/* Copyright (C) 2019 Alessandro Languasco */ /**************** A. LANGUASCO ******************** ************* COMPUTATION OF THE KUMMER RATIO MOD q(PRIME) *******/ \\ Global variables: global(ak_matrix); global(phir); global(pir); global(defaultprecision); /************* COMPUTATION OF THE KUMMER RATIO MOD q(PRIME) ******* ************* FOR r1<=q<=r2 ************* ************* Using the generalised bernoulli numbers ********************/ {global_kummer_Bernoulli(r1,r2,defaultprecision)=local(minutes, millisec, seconds, rows, columns, elaptimefinalcomp, elaptimeprecomp, bernoullichi, Kummer, qminusone, qminustwo, u, v, chi, i, twoPiI, twoPi, logpi, qminusonehalf, S, correction); \\ minutes,millisec,seconds: used just to compute the elapsed computation time; local variables \\ defaultprecision: used to fix the precision used in the computations; global variable print("************ A. LANGUASCO *************"); print("********* COMPUTATION OF THE KUMMER **********"); print("******* RATIO IN ONE INTERVAL ********"); print("********* with the the generalised bernoulli numbers **********"); \\ precision setting if(defaultprecision >90 , print("Internal constant are defined with 100 decimal digits; you cannot set the default precision larger than 90 decimal digits"); return); default(realprecision,defaultprecision); r1=nextprime(r1); r2=precprime(r2); if (r1>r2, print("error: no odd primes in this interval. END PROGRAM");return); if (r1 < 3, r1=3); print("Interval=[",r1,",",r2,"]"); phir=r2-1; pir=primepi(r2)-primepi(r1)+1; gettime(); print("Performing precomputations of g^k%q for this interval"); precompak(r1,r2); elaptimeprecomp=gettime(); twoPi=2*Pi; twoPiI=twoPi*I; logpi=log(Pi); i=0; gettime(); print("Starting computation of KUMMER RATIO"); forprime(q=r1,r2, i+=1; qminusone=q-1; qminustwo=qminusone-1; qminusonehalf= qminusone/2; correction=qminusonehalf*(logpi - (1.5)*log(q)); u=twoPiI/qminusone; S=0; \\ odd characters for(m=1, qminusonehalf, \\\ odd characters v=exp(u*(2*m-1)); bernoullichi=ak_matrix[i,1]; \\ case k=0 chi=v; \\ k=1 for (k=1, qminustwo, bernoullichi += chi * ak_matrix[i,k+1]; \\ a= g^k%q chi*=v; ); S += log(abs(bernoullichi)); ); Kummer = exp(S+correction); print("Kummer-ratio(",q") = ", Kummer); ); elaptimefinalcomp=gettime(); seconds=floor(elaptimeprecomp/1000)%60; minutes=floor(elaptimeprecomp/60000); millisec=elaptimeprecomp- minutes*60000 - seconds*1000; \\print(elaptimeprecomp); print("Precomputation time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); \\print(elaptimefinalcomp); seconds=floor(elaptimefinalcomp/1000)%60; minutes=floor(elaptimefinalcomp/60000); millisec=elaptimefinalcomp- minutes*60000 - seconds*1000; print("Final step Kummer ratio computation time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); seconds=floor((elaptimefinalcomp+elaptimeprecomp)/1000)%60; minutes=floor((elaptimefinalcomp+elaptimeprecomp)/60000); millisec=(elaptimefinalcomp+elaptimeprecomp)- minutes*60000 - seconds*1000; print("Total elapsed time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); print("****** END PROGRAM ********"); } /***************** Precomputations ****************/ {precompak(r1,r2)= local(rows, columns, i, g, g1, a, nhalf); ak_matrix=matrix(pir,phir+1,rows,columns,0); i=0; forprime(q=r1,r2, print("Precomputation for q = ",q); g1=znprimroot(q); \\ needed here to generate the sequence a=g^k%q/q g=lift(g1); i+=1; ak_matrix[i,phir+1]=q; a=1; for(k=0, q-2, ak_matrix[i,k+1]= a; a=(a*g)%q \\a=g^k%q; ); ); } /************************************ **** RESULTS **** gp2.11.2 and gp2c0.0.11 compiled by myself, see below ******************* gp 2.11.2 on Dell Optiplex languasc@languasco1:~/Desktop/EK$ gp2c-run -pmy_ -g -W Kummer-Bernoulli-final.gp GP/PARI