 : 20000100 3639127138 8678145578 0492986819 9199208623 5260301986 6123709629 1765523146 1331615559 0186676649 8934315952 : 30000100 7642986885 5491921770 1995086911 3804430687 7732057566 3602744486 5885476578 3051790029 9732287668 8848992358 : 40000100 0902126036 5728926873 7384327838 0603458744 9954716613 9991864034 4124342791 3521558827 6408501883 4179717633 : 50000100 8180252523 3733480144 7491191287 3132370589 5020156634 3523019841 4566654183 9942471923 7730720983 3042212752 : 60000100 8009533098 4823976467 9238909438 6584582073 5411362902 2521985826 1508229720 8671780536 1762345597 9783632541 : 70000100 4625908469 5933785932 8504264288 4421767119 3499216643 7071208924 2808165102 6886479900 9346237374 9231879383 : 80000100 2124247796 9942966299 1632098584 8506266697 7968326843 2807587697 0320155374 5007237637 2504452739 6910427409 : 90000100 5522594276 6610700642 7736104696 2720701530 9881373952 6435030835 0173451780 6349331786 1982233369 2443242626 : 100000100 7449405490 9406316310 1874499085 0642585848 7590698531 9578748729 1806406626 9075465691 3598765713 0800011305 : 110000100 7092356764 3282533276 4956163566 4072139884 8221079066 8864760281 4517665234 8416405725 9138138307 8621495053 : 120000100 6604815536 3362263521 9420849724 8991291872 5663854671 1115939881 5665992170 0700889429 8524215849 6257186861 : 130000100 0271148979 9053312057 3390196578 3355589378 7062522534 7230204805 4837448414 7259286989 0114382218 5411097624 : 140000100 5949905429 2605520209 0161526176 9583712518 0852814995 3278366288 2626414899 5846410326 9285157752 5953325489 : 150000100 3261234781 3826527063 9754242519 7825979653 7436041689 3060816496 2512396092 8959848262 6643022803 1955413639 : 160000100 4980091853 0270785425 7774091594 1690502754 7436536397 5653743523 7962264143 4208178996 5716487696 4485066749 : 170000100 9337305665 8067721291 0854965621 0366449443 0289305880 6781973508 3396904383 8232213575 2191648471 9017281618 : 180000100 6838456786 7893811091 6489274971 8361794854 9702788229 1720247178 1816599462 0317056257 8737620691 1796458882 : 190000100 2953387943 0161328180 3820863535 6622078019 3929957103 4228664539 0640423904 9100178277 1898859264 9631317483 : 200000100 4887837737 5388819437 4894520468 4683017107 5253295064 7303120928 0182939947 1174790366 3891194575 1447423824 : 210000100 0850261041 8417679595 2584796563 5696283312 0322503099 8305037750 9564637274 0590404669 0045584137 1571998020 : 220000100 8902781668 7899369721 2245728449 5758361346 2616579061 8019262209 9382064553 5756934914 0367812905 7907223113 : 230000100 1959806498 8188712084 8968395940 6547497342 2449489174 7401285736 4090752204 5970491233 8755149399 5697274718 : 240000100 0464919373 8709405816 0751254224 4012177293 5398800602 9899570529 4608000473 2013631614 4728572408 7943245557 : 250000100 6504790315 9342364715 6104490271 0192388024 6003090524 7200335745 0637222632 6008050873 7208734258 0029645032 : 260000100 8329384710 3750324027 3964575557 0014103614 4815821078 8890240482 0928222769 2799994233 7719507467 8394774943 : 270000100 6223912800 3114663452 1188663122 4187806880 7442462485 1580067426 8089022227 6760965954 5118664900 1631756448 : 280000100 6410113384 2518582451 2848979184 1572370855 5853529293 7734837675 9375904487 0918827998 6892258244 8446212763 : 290000100 3594170465 3624947851 6842680759 1976188050 1256273675 6830498223 0694863196 5188070061 6963415907 0592486922 : 300000100 0170430215 7909787950 6664562161 2450154513 7342515237 9447344157 4018721784 3862851139 6369716974 5810208356 : 310000100 9461445452 7497804436 8756579971 3430978563 8541677590 4472743983 4428932507 8890427658 5767050909 4508598449 : 320000100 5393575903 1342690329 9882578865 1685497394 7902161090 5306766198 6512183941 6992172396 1138059608 1500955011 : 330000100 0373065305 4471118854 7983449944 1167624387 4961427698 4651290157 9685765960 2986252714 4740310261 7948813318 : 340000100 3636316923 1458880474 5787933947 0792020849 9662687592 6660189843 4046534226 1032122627 8843146693 2350832619 : 350000100 7149242233 9250096654 