/* Copyright (C) 2019 - 2024 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, H_one, logq, correctionH,qminusthreehalf, 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; qminusthreehalf= qminusonehalf-1; logq = log(q); correction = qminusonehalf * (logpi - (1.5)*logq); correctionH = - qminusthreehalf * (logq + log(2)); 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 = S + correction ; H_one = S + correctionH ; print("R(",q") = ", exp(Kummer)); print("r(",q") = ", Kummer); print("log h_1(",q") = ", H_one); print("------"); ); 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); 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 **** ----------------- 2023 update ----------------- Reading GPRC: /Users/languasc/.gprc GPRC Done. gp2c-run -pmy_ -g -W Kummer-Bernoulli-final.gp Reading GPRC: /Users/languasc/.gprc GPRC Done. GP/PARI CALCULATOR Version 2.15.4 (released) arm64 running darwin (aarch64/GMP-6.3.0 kernel) 64-bit version compiled: Sep 29 2023, Apple clang version 15.0.0 (clang-1500.0.40.1) threading engine: pthread (readline v8.1 enabled, extended help enabled) Copyright (C) 2000-2022 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 ?18 for how to get moral (and possibly technical) support. parisizemax = 2048000000, primelimit = 500000, nbthreads = 8 ----------- Macbook ARM M1 ? init_Kummer_Bernoulli_final(); global_kummer_Bernoulli(3,1000,100) ************ 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 R(3) = 0.6045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896 r(3) = -0.5031885471527643629494405040307298453239410238188126060884184289841036699430873705268003619451084255 log h_1(3) = 3.648571964345393439011203870932202068720938777219982443019391350633349906084007494157719621052504716 E-232 ------ R(5) = 0.7895683520871486895067592799900920908250959525792632501130679500976035857935364194401418722974841785 r(5) = -0.2362688726094550914974318422880812876811129483461570466773696305106888136759686540871578886389611187 log h_1(5) = -1.0151767349262596893 E-115 ------ R(7) = 0.9566751857508418754795073381317377054326685117283356905564520753378709354033169025479045056409141002 r(7) = -0.04429135397019212149863556163242005299982711648820774836624614144074910328630549890654978165037231929 log h_1(7) = 0.E-114 ------ R(11) = 1.109161912870005758969821753166244519576909209441470902797849050305454343616488092920564157096777178 r(11) = 0.1036046966924852041692627197200472811625006132366187607858484133772556536065807655788781813539015187 log h_1(11) = -1.2182120819115116272 E-114 ------ R(13) = 1.077149056209857567485978158918715617931828748944136819793229858880676629353146137459300622977532424 r(13) = 0.07431778804998064773277494914796069104019777025231723400895643364593605985978933913275760287307372885 log h_1(13) = -8.121413879410077514 E-115 ------ R(17) = 0.8553903456876526811590587393660530757514587542055704092666315700303370969290683258191067367054718442 r(17) = -0.1561973695662618421796294283339270082341556190844445855324440511806053661381847570504448890086662632 log h_1(17) = -2.4364241638230232543 E-114 ------ R(19) = 0.7070400490038472907067462197858284459525546330314604325897206955528945678533286332029816944404298960 r(19) = -0.3466679682912584874209477415402535053759315460253623253707335810547209566320926450283550625358125572 log h_1(19) = -8.121413879410077514 E-115 ------ R(23) = 1.273030699396855022344051629606789233655125383739277444944439918672329975936112218992851840841952866 r(23) = 0.2414004350714917109013739096216624436472627970308969566002182611254898827474540622624978988138394556 log h_1(23) = 1.098612288668109691395245236922525704647490557822749451734694333637494293218608966873615754813732089 ------ R(29) = 1.195072258547231417021386923013850810687070707100898101970267662883915052898781303054394279716250406 r(29) = 0.1782066509589351712175206033783542083140262191964934357482403393619804807998619003176020972567035273 log h_1(29) = 2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063 ------ R(31) = 0.8898896210785440789198518157132618895011004554763489611820409908874189269971779363295182339667477266 r(31) = -0.1166578452072867636147463140546322517630757186662857260688487437727761987938503248341839246470649412 log h_1(31) = 2.197224577336219382790490473845051409294981115645498903469388667274988586437217933747231509627464178 ------ R(37) = 0.8961735424518262426393010568398574127548457342934243308056523092743060031241027667989053279802809691 r(37) = -0.1096211989897466046382226501389660081658893679087778655074386828507362915015699871862983791515855202 log h_1(37) = 3.610917912644224444368095671031447163900077587167636163644912681192989746990361065399021533672168661 ------ R(41) = 1.010951492815513373767036516179792595891310687841472575278025689555670420621430277283400627652810425 r(41) = 0.01089195947631960738544738728530415388329735563359763919062501521861668248496205274848947935516387875 log h_1(41) = 4.795790545596741088123887155930258599643413707874834350437135418261147247826473426150109416005269583 ------ R(43) = 1.000328070839879215790843351939301902837536596125034098226748423945180654751572111239539578965626287 r(43) = 0.0003280170364084711608841695548035152142858058704291801830829437760208603968266275149345731400795244016 log h_1(43) = 5.351858133476066495741956265454280118041158173581668370687778799606135752960928683123887335392390711 ------ R(47) = 0.9951041947584376332046179459764681207862124246207843578034645965518208219762572569016781601658495553 r(47) = -0.004907828955939390994403366850205073683649607651347041951337086053965424910486943624177568275460449341 log h_1(47) = 6.543911845564792131789544252101731421968198014623671637586726396317433853227742885657115939401389910 ------ R(53) = 1.002315495560804698088354034974280076229743102582005401258348801603090299195254375025985641265024321 r(53) = 0.002312818931977140245815350565150176540702736281096572470904728050789743597453926488958746239512721724 log h_1(53) = 8.494743062578645555108469125338018256863222782233660802773517212692432052558465981453409766969210351 ------ R(59) = 1.031119959577585883417498689168024148322714710246565939818515052429444052524270430625304943677914035 r(59) = 0.03064555091195193660663228272344417820349129962325737082744367743537587784564370305474650309613613605 log h_1(59) = 10.62718818613952970731352048531111080729981319761411988898990314334596090714836125143526788614803478 ------ R(61) = 0.9154168975763615203860784058478413733685856081264188407131377981333346314792574264595871065820763475 r(61) = -0.08837569170600222156492412623127747112270256714668169536704463078898156272088351334354444521607211453 log h_1(61) = 11.24244132334655856105474946569967664631985703824674579672268129244950721313236703954458826122475508 ------ R(67) = 1.032301963042019681515539763328566110571528464035031763253886414195373680009442767307965761216888393 r(67) = 0.03179122410730252658303901240714042036792498778281856698983503125658382489411584635628500498889791576 log h_1(67) = 13.65711605250075292272476846745970089251251766843253697508973565745464036240614500529008890888028554 ------ R(71) = 0.9465247471036236809290054627120216081818834467645200690617747555821269588497141916681884808473549364 r(71) = -0.05495816276475998783392367572640508178740599779494614114219182796334041087577930464263722614463020235 log h_1(71) = 