CALCULATOR Version 2.11.2 (released) amd64 running linux (x86-64/GMP-6.1.2 kernel) 64-bit version compiled: May 21 2019, gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04) threading engine: single (readline v7.0 enabled, extended help enabled) Copyright (C) 2000-2018 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500000 ? init_Kummer_Bernoulli_final(); global_kummer_Bernoulli(3,1000,30) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[3,997] Performing precomputations of g^k%q for this interval Precomputation for q = 3 Precomputation for q = 5 Precomputation for q = 7 Precomputation for q = 11 Precomputation for q = 13 Precomputation for q = 17 Precomputation for q = 19 Precomputation for q = 23 Precomputation for q = 29 Precomputation for q = 31 Precomputation for q = 37 Precomputation for q = 41 Precomputation for q = 43 Precomputation for q = 47 Precomputation for q = 53 Precomputation for q = 59 Precomputation for q = 61 Precomputation for q = 67 Precomputation for q = 71 Precomputation for q = 73 Precomputation for q = 79 Precomputation for q = 83 Precomputation for q = 89 Precomputation for q = 97 Precomputation for q = 101 Precomputation for q = 103 Precomputation for q = 107 Precomputation for q = 109 Precomputation for q = 113 Precomputation for q = 127 Precomputation for q = 131 Precomputation for q = 137 Precomputation for q = 139 Precomputation for q = 149 Precomputation for q = 151 Precomputation for q = 157 Precomputation for q = 163 Precomputation for q = 167 Precomputation for q = 173 Precomputation for q = 179 Precomputation for q = 181 Precomputation for q = 191 Precomputation for q = 193 Precomputation for q = 197 Precomputation for q = 199 Precomputation for q = 211 Precomputation for q = 223 Precomputation for q = 227 Precomputation for q = 229 Precomputation for q = 233 Precomputation for q = 239 Precomputation for q = 241 Precomputation for q = 251 Precomputation for q = 257 Precomputation for q = 263 Precomputation for q = 269 Precomputation for q = 271 Precomputation for q = 277 Precomputation for q = 281 Precomputation for q = 283 Precomputation for q = 293 Precomputation for q = 307 Precomputation for q = 311 Precomputation for q = 313 Precomputation for q = 317 Precomputation for q = 331 Precomputation for q = 337 Precomputation for q = 347 Precomputation for q = 349 Precomputation for q = 353 Precomputation for q = 359 Precomputation for q = 367 Precomputation for q = 373 Precomputation for q = 379 Precomputation for q = 383 Precomputation for q = 389 Precomputation for q = 397 Precomputation for q = 401 Precomputation for q = 409 Precomputation for q = 419 Precomputation for q = 421 Precomputation for q = 431 Precomputation for q = 433 Precomputation for q = 439 Precomputation for q = 443 Precomputation for q = 449 Precomputation for q = 457 Precomputation for q = 461 Precomputation for q = 463 Precomputation for q = 467 Precomputation for q = 479 Precomputation for q = 487 Precomputation for q = 491 Precomputation for q = 499 Precomputation for q = 503 Precomputation for q = 509 Precomputation for q = 521 Precomputation for q = 523 Precomputation for q = 541 Precomputation for q = 547 Precomputation for q = 557 Precomputation for q = 563 Precomputation for q = 569 Precomputation for q = 571 Precomputation for q = 577 Precomputation for q = 587 Precomputation for q = 593 Precomputation for q = 599 Precomputation for q = 601 Precomputation for q = 607 Precomputation for q = 613 Precomputation for q = 617 Precomputation for q = 619 Precomputation for q = 631 Precomputation for q = 641 Precomputation for q = 643 Precomputation for q = 647 Precomputation for q = 653 Precomputation for q = 659 Precomputation for q = 661 Precomputation for