6069957551 3523146037 9192242454 0748847589 6652114279 3562101533 9009035195 9478532616 : 360000100 8653935581 6579334106 6050503552 8112027307 9175409486 1921839755 6053146792 5780605085 5926679896 2458118642 : 370000100 1882779622 2980816066 9405116026 8259564517 8453117013 5531157406 2939740442 7920495235 0588714321 5520250921 : 380000100 8868475347 1273354802 1156055573 4072627318 4218747981 4277074086 9802510001 3159329617 6705296335 4490249289 : 390000100 1604568580 1318413877 8776641830 8892824013 8721047676 3232131025 8752630691 3214296132 2002651396 4399777139 : 400000100 4874026307 5120239739 7566026264 6577704280 7923815917 9160545175 1395643632 9637840331 2656200422 8033270324 : 410000100 1016606690 7132547525 2991034116 1494260625 6789456338 5432234681 1519634259 5392691590 8024389116 9623684041 : 420000100 2379922637 4301352732 9538581532 7747971909 8958902406 1081538551 4978726377 8408421867 8605248400 9523506411 : 430000100 2840383872 1857395814 6749871948 0418195732 3875360601 4112576305 9184801032 3580262606 4888390347 0357531939 : 440000100 7850572581 6417298176 0082207981 2056211831 9339311915 8019523759 3544281602 8964567472 8444343050 0251418824 : 450000100 0693919776 5391510431 2797436735 4358853071 0525171087 1260115650 8616093772 4059780783 3517426271 0148102906 : 460000100 6600484878 6738482334 1198520428 7236569393 9456117508 7086445633 5155683521 7447125082 2826817024 3180064885 : 470000100 4233063248 6222675665 6212973517 0718389836 9622550316 9360596041 1058194294 4906913134 1700694125 4420569636 : 480000100 4663814430 6107690951 3444917029 8509279052 7459015574 0115424579 8645082492 5074848201 3273705991 2575618751 : 490000100 6410249120 6886300892 7327592489 8984936579 0200498352 9001492423 2770321830 0985286342 8054485265 4692013108 : 500000100 7529594681 3961453272 0346749298 4789549860 2805681118 2804122493 2947830857 2113028728 0875640249 8020676090 : 510000100 4813332539 3513582630 2388572169 4424238105 2068036313 5092009008 3155819528 9696010507 9528404380 5809564117 : 520000100 5602016812 9487474711 4790078113 2462109878 1221730616 1866893318 4174311589 4300220517 4483799324 0091247600 : 530000100 9607382169 4824497855 2632637331 0323656190 0421555640 8793525890 7986443048 3535072275 2598327387 7139454401 : 540000100 3580105252 0048648136 8462273117 3039798089 6720103787 1178812276 2443447450 5254254535 9699853589 9738189509 : 550000100 7753521259 3709686702 5281593655 0172042358 1661346307 8590910178 9781813811 4821535150 9306265917 3273666730 : 560000100 1382130357 2859297444 5316082286 9970897658 8185429434 6125933898 3923022287 1209202395 4754578809 2955914063 : 570000100 1294349318 9360928736 0701889961 6323924025 9341382860 4852909192 8365472127 7663834165 3113872011 9254592065 : 580000100 6878374915 0977398033 8884141877 5418243096 4420544298 4068283000 0465741088 5447923759 6754435634 8337557162 : 590000100 9891386327 3547320013 4299876746 6607876666 8065412784 9548282669 7093170354 9343572886 7648844605 7391618499 : 600000100 6115058423 4941835903 7246900776 9151038979 2136627249 3225532847 0138770888 6868065130 8851127533 4174414008 : 610000100 3752046481 3823165129 6549987298 6851111652 4016393043 1792761926 0822628181 9698490437 5788247222 8244481907 : 620000100 5005855581 6825489196 7609982325 8656269785 5304778916 2072318884 3590819064 8625487456 7433399434 7370149113 : 630000100 1957613243 3464374226 5299615194 5815462437 4469148080 5134126796 7061861838 9968780398 6273808995 5496464399 : 640000100 5952912685 2286667641 9696937414 0356720742 6766520045 4552530422 5579460011 7164210538 6767468148 0142136810 : 650000100 7225296394 3621287734 6093058353 9064247981 9637847871 9627524466 1441582230 2621240687 4591261455 3090865427 : 660000100 5828291148 8163613457 5818076886 1681401257 0897483619 3568071714 1203551148 2573385718 6059953891 8779989606 : 670000100 2571034084 5838062047 2776276678 3741751687 5292004074 7278145920 3935087637 6896333067 7930896841 3808114794 : 680000100 8780159359 6560040816 7898152979 1630829337 