15.17206941873242869155513574882966748604243672606594321602425768524245737243394934270669669537874462 ------ R(73) = 1.282177932307605383822467611851369711957384306038708800677872538316728327553187710002746147320062771 r(73) = 0.2485601416192977233485985949901078227023458579824294238063800780055556302080978170134273590929261635 log h_1(73) = 16.29686231326223707733215798657612316794753194298263458241042745881382868790639247084803031816296161 ------ R(79) = 0.8457945961200297550455294076382423795384793601444685981418794787477439435479084808903348288474431623 r(79) = -0.1674787430395554396980327371073421502408683411911757990727011018337644485818585356494196261517341478 log h_1(79) = 18.42214382312985664139886354508158241320322395008547057625966665997992914411238686175824390560888906 ------ R(83) = 1.223269265484414616195076213901577695755917327589649354931437707543554073921893541041152140758969079 r(83) = 0.2015270004883066457927627413784520393410351522432338981327646086933168343453903982224590948878662748 log h_1(83) = 20.54678752589194248394560927834044687891217316656483126028780275177937167289303052838555881860336959 ------ R(89) = 1.286321474619223462344536945899736181877726106678888331734549067451154326113810359531967907969874749 r(89) = 0.2517865748393494807866229914137070346144957827823578535492411947117267164699529372018534214905473277 log h_1(89) = 23.31697933722730979484459574458406334119420714107573815421462991934467863076822632395276816336093871 ------ R(97) = 0.9046761428702376506678185793342554386262237883278983016821697804084706089911402229182866873374592046 r(97) = -0.1001782525089982588614145372590390656217641079318948709833067761799559966195359896079800996954931279 log h_1(97) = 26.74264420298693439256220343185888692098245112367068665209668456597236311078783868085959518410618634 ------ R(101) = 1.110499587535864480519238880822910803010038576148544226903565842698872172244491378024387382169164450 r(101) = 0.1048099929302720343206720795916380224845002449739497997683399193094760169112848583702960106101653113 log h_1(101) = 28.89723729089568888869408550022576868358320028269358727038676528477860901981588046362351224930203994 ------ R(103) = 1.055651988337187431861637134816806476756165229326968759047548467670306579150635000536559161240206211 r(103) = 0.05415857446951749964690200169099684076144220209025010103852001091855978167102572292271696512936718790 log h_1(103) = 29.83589355623819107283846235869235588962331939722288105700736330547892933692670801784603217013934372 ------ R(107) = 0.9926076779267250130951961237566693979657735137486072408761129764951318188183413638373032219131511214 r(107) = -0.007419780691752351116165571477986535402169866438541443359175603241031879025121158792907946766098015177 log h_1(107) = 31.78103582787530209545708195912483775035172065593037977504665089436392563672976945649643007111346860 ------ R(109) = 0.9155428388523018685066750024637759142119621635051256177491556574654526154930635518327336163684178913 r(109) = -0.08823812311436882921749324843559625370365208802808522695861814708651063618684838742257966017361141947 log h_1(109) = 32.71728817375694397848898800307352143903068963154203372900615502133347978310328686598318302906043155 ------ R(113) = 1.161855736350618080577611145899844750232670406516443122403208800176464193827909426127465164317182843 r(113) = 0.1500184995610680310371208097241619873841365799027436140804123907802706223079310740160077689282077568 log h_1(113) = 35.01629670385554274993923576861779137983370093921807668015021289116754956163691330206013462119404657 ------ R(127) = 1.062698354997176354079801908884500306649236075514951773381953311038408733015629412028780258826453436 r(127) = 0.06081129144874085861375585680632583649905932326271467921474595380476148540302253094746159042737666883 log h_1(127) = 42.40378359812413707775126744193618261544052430686004721286045111782819842697660691375726067637492481 ------ R(131) = 1.278976993897628672705929882468300785284903709438492543639972261471143278312625989338201151038500460 r(131) = 0.2460605348646604854850129462032664668955364127441946414157042770102166368161755148036545511610484025 log h_1(131) = 44.79630872605572609246508370911026727210435012965206878986603176984653718281843490232788329107658952 ------ R(137) = 1.001888536504207928515711428333312180688988502643736519631034611083049274360374531557864965624004681 r(137) = 0.001886755461168050668360941089586140071009205372217728990991000554512642908053536966958031537113906770 log h_1(137) = 47.91872582416999251399240339067905235670618401655371486673865294044685987897128630216271330981856353 ------ R(139) = 0.8716611518739232788670854213024296668791137011185749631439974986107156909801958540223225374239110443 r(139) = -0.1373545178933138028904818003462952347867942344665232547827971464948061753636612387041934832972945931 log h_1(139) = 48.91609970656135052104311131721845130912803598048758503700075719831246913460712039148495780837891043 ------ R(149) = 1.048865276426911945647910064493683938342422445829615961241833744172516652783826776900693842842444723 r(149) = 0.04770889068667165295680733758333180897024931444580294315296995708813868613733205530672304249949614333 log h_1(149) = 54.88791278288249853659751292496667621609014465574686471633774564239713901380881405019354300802083671 ------ R(151) = 1.096135260505308120356032329215248864952672537980057148922751066825209158780210119727472470709712115 r(151) = 0.09179059378088875201912448173651113284034885724460193485606833154087662289409258862819130979425886350 log h_1(151) = 56.10943151101450945304199367250437889662907906025925546478773782680835057408779239696327101143176927 ------ R(157) = 0.7430450532910889660052300286210995504858412553970559370872349766809954404687132600415838521736305913 r(157) = -0.2969985991026067257994545031778206878580587464398644016714893245181460765664269799909219737372869013 log h_1(157) = 59.29156961546071316286414819514301401152918073497219269010099027893282091675551040809537726713940880 ------ R(163) = 0.9516739236944299288308183830698364178533681330002947783720719515152356248711424108604706467152146912 r(163) = -0.04953281999086069149635972908638800565867760730580913768625553274073229342034829026206553263718289505 log h_1(163) = 63.16620531489273731320214828867285078509374324103760248097763604487956232943652623002714977442704527 ------ R(167) = 0.8540489171409883518683860745104309672314966634213763791144768767499375605581837249110944279587267384 r(167) = -0.1577668068094286295019388489012609510233156018598345095102193145899380885073770061516389219741657849 log h_1(167) = 65.50632088948693360291996504087751044644641451959551721099990671352454777491715699096063875070130206 ------ R(173) = 1.257503111006048632564766522323442137564247122686449453100234216917615549526866424719848177051075215 r(173) = 0.2291280969497694541259987476788419403325398106944031344363071373393892044404113682004073374904373663 log h_1(173) = 69.60967772420827684240053329877440909642520241877517998635205101333519725129152756939126596313459291 ------ R(179) = 1.318989552186990085406721205475389313857531781335293052660401099869295491007573060546512675273320283 r(179) = 0.2768659526931845578821149806614818207287226527944282731971650606306427967653784909864072456243340944 log h_1(179) = 73.42500838857573415649075558789351748340894300687144968581961983350619680496678665426151181519852219 ------ R(181) = 1.016467253079017832408564387974785972294133677747041565851626072531984908240252285042687796037598322 r(181) = 0.01633313819991843603706760580342101158478505996333320455723518378169555210076586440044353898992168207 log h_1(181) = 74.43140778014675457959295095136883305766403224863659957639613069085179242104162054728738997550582602 ------ R(191) = 1.298509553472467636761552717150432608458003870773163655512756685235572292742127078942966828384617032 r(191) = 0.2612171094077849584265765292350920343207287323395619047506494339355068805091824984286149199624918984 log h_1(191) = 81.09130424134145816280373141544072363924012048419699975042706329684505007779556048818800459370598738 ------ R(193) = 1.173849566142805236836251761084146867755611285414726296736147400493544606298569246819444086673962580 r(193) = 0.1602885753274080609355275215706327655823408396549067122444764396355220882681663019680524725005082107 log h_1(193) = 82.28905663692957898942107186892010192558283317193103983977366710295509515859527306226679305859917798 ------ R(197) = 0.8714268580587022585427508674145770632292302440882104559710750970180133611346618651220339293109139570 r(197) = -0.1376233440401255889089035183692665463592647737061241377367062082693495748739968001665232583514797186 log h_1(197) = 84.60375776530338767055356672572940135931453364555056749415249046237703644779510780405752126877851119 ------ R(199) = 0.7977576598180326170333641097002519449007023470313238106627818094946338221390893263699901276292175009 r(199) = -0.2259504120923408298128647138462609062444787428932929460706229960810491225723954425494528880853791173 log h_1(199) = 85.82926084252926757531020483866075166477513827994420776503033188900174657753599348585944316105211984 ------ R(211) = 0.7096581038457700773915382688127205105235892454312320345381985871918905674106433951514915314211739285 r(211) = -0.3429719688054611197768351474537419285351684115126584482875246580541432498194929760098556226258947208 log h_1(211) = 93.69749337974276593796581253062871658384882454553233794602749762898671701083520277082788743431304551 ------ R(223) = 0.9001673677400910738942007486095130304962683749282789638201932898271174840849251856970309304249905770 r(223) = -0.1051745687913597337975915590197769041375994163602149868461046273327828903589361800491160099033277339 log h_1(223) = 102.0888233278279467633263834297486951231801590780228781276029718817813259108922118563046684377533798 ------ R(227) = 0.7629883976313712260376287117080471290023903973055326769131580344402924663971494429532754403905696313 r(227) = -0.2705124540631373549639032800201629031080863155814860409623396831227400461416207468587464662610088975 log h_1(227) = 104.6771522274214221253682490816945071526983301144806946345352177662408495581077907025337668283881869 ------ R(229) = 0.7241457414201049462008640419682792790263073995404659340448964914913780082847688929513229702464761983 r(229) = -0.3227626065505959202031072316660449461089618686796977415237794935359483834140820792251225834068878438 log h_1(229) = 106.0082752094898638728906100174025921333991709700307985191118820014124641395967640665042251454446047 ------ R(233) = 1.431022167310580634695837702637477356830046422510857435265617948029074058226590900044431556351431622 r(233) = 0.3583889912374081316179199844975820325256278492623177580132937453320422118168416599715257408755698778 log h_1(233) = 109.4690652286896107008299208653168273895064892768508597954019710091305178588572877353333619046990812 ------ R(239) = 1.185202592210183810285265788710899007983217617138405924310360623631534190175141051333564964890650041 r(239) = 0.1699137238648845297432097656085625919371256853789997259766062907031446821856700166049096924810546296 log h_1(239) = 113.4817348935691126467133620753752412074871906200264678111386800217228179105907788508454745888341560 ------ R(241) = 1.119081926996513254811207690779420063153586980612309284127404310761872215104461038734979765943116421 r(241) = 0.1125086411238352323588029212405822378482806622167209902684765135388208588932404892391713995165502859 log h_1(241) = 114.8330207954922772379479887467102850752412327285000752662283247410819346845138160071679945209448006 ------ R(251) = 1.180416944253928591703875835088579564236309076805229781323060677784918945460788241229701810518127066 r(251) = 0.1658677186548573567424998259613234720728030573997580772009381604924756425894731224917882509363404157 log h_1(251) = 121.9906432329148939959574605230391708960373813718815190399002688663814746449101325925927088333813342 ------ R(257) = 0.9055962573549657664091346453876335287288350791071366057623010168176372106092550047049494665459037113 r(257) = -0.09916170432116911432679914458891818631195808911428431450875607719491993592984837194309179252313506920 log h_1(257) = 126.0356664940318411921926509165127469322029026781248317312239182728243075290493425072663621787703998 ------ R(263) = 0.9371707816685296065406493231972128289458602694583030267914142209934065841167075684014444275853787993 r(263) = -0.06488974903110218320973181913018507757354250128921224257160642575568642990356897698710486334496669911 log h_1(263) = 130.4146048717259274269752059336934638574504949042653798955062533811429428560354403750805337064493177 ------ R(269) = 1.010524299413428660411048830135142525023840396277432529543162601314219818124025091983281437669591766 r(269) = 0.01046930449317675812414112953929925442169374190907329643621744928218703908659745608774798195734463881 log h_1(269) = 134.8684633991258864933698201679680723548739425004299316809971616113771708603603000725201551690165352 ------ R(271) = 0.8412088090144110303458717890667285657753956266972254706481064176948264195975546562385613142649397469 r(271) = -0.1729153632464507276558629858429677162237473007723687107241391465145153136432798651656320502453906075 log h_1(271) = 136.1519670823113659577079403206999957543033721164182358290168110035132387161654610717683788465673538 ------ R(277) = 1.222871677008036599963253470457980992318270948187844137545573845854395071808444252278658102374390147 r(277) = 0.2012019264314748243313634728666834969147083910851968503079120676513432616094542909100066388703685612 log h_1(277) = 140.9485393756154383932354168130971502046559021454761944502358498856068408898441272372132066865543283 ------ R(281) = 1.090723126714464115074577568205961616317281311636414512039595851303559679118672041061618610457265775 r(281) = 0.08684089528004099808220060175337243485758999752091735368621367641276828220909562353740781342998425655 log h_1(281) = 143.8003803012275679315784275828538176399536868999358134798567154204886514847348740559445477264998241 ------ R(283) = 0.9873004592498935117673519297087714914553771518221292665800753071654015447409579869453062969997401355 r(283) = -0.01278086920744889785221212882356077278727082766079272121725223349882429851503993159630780069792401823 log h_1(283) = 145.1891531291262733917001153221029659976051371982986877718445260288378271030662229908452895041422203 ------ R(293) = 1.288430235952372831910567984550110475667221709592238059499825952189868970815116170005096654559361696 r(293) = 0.2534246060296973676483988124070491540033572194228164262994406022000310687198554616029377470983182623 log h_1(293) = 152.9492931580881827167575273515497409595541197355367772563082545316340219337671993777452142754783945 ------ R(307) = 0.9135872522019948222051491689937550823877147771086375517920025017683514919742739542195895540328011735 