q = 673 Precomputation for q = 677 Precomputation for q = 683 Precomputation for q = 691 Precomputation for q = 701 Precomputation for q = 709 Precomputation for q = 719 Precomputation for q = 727 Precomputation for q = 733 Precomputation for q = 739 Precomputation for q = 743 Precomputation for q = 751 Precomputation for q = 757 Precomputation for q = 761 Precomputation for q = 769 Precomputation for q = 773 Precomputation for q = 787 Precomputation for q = 797 Precomputation for q = 809 Precomputation for q = 811 Precomputation for q = 821 Precomputation for q = 823 Precomputation for q = 827 Precomputation for q = 829 Precomputation for q = 839 Precomputation for q = 853 Precomputation for q = 857 Precomputation for q = 859 Precomputation for q = 863 Precomputation for q = 877 Precomputation for q = 881 Precomputation for q = 883 Precomputation for q = 887 Precomputation for q = 907 Precomputation for q = 911 Precomputation for q = 919 Precomputation for q = 929 Precomputation for q = 937 Precomputation for q = 941 Precomputation for q = 947 Precomputation for q = 953 Precomputation for q = 967 Precomputation for q = 971 Precomputation for q = 977 Precomputation for q = 983 Precomputation for q = 991 Precomputation for q = 997 Starting computation of KUMMER RATIO Kummer-ratio(3) = 0.604599788078072616864692752547 Kummer-ratio(5) = 0.789568352087148689506759279990 Kummer-ratio(7) = 0.956675185750841875479507338132 Kummer-ratio(11) = 1.10916191287000575896982175317 Kummer-ratio(13) = 1.07714905620985756748597815892 Kummer-ratio(17) = 0.855390345687652681159058739366 Kummer-ratio(19) = 0.707040049003847290706746219786 Kummer-ratio(23) = 1.27303069939685502234405162961 Kummer-ratio(29) = 1.19507225854723141702138692301 Kummer-ratio(31) = 0.889889621078544078919851815713 Kummer-ratio(37) = 0.896173542451826242639301056840 Kummer-ratio(41) = 1.01095149281551337376703651618 Kummer-ratio(43) = 1.00032807083987921579084335194 Kummer-ratio(47) = 0.995104194758437633204617945976 Kummer-ratio(53) = 1.00231549556080469808835403497 Kummer-ratio(59) = 1.03111995957758588341749868917 Kummer-ratio(61) = 0.915416897576361520386078405848 Kummer-ratio(67) = 1.03230196304201968151553976333 Kummer-ratio(71) = 0.946524747103623680929005462712 Kummer-ratio(73) = 1.28217793230760538382246761185 Kummer-ratio(79) = 0.845794596120029755045529407638 Kummer-ratio(83) = 1.22326926548441461619507621390 Kummer-ratio(89) = 1.28632147461922346234453694590 Kummer-ratio(97) = 0.904676142870237650667818579334 Kummer-ratio(101) = 1.11049958753586448051923888082 Kummer-ratio(103) = 1.05565198833718743186163713482 Kummer-ratio(107) = 0.992607677926725013095196123757 Kummer-ratio(109) = 0.915542838852301868506675002464 Kummer-ratio(113) = 1.16185573635061808057761114590 Kummer-ratio(127) = 1.06269835499717635407980190888 Kummer-ratio(131) = 1.27897699389762867270592988247 Kummer-ratio(137) = 1.00188853650420792851571142833 Kummer-ratio(139) = 0.871661151873923278867085421302 Kummer-ratio(149) = 1.04886527642691194564791006449 Kummer-ratio(151) = 1.09613526050530812035603232922 Kummer-ratio(157) = 0.743045053291088966005230028621 Kummer-ratio(163) = 0.951673923694429928830818383070 Kummer-ratio(167) = 0.854048917140988351868386074510 Kummer-ratio(173) = 1.25750311100604863256476652232 Kummer-ratio(179) = 1.31898955218699008540672120548 Kummer-ratio(181) = 1.01646725307901783240856438797 Kummer-ratio(191) = 1.29850955347246763676155271715 Kummer-ratio(193) = 1.17384956614280523683625176108 Kummer-ratio(197) = 0.871426858058702258542750867415 Kummer-ratio(199) = 0.797757659818032617033364109700 Kummer-ratio(211) = 0.709658103845770077391538268813 Kummer-ratio(223) = 0.900167367740091073894200748610 Kummer-ratio(227) = 