7278090535 9027344953 1208964461 0372998494 6464845860 5261225098 : 690000100 1141980565 4249937159 3070139265 2186178202 2107783514 2851824399 4458939913 8989183593 2837818426 9531835069 : 700000100 1563361181 3014822348 9697271313 7763089677 6941485660 9359242275 1469338119 7617819071 1883115212 7289181794 : 710000100 7703691348 6896621209 5846510325 4593634084 5589292315 4476065580 9965795152 5770489435 8507926489 5196070948 : 720000100 7054923680 3677287824 4669422185 9939256881 7060051957 8118008116 7817939368 9169971084 6363288234 7014887482 : 730000100 0746944878 4221738868 4505555886 7660410425 8348827433 6366137187 0701988292 9112352733 7189448444 0884528636 : 740000100 2528720041 7197152240 5093362713 7055075029 9452799704 6272114854 0621221940 0541630278 3806324228 8336869772 : 750000100 3413277042 2093562962 1340806795 5656527241 8796542259 8148218100 3605932116 9528923534 1679631939 1085384159 : 760000100 3284426168 1037790696 5240242714 3955563970 2624531783 0516104870 4897127912 9586203821 5882991756 1465952232 : 770000100 5186341423 7560324143 8481083495 0149884186 1755721979 1722371140 4733919530 2472478417 1489117547 4095140791 : 780000100 5590774699 3052847260 8348743307 0205741891 4786658608 2040113202 5743133423 2520556178 5058552391 7069509854 : 790000100 0578022700 0872108167 1492658524 4350099801 7697164561 5656271219 8574342925 0223546811 2771715188 3236185087 : 800000100 8512444521 6539855322 8238116590 9972589266 8100471103 4255450385 6217734391 0144341712 8832963487 3206466245 : 810000100 4151085154 1127481147 9411126257 1766503639 3666207663 8625500424 7364240169 7390187762 2266545079 0004653892 : 820000100 9052269699 5049564536 8803161052 1378340213 7440495537 4645375806 5815133482 5902152269 7541232855 5973674161 : 830000100 6065941839 4888304528 3404454376 5214202769 5485314570 4803644530 9918238854 6366543022 5844730470 6814450274 : 840000100 6486371553 6318847998 7162354993 8616810052 3605489745 7126990079 0380133454 5961955122 3776103346 9917786776 : 850000100 6605617834 1685093169 8000779723 1123926292 3846898102 4374907721 2083637819 8526507523 9678568045 3337736406 : 860000100 0140298176 9726096407 4650185329 0076522178 0860506864 1410944229 6908345643 3042472237 6576096554 0895346928 : 870000100 6602434359 7377082303 4072403603 6218735798 7262390221 0796360960 8511002481 9550948049 4242079924 4825776389 : 880000100 3508203527 3533134141 9130401510 5728057153 6720406657 6910750174 5385897150 2240418549 8321251404 5401316711 : 890000100 8716409493 1615888255 0931394031 1229898426 9656513504 3805574727 9509095332 3123784256 8368096862 3847575138 : 900000100 5291103165 9871411933 9275554487 6315725803 3488031976 3153761332 6622616686 6116421914 9740174717 0564822286 : 910000100 3908148975 6305112941 0178058820 3255890993 1219053411 6878487494 0872509612 5511371121 0409722669 6738156542 : 920000100 3608886142 7282250820 0282221914 0095209271 6569340659 0212921040 6219555577 5909255905 3431973572 6967771460 : 930000100 9358015502 5094285360 8589011615 9330845968 8804434267 1944171595 2922264802 7404879775 7436266519 6627276333 : 940000100 5945395429 7208867561 4123826841 7080746073 9620223861 5292326531 5248698718 7738107325 3081178327 2135828241 : 950000100 1014068651 7571873750 3256945292 2571832111 3391949829 0837053091 2526345648 9511479349 0646976187 1027477255 : 960000100 5560719939 5943529647 5188964062 0418398590 4740580052 3667244084 0891306851 0762186574 2769017013 5637971610 : 970000100 6584992842 8820565216 3766916627 0728256752 1088226321 9238144706 2406556796 1493960005 9713434200 5986980311 : 980000100 3729344411 1441148551 8592373395 5793954860 0817816645 2133011626 0563280718 6100423477 5198311802 0182686724 : 990000100 6159331468 1378261421 2881714180 5062392118 5418287056 4612864696 5417548539 2969391229 2309832507 6553436920 : 1000000100 9831774467 7566337552 3465524050 6273342407 1730621730 3645135056 5912892343 4096853244 2498050084 1877841894 : 1010000100 8164054124 9864961184 0408454188 2476478666 2979292577 1437794614 