r(307) = -0.09037639354683542624185588242742955247632922099303514439157869594663415442092326759997613625642742497 log h_1(307) = 163.2382800643910303022753258633049012160539282966458543157896441483286924841006753172047857776628646 ------ R(311) = 1.145893745426473022134441426871823178333706292400792292936590578234620063814679924267381733580992714 r(311) = 0.1361848962160421919146070417665185784812464323903180290616922714452721536588626727192058353084413577 log h_1(311) = 166.5321303893973170838152727639487854649375961201218354058198394466312278683972528961757099511276574 ------ R(313) = 0.9389331767581916618067398442288986436184435690846354610606150910504293383015595004101642329050101779 r(313) = -0.06301096656794707912303716110277084725136581485322474610563362431653518191561608929966044333944169376 log h_1(313) = 167.8713659068062069124537766331223200163429647431629300327844937065490336537623956042937239393697519 ------ R(317) = 0.8067182318898481284945719857774442930570488961026078016713558583037961529639069812556216042270703759 r(317) = -0.2147808267121654002952767544013515762618564148629438474741807450463579935206362022044133123046511010 log h_1(317) = 170.8059317713536426425187322067705555458186952249236249353993966680695181481769219013647456670194131 ------ R(331) = 0.8135627495605184590233164933650585889242407727813613961377969239063645740977903927449432187752259914 r(331) = -0.2063322200052325814387359187594663729910669430135623693705676693257050069617152289787651523828428671 log h_1(331) = 181.7139833469974601559851892934940624000275343831326559898760449656685657482849476072309922043807565 ------ R(337) = 0.8611151152192259126883225579098310250466304824046327040712615817933444899782860237592618917528612877 r(337) = -0.1495270840497703703021745776649619596001014062816181340255394816580988118107601853896364306635270684 log h_1(337) = 186.4873220196908802838379232566223407113210879681996307528346837737910967089922804001118990706341613 ------ R(347) = 1.085179417581052674464833130583284014081413020281332371703161668835106391076138182655378293112194760 r(347) = 0.08174533514751820845847101915301590371878018386922110463358239696897150406543265483594590748529229644 log h_1(347) = 194.6380782722408741928283384030301983567413536257741158277582728990727112036437206798112327884174360 ------ R(349) = 0.9839573134487701044559123913262739100861728497848979511432017485279607486354621737043264522800089344 r(349) = -0.01617276351234466630497063549013050923709265077170472555789176543969921812640205268624346846449818246 log h_1(349) = 196.1326940156350768651118871966934017921739963541518131201162488871457483250985405628478071031472659 ------ R(353) = 0.8860350566174460450308781577592291780666364177742813342643076161277480463289818021564829966410114006 r(353) = -0.1209987618715067883841527643200228745275188692595982489266859524869788867838373038257699474499012959 log h_1(353) = 199.2214417977070457125761025720900746840590967886607904224871629108260299717356485281861453585133646 ------ R(359) = 1.160026444467082545669164327352704386863454730082365439990562271053297231072396721561963253757981189 r(359) = 0.1484428018128081674662126313199636912149154335872780177136709124724035374987094532042919491051138401 log h_1(359) = 204.3022712532891499303778052667905631721127209421160195475261597346264145513351432042594749882085315 ------ R(367) = 0.9086410187793691206326531982541421348425085497237493722112450853424520982181207271967967371380369534 r(367) = -0.09580518164021530872659907544130773601119004176938651478170929777024993105514235922268061172717016942 log h_1(367) = 210.5118097910572566338874038937155104542869799798765040668979475856861656779505238626217735696961387 ------ R(373) = 1.075076144201332576462670355346585113292976182928986159454102213060844064450604521337480572500040639 r(373) = 0.07239149088636464479113045470168919176692532309204530797630444259986838227388717758099932131018544063 log h_1(373) = 215.5487757658267108974847535128265326957714217911030739553135478335738168169504629459016132454776409 ------ R(379) = 0.7214461864713844469442699568877373949658859966990438659741455995085698739502042010387392492298306740 r(379) = -0.3264974891751043328966136732499731931173644122422295855677064987419286734938050014157529169801100379 log h_1(379) = 220.0425917253902608001020718258062007298303679820022460629084212255341243148417890562717519055257844 ------ R(383) = 0.8324380926742871396047076038085164489428524335781087284566345232408333710975870511258615025589220659 r(383) = -0.1833964230466240194626296683875497109576634540160867754346875454497040980643101727587393895219536683 log h_1(383) = 223.4606075292606719641984059813380677655638075881474495572739315748556611499217099626644034065448925 ------ R(389) = 0.8499778289685450397162756349668002541637823771157382635082250190331184233847979554114521350577059629 r(389) = -0.1625450134043746475052150474423674201630123678445728286139997918172093037051552883123517574154351609 log h_1(389) = 228.4132269583502835503088622445890846913317289794841899740753787102669030934836438144425360881708294 ------ R(397) = 0.9975778112015857909425324661679831496845979499584767369728126274473821638343346936864424048885609148 r(397) = -0.002425127043316660673333631915373793583512459813320797797646055822166096400421315906739284270400115139 log h_1(397) = 235.1846909731852642120765258906041457376073632265291172251409649441364305525934546477511758440624003 ------ R(401) = 1.139983283164470706313842789312812288040206393307108276296087592938866790947134657386343330056214768 r(401) = 0.1310135984080730248968948270908750698868197599041656400298993040874457904344782216972443986519589525 log h_1(401) = 238.6388516550624093931008202085863185120749998540690180813068808576698087074838994527821033664667184 ------ R(409) = 1.199198097439095407487442447976822546223574586071189457743096211345999173639004505004859015631366056 r(409) = 0.1816530812798884675141132698153134456099994343479410137382325846756209673025411163045931669473924263 log h_1(409) = 245.3605397867419349390496004780052599434527685526152836995629478834981474310298840061661357177148170 ------ R(419) = 1.189744588823769359267669710017663436245699906056023038676515113126723824128500573186674331362945483 r(419) = 0.1737386528463270558077744808880133074086501275532113944375831650915094114338882068140714833030162376 log h_1(419) = 253.7459610056385969250217055226172525512658268294180542747428022805397516936386914562062901302568031 ------ R(421) = 0.8645796653071174117734286953546534608861148898938223168058525024949309643317272438429524323760464019 r(421) = -0.1455118262165840239256577690847754854305395157382629691287937779983179076192150347057706747540311441 log h_1(421) = 255.1125317787132400021570052721063245341414802600672325850840003841884173733223425620935518464196569 ------ R(431) = 1.137542611035934624617170856234859212415010376390005297944274978740134736293802244250951065107102636 r(431) = 0.1288703313927218328561929368350423993759373278054120395695876408159381936538527406153111959765503304 log h_1(431) = 263.8511760102000231473534868901455863635340840315748858852218656597668376160153248929661251109690496 ------ R(433) = 1.071761351820417713854505952047721894611940148211533665389154286535856789470480585761637115498160330 r(433) = 