0.762988397631371226037628711708 Kummer-ratio(229) = 0.724145741420104946200864041968 Kummer-ratio(233) = 1.43102216731058063469583770264 Kummer-ratio(239) = 1.18520259221018381028526578871 Kummer-ratio(241) = 1.11908192699651325481120769078 Kummer-ratio(251) = 1.18041694425392859170387583509 Kummer-ratio(257) = 0.905596257354965766409134645388 Kummer-ratio(263) = 0.937170781668529606540649323197 Kummer-ratio(269) = 1.01052429941342866041104883014 Kummer-ratio(271) = 0.841208809014411030345871789067 Kummer-ratio(277) = 1.22287167700803659996325347046 Kummer-ratio(281) = 1.09072312671446411507457756821 Kummer-ratio(283) = 0.987300459249893511767351929709 Kummer-ratio(293) = 1.28843023595237283191056798455 Kummer-ratio(307) = 0.913587252201994822205149168994 Kummer-ratio(311) = 1.14589374542647302213444142687 Kummer-ratio(313) = 0.938933176758191661806739844229 Kummer-ratio(317) = 0.806718231889848128494571985777 Kummer-ratio(331) = 0.813562749560518459023316493365 Kummer-ratio(337) = 0.861115115219225912688322557910 Kummer-ratio(347) = 1.08517941758105267446483313058 Kummer-ratio(349) = 0.983957313448770104455912391326 Kummer-ratio(353) = 0.886035056617446045030878157759 Kummer-ratio(359) = 1.16002644446708254566916432735 Kummer-ratio(367) = 0.908641018779369120632653198254 Kummer-ratio(373) = 1.07507614420133257646267035535 Kummer-ratio(379) = 0.721446186471384446944269956888 Kummer-ratio(383) = 0.832438092674287139604707603809 Kummer-ratio(389) = 0.849977828968545039716275634967 Kummer-ratio(397) = 0.997577811201585790942532466168 Kummer-ratio(401) = 1.13998328316447070631384278931 Kummer-ratio(409) = 1.19919809743909540748744244798 Kummer-ratio(419) = 1.18974458882376935926766971002 Kummer-ratio(421) = 0.864579665307117411773428695355 Kummer-ratio(431) = 1.13754261103593462461717085623 Kummer-ratio(433) = 1.07176135182041771385450595205 Kummer-ratio(439) = 0.684841340617297620550058956264 Kummer-ratio(443) = 1.41089988430397986980906568345 Kummer-ratio(449) = 0.905396436586144248958915474607 Kummer-ratio(457) = 0.837346341905856217786367913436 Kummer-ratio(461) = 1.03119557377397403645284724907 Kummer-ratio(463) = 0.961346251119598417786866352317 Kummer-ratio(467) = 0.897404548591928368706570837377 Kummer-ratio(479) = 1.10506715780642069705910978939 Kummer-ratio(487) = 1.13041022782656063139453697156 Kummer-ratio(491) = 1.27221465691304968352754354985 Kummer-ratio(499) = 0.829790249594650636698813826805 Kummer-ratio(503) = 1.09956174719578329093362210466 Kummer-ratio(509) = 1.39692082719612661320417410651 Kummer-ratio(521) = 0.744885791819182728609101592480 Kummer-ratio(523) = 0.995148478739928942038026932208 Kummer-ratio(541) = 0.944726557829529815213457795295 Kummer-ratio(547) = 0.738685054761954589961666132019 Kummer-ratio(557) = 1.01800618130970440243478675140 Kummer-ratio(563) = 0.923221250913375236441620018462 Kummer-ratio(569) = 0.866443845143573852727051682848 Kummer-ratio(571) = 0.996624806368519727623091513498 Kummer-ratio(577) = 0.913702938040185102392773892046 Kummer-ratio(587) = 0.812524598506721216603741739545 Kummer-ratio(593) = 1.07734617489664930780759188442 Kummer-ratio(599) = 0.964087738347230697795712684717 Kummer-ratio(601) = 0.928273397518240972508543005502 Kummer-ratio(607) = 0.836373127052514432476677991017 Kummer-ratio(613) = 0.877036593034721489103550202941 Kummer-ratio(617) = 0.842460845419467161414453788481 Kummer-ratio(619) = 0.804639186365482318180970492383 Kummer-ratio(631) = 1.13964698072442766479584447731 Kummer-ratio(641) = 1.34299156432328475445263673245 Kummer-ratio(643) = 1.01836205611360685304417553494 Kummer-ratio(647) = 0.902336673171188755954907722093 Kummer-ratio(653) = 1.27087727805772466468796098338 Kummer-ratio(659) = 1.39106317898226550143998268526 