2214896825 5149616101 3763372244 1583988085 : 1020000100 6516236951 1133610112 1432604136 3428675844 3979299564 8051372550 4094206260 2209634526 9249562646 4619507119 : 1030000100 5932338001 0640715855 8379885078 2430274998 4176001748 7995113335 0347627061 1214120665 1508334091 6254385497 : 1040000100 2100989582 6200726631 6773445146 3821239004 1246120178 3168384507 2285436913 0730090925 3944631501 3853502497 : 1050000100 7717538310 2920102179 6231702350 8269368981 6339362202 4341539710 1277360171 1774203011 6324950237 1071535055 : 1060000100 0163723237 7562792988 8774516717 3246844743 1118895088 9368901866 3684111195 1014701385 2968796305 4841589720 : 1070000100 5212020935 4231202051 5386289808 7924021399 5242838163 3785779936 1665826182 4750635279 3304353776 4245138165 : 1080000100 9764762422 8460507078 3413905810 2047758354 7597376290 6449914131 6252343831 6850263177 2280004094 7963855615 : 1090000100 6340109869 4627837748 0077851133 6471999156 9820545293 6189535902 2023905458 3215495746 4368272600 1705822768 : 1100000100 2750408614 0981582633 1619780116 1802987896 8300290636 4776098152 7369847676 8025160280 9091353048 5044384398 : 1110000100 9689323402 0662135172 7085547097 6489629153 2553760313 8774746411 2940294697 7142017317 1275134156 6272521912 : 1120000100 8361071478 9686324810 8049360338 2922283958 9134104202 3044997052 6876198712 7478582333 0967068235 5562225889 : 1130000100 2075646401 3253976995 6694023527 8314230183 7583451721 0211717092 9399664982 1703105411 5454378058 9376057053 : 1140000100 7522955729 6141233061 1822302293 4683674274 2984199888 0186744305 1776757052 0163583436 2704923450 6645733786 : 1150000100 0767882914 5764155307 2244966808 1161602710 3447672429 1190946037 4373376987 5427387552 1616205356 1511335010 : 1160000100 4799835235 5665845078 7590652386 6336958525 3158168074 2105968211 6318544556 1533791957 3907310303 5803084371 : 1170000100 6022142654 6886478032 9624088811 8578934412 3075908878 0461341291 9710300561 6455717934 8477229359 0794718010 : 1180000100 3542485176 0893717084 7452651525 2919723290 7721130753 3513199997 0197380826 6487315837 6890788407 7436195174 : 1190000100 8529779072 3327164031 8235719064 5145651223 2777970535 3574502555 8955834848 4043520466 3477733747 6932455847 : 1200000100 2727252394 6947515266 2756730892 7624875955 0167234441 2134563038 1446664164 7754042569 6979028215 7639778284 : 1210000100 4659765524 2292776428 8666665063 8406020317 1457213602 0147023417 0721892212 2997378933 8571862709 6310230036 : 1220000100 8774148106 0106565862 9839616061 8500841728 6699649052 8928480707 2841798187 8242952756 1782440685 7439908225 : 1230000100 5934850684 8110668385 4596616827 3138022957 3405871663 3605441369 7080868499 8047018884 9715340080 9973367157 : 1240000100 Last digits of the computation (until index 1,250,000,000) 2892425365 5002482483 0343444601 0083754535 7304704974 8611813719 7188730476 3575213870 1414050025 1835828906 : 1249999100 7782861120 0025817999 4108067182 9227642790 6733332938 5871285723 8695500907 3350723835 7540448093 3192380600 : 1249999200 8802719393 3348917610 1042077548 9208983277 6290440855 7531092131 8472182643 4837533875 0074781633 7399956257 : 1249999300 6513752969 7512158030 9968982403 8091868359 9251977923 5183842709 8458943888 4372214078 1362497256 0357999061 : 1249999400 4227815646 3046611433 9688086972 7043804853 2119023015 4260075440 1908927412 2523140949 7423022760 5438692268 : 1249999500 0023814739 5421029169 4569058423 1199799425 1691458430 7404621916 1814704718 2487409453 2507160404 6661658172 : 1249999600 1186982906 6646353895 6111781239 3718947341 9772989399 0277340850 8415345334 3370739515 6949048364 4370788982 : 1249999700 5047617124 0967837687 3051834390 8856669418 4222444578 4366057631 2434620301 0243392769 9373477716 2663177590 : 1249999800 3517326547 0134844944 4296998492 9956221953 6215457492 2257023102 3074065566 3344330587 6775155307 3296090403 : 1249999900 9498382185 9708896579 6113706959 0634790588 0998142929 8515954435 6558102125 4999555888 8769683294 4475044854 : 1250000000