0.06930341829468954488024951965102398740929246203996768078327950383740781153348920415856747078179206108 log h_1(433) = 265.4914166067074036977060540768896728138053160882860428123692053921099089054405187754949756235828419 ------ R(439) = 0.6848413406172976205500589562641432789833003765778232254569757262119178545989548929995377666520234187 r(439) = -0.3785680870846511332738924797707621261900712932007989412149406911361612286303217030691356718235450335 log h_1(439) = 270.1566866946350674506088554326011720719479729367888672315753720572177481921117247976462231271013444 ------ R(443) = 1.410899884303979869809065683449783171567587543093065742601464446384455145506909589727101901754213208 r(443) = 0.3442277166353980174021778042258834939643129193619636413738749337324333121959507488862942156107271421 log h_1(443) = 274.2995725807626151065034756841493041467278606408162719829495903513075046399650110874746059166201205 ------ R(449) = 0.9053964365861442489589154746074225677099852352385750978627641344915473640657615249744234793016481200 r(449) = -0.09938237972234653630701025599976081043253466696831384775582373962263923204368516361491330784375753696 log h_1(449) = 279.0028882400189455931704623190921432600357586605904750331278236391385556031463103406568454648139243 ------ R(457) = 0.8373463419058562177863679134357931995713685157546844212052772143129503485964935124514694089289808984 r(457) = -0.1775175044672428111825397625096048639330137122416051829680198198544943040616803494984450629333393904 log h_1(457) = 285.8182484875954726536144413906375343854522374939228932563055275255409798527645368199140050173803667 ------ R(461) = 1.031195573773974036452847249066595969966168565628023365180272494918609674346988845216963008886450880 r(461) = 0.03071888032755222722920784129683861098911166015378085407106536346874388272178662030853924437442718108 log h_1(461) = 289.4863137743492644489308836360159794427581443627553414234888155996866457584550445740675306948506777 ------ R(463) = 0.9613462511195984177868663523170783907260879357241346385492941254325802319091680605287252799820845921 r(463) = -0.03942063197133420380911035530067068322709287352869429420446515463890311817451077655334613661074749509 log h_1(463) = 291.1493260090760740333979801634018644611841649187770807420855006410755036410656743948010357523717529 ------ R(467) = 0.8974045485919283687065708373771781001450205605388534074315253935478979113440498328818466846235664329 r(467) = -0.1082485168193483157534916747531697129453257127058912183979786530824169418512335903154691747840519601 log h_1(467) = 294.5532299664585334287500097079637925524428855973144518000853847149215294529643544389055256273472843 ------ R(479) = 1.105067157806420697059109789394804190393015911449014566329169279380057510850134000312720873036545853 r(479) = 0.09990610941831527069395404514499487884822579708001684898750755690694894531566986874164315661316620313 log h_1(479) = 305.2303564061599758406732508483717415665851610034263888052640801206442236280944935930358534463324403 ------ R(487) = 1.130410227826560631394536971555942483388862501331247203556883466695876591879044019905198551353038320 r(487) = 0.1225806003183738125038167447664067327691670693862713669350500383862100698313705861957870238248583528 log h_1(487) = 312.2739557210246159810948738012862347005658632776825207672597653055259375231509804273502752259924506 ------ R(491) = 1.272214656913049683527543549849645032164005434982986903004783856631910352293614287026378719010424675 r(491) = 0.2407592061151212623127680344944161784536140063820609931337140232817885418189372608885436463998831051 log h_1(491) = 315.9148748990087145730409344565629038603820938986430833398854765513834332024785372959294752546569489 ------ R(499) = 0.8297902495946506366988138268051999202168905526529697617021609853614314772789820983770011254795676136 r(499) = -0.1865823214601289177483648032962771506479050667417088383185678889570226940864500599085057768827434412 log h_1(499) = 322.5572403399883834783435964052056391540127460322245172752528905995607609636545691767060745251052362 ------ R(503) = 1.099561747195783290933622104658754638059130507595338581775776525266753057913360691889719047365294304 r(503) = 0.09491168877712382502557349034228042093786589316258023754500711991409291901188323142056899410585877798 log h_1(503) = 326.3855717182083623726905590640850708045444327055159566258318727174997987520767729161309695861758856 ------ R(509) = 1.396920827196126613204174106509056402194802808800814878436351907498074313033065408315445915001017984 r(509) = 0.3342704052229593663090001696153265365140478608274090623624711082064921242278683750156592489531640764 log h_1(509) = 331.9599888362760629341812997726688490663948215864508294261373741627674735589286066330905116943579210 ------ R(521) = 0.7448857918191827286091015924803868220890700810921594822949427280415640840601791017075216202484541900 r(521) = -0.2945243719257246884790349878718502932938086480749888972102176723419983325768444047343929888453386498 log h_1(521) = 342.0538410118954611648120717565017302681329731115939229241019103703354307848104303052811393198445019 ------ R(523) = 0.9951484787399289420380269322079374646504497475670484134848710423531339915818710380492602312083262786 r(523) = -0.004863328092216618820443539412381844328211833317035982103926608552694099269551942326779619180007526227 log h_1(523) = 344.1373320441659189377984939423854178400596533625228122370714175055704854104311898114944191102471199 ------ R(541) = 0.9447265578295298152134577952949957448302473532267567700541586620882228133112570150407309483716227860 r(541) = -0.05685975015429724767115260953590209926487888010457461261204762737944531914947222120107411498107506749 log h_1(541) = 360.3145014230038544339132651644011675493654484596299885882834749092368314937194631561066855989525083 ------ R(547) = 0.7386850547619545899616661320191055714042517207351723421958264130737470914387986866216583077034248639 r(547) = -0.3028836264621086503065538709231821853655558195747866405838819057866900375606225791841543374110745852 log h_1(547) = 365.5115347573689354215513776427461165269973782090557974807594291542186806144778048634024957874315867 ------ R(557) = 1.018006181309704402434786751402581242206083762842114062712588146266426421913972166555221333776686755 r(557) = 0.01784599012335848598610665140300491989534264549310351334762522534993153051328157345021114014824490126 log h_1(557) = 374.9403022987050224331397764896072207201383896094962211393289531400029663120561032380840113545255098 ------ R(563) = 0.9232212509133752364416200184615250421726918014071928857005016382345575848801502613402794137884930039 r(563) = -0.07988636473957336266351246524844445452558958378677557650422104653230771651646313496442251691149845252 log h_1(563) = 380.3288725365604801193268416856629675397189868163909322564688571165046385135495112319805207160224274 ------ R(569) = 0.8664438451435738527270516828484225160574569471468748507296399841925674989415227023948758824752928689 r(569) = -0.1433579784544109620172877125511876587767702082522193896668474564606294734941111177057787631667063706 log h_1(569) = 385.7676044219167387166250818724479718353856658534495352813503094599345799916648372034036732942215331 ------ R(571) = 0.9966248063685197276230915134980699083934785820192977357099270110462162406519135665546840613355693826 r(571) = -0.003380902446695979969907711208460249607678613691586933231314733256224064950923564299465263391091593181 