Kummer-ratio(661) = 0.835444309752321465693683859729 Kummer-ratio(673) = 1.03660206982398637181187353211 Kummer-ratio(677) = 0.924240133124973644017920446624 Kummer-ratio(683) = 1.13528281402409476998254691134 Kummer-ratio(691) = 0.769214279574540506964064110369 Kummer-ratio(701) = 0.920898828679698610416262543827 Kummer-ratio(709) = 1.05648934917801861606174800341 Kummer-ratio(719) = 1.20306325855333927681117243729 Kummer-ratio(727) = 0.998569214227803286313406396076 Kummer-ratio(733) = 0.980149101772619867360780226218 Kummer-ratio(739) = 1.10263546824053086630671245464 Kummer-ratio(743) = 1.03495494096205775904091176831 Kummer-ratio(751) = 1.01856200583585073878095848975 Kummer-ratio(757) = 0.967068761187085985455414554477 Kummer-ratio(761) = 1.46958285813141552491322656984 Kummer-ratio(769) = 0.898922303673921113149727164747 Kummer-ratio(773) = 1.06810947197031447130333305035 Kummer-ratio(787) = 0.971782328439863366864515564748 Kummer-ratio(797) = 1.03075130387360942943641986527 Kummer-ratio(809) = 1.31970441406018712251949567645 Kummer-ratio(811) = 0.802838172648154207078568189058 Kummer-ratio(821) = 1.06528437036549643319352814656 Kummer-ratio(823) = 0.967693184761820486564657059185 Kummer-ratio(827) = 0.865559936757584420576919699532 Kummer-ratio(829) = 0.822500335416155497484009196449 Kummer-ratio(839) = 0.918710905407657616100443176668 Kummer-ratio(853) = 1.08223582880253347548004283614 Kummer-ratio(857) = 1.05075311490694694576396382027 Kummer-ratio(859) = 0.880800941805681784763976757216 Kummer-ratio(863) = 1.05694231206444764180240401289 Kummer-ratio(877) = 0.722893985227057412182846378541 Kummer-ratio(881) = 1.09738994199075350184435336371 Kummer-ratio(883) = 1.13318227639393214982039012685 Kummer-ratio(887) = 0.969179741967908231084177199367 Kummer-ratio(907) = 0.902625588663114804780516236075 Kummer-ratio(911) = 1.07798557536304873099351043701 Kummer-ratio(919) = 1.04003346554199950901317303457 Kummer-ratio(929) = 1.04414904452989167744813201725 Kummer-ratio(937) = 0.900179348577500197841322625238 Kummer-ratio(941) = 1.09400867179752235523397214847 Kummer-ratio(947) = 1.22587448270510743026091490433 Kummer-ratio(953) = 1.16083173031283885682226845601 Kummer-ratio(967) = 0.728600044046688614814368250479 Kummer-ratio(971) = 1.07939115916440046258710381604 Kummer-ratio(977) = 0.838908858803712823541254724752 Kummer-ratio(983) = 0.788676772029738540472465667637 Kummer-ratio(991) = 0.909439361535051297600696397509 Kummer-ratio(997) = 0.855757544913506544665452178650 Precomputation time: 0 min, 0 sec, 19 millisec Final step Kummer ratio computation time: 0 min, 8 sec, 928 millisec Total elapsed time: 0 min, 8 sec, 947 millisec ****** END PROGRAM ******** ? v=[439,761,1451,2741,3331,4349,4391,5231,6101,6379,7219,8209,9049,9689] ? for (i=1,length(v),init_Kummer_Bernoulli_final();global_kummer_Bernoulli(v[i],v[i],10)) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[439,439] Performing precomputations of g^k%q for this interval Precomputation for q = 439 Starting computation of KUMMER RATIO Kummer-ratio(439) = 0.6848413406 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 29 millisec Total elapsed time: 0 min, 0 sec, 29 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[761,761] Performing precomputations of g^k%q for this interval Precomputation for q = 761 Starting computation of KUMMER RATIO Kummer-ratio(761) = 1.469582858 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 84 millisec Total elapsed time: 0 min, 0 sec, 84 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[1451,1451] Performing precomputations of g^k%q for this interval Precomputation for q = 1451 Starting computation of KUMMER RATIO Kummer-ratio(1451) = 1.489316072 