log h_1(571) = 387.7451539370872526154095237763844466570082623924943160065123687274145359526807512509636731707579177 ------ R(577) = 0.9137029380401851023927738920458943111155625226822535774357016373587734443081960787127510185199656660 r(577) = -0.09024977344132332876738145183818092400423494900106726431935200258424930722244073140368105401478258505 log h_1(577) = 393.1814309249015591292833420542340540992094906845879928061234542124120212976280077524859840458566760 ------ R(587) = 0.8125245985067212166037417395454322072975722437150646296605939604786932565985320112973590236065565803 r(587) = -0.2076090901513303002520519865163954733545718349814990179328378436588143942225608435256879519994848111 log h_1(587) = 402.3037185571146144501862779927765641065146647979565171789070548798081324716693474722760323082449473 ------ R(593) = 1.077346174896649307807591884417183046154854523516579642618940036344301765294286112618082629467205606 r(593) = 0.07450077168762881704609004989367730156818024666656807594813051982336673551288935897294078501094020370 log h_1(593) = 408.1500017457424034219544600230983583220072345515147017908217365607580434027327355882183112501152553 ------ R(599) = 0.9640877383472306977957126847174917068153125275601950094951930079310170157733737250128223159725660283 r(599) = -0.03657297363017521077056347571067973001449428729950976903752022105424614155305512145323387544933401231 log h_1(599) = 413.6182018669105975099444617938473795829335791175045027980180417390992916095850441995379287935692377 ------ R(601) = 0.9282733975182409725085430055023724335029937867496534233225480389996637503729738802730413380113857403 r(601) = -0.07442898019010633925495846508995724855401141415124429127166581075793493087733140015404818409695276142 log h_1(601) = 415.4434333923825383721995883064487013521786575489799743872638630684245956805040739401824507578028009 ------ R(607) = 0.8363731270525144324766779910174741985583386988525834264734600926959280360642933372637590371411235669 r(607) = -0.1786804412449700266508512543599575258790094528794461377218491715824502174167330497554911354143521053 log h_1(607) = 420.9383562538567787417573264194144236812700764630674384532873492638217551756373542874969312684574973 ------ R(613) = 0.8770365930347214891035502029408569765118181131162512311544942468164698607840141288665886161213191852 r(613) = -0.1312065622412924812733307429262897499308005610337650378800922445574868366143721979381489760614270490 log h_1(613) = 426.5997584312354903723009221090057354727800206819749679341704108649046533369409050731857414486013886 ------ R(617) = 0.8424608454194671614144537884806184691936094209953385761233677966241573777275193359449723548823190436 r(617) = -0.1714280921151472534308671064796055748650273064928141416595268480039338659192966354060433607096896942 log h_1(617) = 430.3102813397015002889268575736976826693270411460166549264337468907142420703157196332069488578372231 ------ R(619) = 0.8046391863654823181809704923832998782589264059445877896016522629176633205529504867111182054190977672 r(619) = -0.2173613177373065509566789602889391460743373021214358491393867908373624441454530833078263849828383755 log h_1(619) = 432.1421422447895782346762792307795759835303094991254880846820723058126090277882790235509839222311318 ------ R(631) = 1.139646980724427664795844477306395537775596420845209540590755683769507629227314116989366266521001369 r(631) = 0.1307185484187742762831159249494009930774221473246067179183349841554086077548374731137676792438255989 log h_1(631) = 443.7905690228171776927725924468651267275569033966960289895438880223273849040930740593451143880454668 ------ R(641) = 1.342991564323284754452636732454451887293086218229413643275387564778419824502333998401955206840909938 r(641) = 0.2948996363026727326019105578819342053503641571391053557191299972206147734862316833007610001925031245 log h_1(641) = 453.4151281300999397157865639980024511869171276959062587642346458938040804931072326397486632931222434 ------ R(643) = 1.018362056113606853044175534937774061951499530225231305159793556358019692899554873923930352661517730 r(643) = 0.01819550923388602407320655980216466362130100110023919070694272032036301996543186638128443094473746698 log h_1(643) = 455.0351773367694621537465612318611599709874779013140284614192337734417586640197553536135227558343889 ------ R(647) = 0.9023366731711887559549077220931761127953933988246795850976095915390230989308148960394946793317572977 r(647) = -0.1027675767078984506687441752943372664408679592133207114337341796372382557164754697822495882483910128 log h_1(647) = 458.7123600100297974251042257724851608708560264428573006151891223972448656192689470191515764374028672 ------ R(653) = 1.270877278057724664687960983383727137704483996143924046255845594669239383689005284985743445560900040 r(653) = 0.2397074321228234915396816481072066274010252391317000797102523958553079469601605061420328798541724598 log h_1(653) = 464.7635801645708352319059013096115954217495158185318594495028361037448570282275824227418248652108058 ------ R(659) = 1.391063178982265501439982685260788597885564063552479813749213030905196336098530255285429118246556489 r(659) = 0.3300683317389475948286134071886408588546143983231248900748205444858852598853454833494511298626006088 log h_1(659) = 470.5764056237891735910300012844312991207504566768899234428806037698826473851814327470410354068477690 ------ R(661) = 0.8354443097523214656936838597289916332025941115947209566629665368631118264720588132892066752199806196 r(661) = -0.1797915881624607396438846699955941236154524215918006697045940843964569000078981140172950842324611413 log h_1(661) = 471.9770610926933984443101931112203480212040953312784060559598922256365020017536187709287982157966367 ------ R(673) = 1.036602069823986371811873532108773773230786381968069268630019689704281625366314919961348373662245231 r(673) = 0.03594812351466140728254044998520852529232737554827068529182274326942067786485076536063283842840573228 log h_1(673) = 483.6873617004935232141541764830894878307465378799501716204461163761647772334536062457934072698375924 ------ R(677) = 0.9242401331249736440179204466235681915970115256992006548432261124465663675608620118468603709649011681 r(677) = -0.07878335677014975095651875505186139731622953399483422827868941463671373704866210462409471819164794737 log h_1(677) = 487.4160317725177709658754405606958352979205907814646263154835518514663574008448329056132907210784540 ------ R(683) = 1.135282814024094769982546911342267330276686384803097952961758353063495640231481811335989750370036375 r(683) = 0.1268817952459694158398114948142184006424273462275918658227473686141841492133956030784644045841237549 log h_1(683) = 493.3978177466096007012301064085667896999607052853702670182373789533591354432422851534938937473745237 ------ R(691) = 0.7692142795745405069640641103691671529274100031613757327536207394060071391791846153324997553066453480 r(691) = -0.2623857012503545809323626395328591176487583726767603497606723557205854901856891742409400845855782164 log h_1(691) = 500.7304329917761259615033764763318689843335364731140026746723885747606796718814316905870037708555643 ------ R(701) = 0.9208988286796986104162625438273950741220198857349761310290753103532745249135886347443397280738869772 r(701) = -0.08240509818558594907431593660048257546800810707449685557053553613604703815863385732040864257319775632 log h_1(701) = 510.5951569571912960421734233298534469348820123565851404903432760745240455297075898143642688077977678 ------ R(709) = 1.056489349178018616061748003412687144528016447494990577505368694303224708944096983218390662283012474 r(709) = 0.05495147679080495753643115327436319621637171305950924260305700095132912579170692714307429336921755700 log h_1(709) = 518.5059008711611439378852860098169631925825244175741941171167285390533502218088423282946388461721709 ------ R(719) = 1.203063258553339276811172437289348533829160575829628399088857368008857311545285623407421572933184454 r(719) = 0.1848710196111649838208668402218769519776013145674599313799988826665746100544080575242205416341380646 log h_1(719) = 528.3841264278650089739920823929947651036853616356010723589299624880715514358829023407246632104756575 ------ R(727) = 0.9985692142278032863134063960758987433884779764495074381232399999496857901401074901307597578042968988 r(727) = -0.001431810323551953610304460369186967496202533114727206125046988264992507473009467571947004135579019311 log h_1(727) = 536.0214224135112671023102058445069728089278151717698530745194547428970765580408178382496321989585580 ------ R(733) = 0.9801491017726198673607802262180185548575569083268428660260284255483287021055691943168485013448878070 r(733) = -0.02005057422440668076474244033226479306455534357336646771966685795704605068767096486228873070074542804 log h_1(733) = 541.8848994476589662229744832086718877700832059236197452523304399284231104842589219920560316486614821 ------ R(739) = 1.102635468240530866306712454643868064544922637848069637799483854548800400982442612663183515564997669 r(739) = 0.09770319447806712360477832046359414525372413909306892863584435488072206940091718286534281553273845178 log h_1(739) = 547.8969772058809581164216363020158136816585812640965479947633963092327482654097171332626123881003208 ------ R(743) = 1.034954940962057759040911768305799112216659615821546088785900432743506302843113918119563230456587841 r(743) = 0.03435789046728396189204146840261127867838392692593392537124518014758008678020236151061721499251084078 log h_1(743) = 551.7699257730221391164370227687291518962634184101681228442691554501649871217520871412660611633785297 ------ R(751) = 1.018562005835850738780958489753646041102797557369284509597561717870320178583194868497355514144290497 r(751) = 0.01839183439555626648371160273692471992134660701035653121416349343530808856167211932943792341749557372 log h_1(751) = 559.6426044689903445328179178603989168720434955423818238765510918699514042910351505486983096685876066 ------ R(757) = 0.9670687611870859854554145544772664532165877060773598761129540326758872745565277416187259881804314231 r(757) = -0.03348567831428690817645912165610501108370458452139930216412532246661734038189781464673258145692582924 log h_1(757) = 565.5211485526283331089533889368140306762159083939280476972395383741217430934858991444128020825129998 ------ R(761) = 1.469582858131415524913226569841312009167105693745979875895487742512449405710194222484970666391467311 r(761) = 0.3849785905416971348930257146146982531796000884837153780033224185880192273751533946008315658983278917 log h_1(761) = 569.8998118871323534351369990084464657861538351899655956066820527015310821055012791139097648383151527 ------ R(769) = 0.8989223036739211131497271647474289002673567555360138370735734348232825688838190571060798509745264724 r(769) = -0.1065586735246832820858759222915594705200994707117324633520171722769975368098345196624490345979188791 log h_1(769) = 577.3443521204358834247829899024202551982115455678781948360232893659983385728620223403751816976620399 ------ R(773) = 1.068109471970314471303333050352021879900722805565093006342107680930850946294340467694273562497172871 r(773) = 0.06589023712991168876370351042405780229901239656085314276495076437979212222264152584798572270314475209 log h_1(773) = 581.4926252096423608972997821133928597293813533909757585052909164184270925057757046270734788245578225 ------ R(787) = 0.9717823284398633668645155647478569676228286375082200985563699216210515716930279198195376877945792246 r(787) = -0.02862344153473408302791572221327499134589763166124718169324027045749509683948657809832595721261696577 log h_1(787) = 595.3539157025895735465241896601015556369493301348785638516072244944869514145831949025249507117742691 ------ R(797) = 1.030751303873609429436419865265522058501246636476678610598593325988505450147683962143537907726292880 r(797) = 0.03028795757968169836517584328163309139101536356774117051432936864628649250705620395158613342474105523 log h_1(797) = 605.4192984652631324136586984578558831951644613945870256327311333013663718396915792190388096047828749 ------ R(809) = 1.319704414060187122519495676448014184825508393453089814531975928833002683577157581958419243159277669 r(809) = 0.2774077827802126641119925028791064413353753507837008231119614094269892024707670875031365713708570357 log h_1(809) = 617.7154002968383955703790475339750212141653319958651580310569799785167628282947179605730664300171408 ------ R(811) = 0.8028381726481542070785681890576791964920689728433202927811272610992436316135609831451729158242808879 r(811) = -0.2196021138030531805787416878272833924996102153785387441767536510890995707015791185630636058378173324 log h_1(811) = 619.2308821834585007510488710489118529977753305070116236758933617285311603214131743287810839511866680 ------ R(821) = 1.065284370365496433193528146558398140954751382954684060584579582210059953082664920795479770457729754 r(821) = 0.06324177794707679677220457506582336344398248430390287100384509495786964991926848485599004532166092646 log h_1(821) = 629.5945522778755874989983881913203625112874402234759445648183026831280346769021478915598393902730861 ------ R(823) = 0.9676931847618204865646570591849633119800487701442729214835308307589660077830603214191677725101511394 r(823) = -0.03284019983801021429918715270910118721953623448033873820488086909071912851865808850276682260235098147 log h_1(823) = 631.5182881262958575110757690078019621773292972619369689113885723906512238082454186975099102757169348 ------ R(827) = 0.8655599367575844205769196995319298429463589071606259342634178860019739958338163213206779097520578668 r(827) = -0.1443786557438749141659830129276590851728592881643978679958983426004655309180827348729913287779878357 log h_1(827) = 635.4500143151723334401989900657834878056767059968241584449428157381552757557890836503870823916392172 ------ R(829) = 0.8225003354161554974840091964490409585382376347069555078757550663551833284093027568642099941419294742 r(829) = -0.1954063885418962882391705381888806918484681089233313109351868364418516142863750801222150635742524675 log h_1(829) = 637.4224275666903735416682977266749428153753577597079132763909356503470978754920191063952498876123469 ------ R(839) = 0.9187109054076576161004431766675830753281882634631602783320782642568366521686559996422127840096985303 r(839) = -0.08478378129769339599936100606994511159817384019795554575436473520002765097901829719238400070846535834 log h_1(839) = 647.6682262850933946033285161756017179598461320139628919475674440102768035766410131561116737325098720 ------ R(853) = 1.082235828802533475480042836136154761473707637768352185586034326318264211013672541989031668366137318 r(853) = 0.07902911305466984476747942540215013786785925712587250350180552712488973141781552040541112504028494125 log h_1(853) = 662.0710891664527703986009576266145590174617103019074812451871111088149359488187989831300552395496228 ------ R(857) = 