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 299 millisec Total elapsed time: 0 min, 0 sec, 299 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[2741,2741] Performing precomputations of g^k%q for this interval Precomputation for q = 2741 Starting computation of KUMMER RATIO Kummer-ratio(2741) = 1.498121015 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 1 sec, 61 millisec Total elapsed time: 0 min, 1 sec, 61 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[3331,3331] Performing precomputations of g^k%q for this interval Precomputation for q = 3331 Starting computation of KUMMER RATIO Kummer-ratio(3331) = 0.6424292976 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 1 sec, 566 millisec Total elapsed time: 0 min, 1 sec, 566 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[4349,4349] Performing precomputations of g^k%q for this interval Precomputation for q = 4349 Starting computation of KUMMER RATIO Kummer-ratio(4349) = 1.518570512 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 2 sec, 656 millisec Total elapsed time: 0 min, 2 sec, 657 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[4391,4391] Performing precomputations of g^k%q for this interval Precomputation for q = 4391 Starting computation of KUMMER RATIO Kummer-ratio(4391) = 1.507776410 Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 2 sec, 710 millisec Total elapsed time: 0 min, 2 sec, 710 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[5231,5231] Performing precomputations of g^k%q for this interval Precomputation for q = 5231 Starting computation of KUMMER RATIO Kummer-ratio(5231) = 1.556562248 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 3 sec, 834 millisec Total elapsed time: 0 min, 3 sec, 835 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[6101,6101] Performing precomputations of g^k%q for this interval Precomputation for q = 6101 Starting computation of KUMMER RATIO Kummer-ratio(6101) = 1.511405291 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 5 sec, 215 millisec Total elapsed time: 0 min, 5 sec, 216 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[6379,6379] Performing precomputations of g^k%q for this interval Precomputation for q = 6379 Starting computation of KUMMER RATIO Kummer-ratio(6379) = 0.6735230263 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 5 sec, 698 millisec Total elapsed time: 0 min, 5 sec, 699 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[7219,7219] Performing precomputations of g^k%q for this interval Precomputation for q = 7219 Starting computation of KUMMER RATIO Kummer-ratio(7219) = 0.6580840901 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 7 sec, 292 millisec Total elapsed time: 0 min, 7 sec, 293 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[8209,8209] Performing precomputations of g^k%q for this interval Precomputation for q = 8209 Starting computation of KUMMER RATIO Kummer-ratio(8209) = 0.6720450390 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 9 sec, 415 millisec Total elapsed time: 0 min, 9 sec, 416 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[9049,9049] Performing precomputations of g^k%q for this interval Precomputation for q = 9049 Starting computation of KUMMER RATIO Kummer-ratio(9049) = 0.6676142442 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 11 sec, 460 millisec Total elapsed time: 0 min, 11 sec, 461 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[9689,9689] Performing precomputations of g^k%q for this interval Precomputation for q = 9689 Starting computation of KUMMER RATIO Kummer-ratio(9689) = 1.524371504 Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 13 sec, 112 millisec Total elapsed time: 0 min, 13 sec, 113 millisec ****** END PROGRAM ******** *******/