1.050753114906946945763963820272607649744870269326320468963922855226676506224577200398518018018306392 r(857) = 0.04950715935926789135675862442009814848118736266226738413653713019897645119041272164905707857667966159 log h_1(857) = 666.1204224152420009582507466656105207014177721394735067901315331786874591076140715628768999714244498 ------ R(859) = 0.8808009418056817847639767572161194001026632434234854393782496836036778182395953423598111830608093758 r(859) = -0.1269236243125652703694868187379167919260839345341265966038920608003823590700411451168394640434093537 log h_1(859) = 667.9851647542452001103332536771864968583903054518782077264496148870719139114521816060253323815651022 ------ R(863) = 1.056942312064447641802404012885112282020641656519174748553263508922596464617266515247432845133229650 r(863) = 0.05538012835467755661639926243622062916921494755478098502669218826316353808725907222760887536803203234 log h_1(863) = 672.2532923060488696377642110809905526105355878250398891247326814159691787165727095973981747948610853 ------ R(877) = 0.7228939852270574121828463785409419554920239844051668739871540041537229599050699543313017402384380482 r(877) = -0.3244926993493331027946906208533383694867286675777468229361594339545329427009641576598058842710078468 log h_1(877) = 686.2100377157975592492997665224301471665554423090673860800569462834844574132881772347361379998840457 ------ R(881) = 1.097389941990753501844353363706203335467698475699987035016149826534617010787595153452637071717964200 r(881) = 0.09293458029251091806514701187307678983282262267888795847384664267732634885722688228499205886746936431 log h_1(881) = 690.7339078726361148092920866776221037496203613092940539362247733194719008021481639820898587047148001 ------ R(883) = 1.133182276393932149820390126847980220189832902958837281177924286544717762157891744603254140644684646 r(883) = 0.1250298485359340222824653605773065028894140032444680412497747344324333440845789182081854721866933770 log h_1(883) = 692.8209226763516640771470791034383481961603114933294675448596489530087338680518578325643189162215220 ------ R(887) = 0.9691797419679082310841771993673360473362366016434053541665244638233156522614244968552986621865511991 r(887) = -0.03130519206598333262075053980979844712664630577751465950710292465258532231437185834413170450322453269 log h_1(887) = 696.7778101330353493741026603771032975935804421497753094631304111573256442943524827714796414295556233 ------ R(907) = 0.9026255886631148047805162360749177002973054082675283299611499504530707027214220195671124644576347865 r(907) = -0.1024474420430330208447986792288795143504394143329670787353157620371318389380484624259490261433256742 log h_1(907) = 717.3397960562769854035266294383421027822992050233081366747229745839339623720470368301063725932334685 ------ R(911) = 1.077985575363048730993510437007042423824311133153234637470651700625734779743544257001418603441306500 r(911) = 0.07509409147246439012344853570365982657026942990996555447800731464160050891240374411237735627160273091 log h_1(911) = 721.6572280796806910149964123627185828988878517968117565386135993905094974822454776731462630664948078 ------ R(919) = 1.040033465541999509013173034573574422382242322360689501829736868041813305460863950597529974337431674 r(919) = 0.03925289104133626213282320996533058480633305365432319818894969263452974852066670165774715248885158774 log h_1(919) = 729.9142777926975695436958141608708703779906270929496784153026504985012900136753000480427350211860628 ------ R(929) = 1.044149044529891677448132017254925512711450293648381225855627811864984232818981432010629916813022886 r(929) = 0.04320224223045180073639001074135177163726525500151667142449677059047589351196906016182768947002205589 log h_1(929) = 740.3087267524784672667632688341791576556293753172833168641395643515541214137737123065520024391695254 ------ R(937) = 0.9001793485775001978413226252376923128687518512882950136486785231942125548936711709012872256765460061 r(937) = -0.1051612593134678238712333267364302510800045363217059773084848490684789495481871326050537513916922943 log h_1(937) = 748.4920901677020053112974058529158897098828061819166131836136854670009480647655415734466276781957038 ------ R(941) = 1.094008671797522355233972148468333604101469731321189443237034805979580791697171879079683228585324323 r(941) = 0.08984863065715072150281788229953620631905669219469553202765528003169881122802722417052833238864474741 log h_1(941) = 752.8593555409743125729320985322079600724744466234204994113592309634160184994136369042429836800942900 ------ R(947) = 1.225874482705107430260914904334844365943312503886016762782595350476089031678195738981009072142160088 r(947) = 0.2036544527501277181778522861137500571625532962375917667055688217333928477813229351464664829021001081 log h_1(947) = 759.2394838533832422460712160520261155410315325338455880545625079589991235885374600819938038077480836 ------ R(953) = 1.160831730312838856822268456013043133221135229480606503504798477964215408435483628518759434920589787 r(953) = 0.1491367570805829662790757788835418920152399480432079381658359778544840557540257098838728747591148084 log h_1(953) = 765.4607623001463829371802970742478384193192886255863850769266107282608812546333861816740283297094020 ------ R(967) = 0.7286000440466886148143682504791656287376605867111737021560334922744581935784605631716788702747265795 r(967) = -0.3166303339538069224219931699412646951694749817441141076486842179538222185127764104159385247631822525 log h_1(967) = 779.6750289183161438221502522137454085291649487143350942193461804509004774599368836133306416004882698 ------ R(971) = 1.079391159164400462587103816038585286634988626277050147346158884708080457271485070910724338235194030 r(971) = 0.07639714065969163127120681553385088281438826414188115604993458865621084011324441159341345724575756092 log h_1(971) = 784.2716621416246933054844446967366202986650255622922719041974912595341446154092864504557547401071956 ------ R(977) = 0.8389088588037128235412547247521113110098740031324507228637614611873150395830782253142754216965640136 r(977) = -0.1756532091621463780074051494680588989981087436904089918232161317421263851972022142817708822916545158 log h_1(977) = 790.3327153141189743402894929788236226757708564064019553310892677798670342760464519836541967658459543 ------ R(983) = 0.7886767720297385404724656676372853032180930005207262341162363103228486599586696823599514365864730647 r(983) = -0.2373987099694827668641811769365502357509543632578631137299141058277351489507820902084763501594819872 log h_1(983) = 796.5932569798783851968958867150387124759425754265643146228809357826715127233883420364810466147063885 ------ R(991) = 0.9094393615350512976006963975090972967702253280873744537844840704469053402941745713978151671678988059 r(991) = -0.09492695555556185628594269339544386175503257603847752863359561914998034997538249605837272667617552585 log h_1(991) = 805.1796341333803845679447372060219020939677553147513825523431535684005863332292076308928427671026393 ------ R(997) = 0.8557575449135065446654521786495578959805332500560536042382441383964281143781102272110332242328991730 r(997) = -0.1557681848844382835292131088140839426155248543028794239372195408163739639587151461468551381900370419 log h_1(997) = 811.4622924142810387728604702285170512664512073515702855202219606510545953567468726714143740457130137 ------ Precomputation time: 0 min, 0 sec, 15 millisec Final step Kummer ratio computation time: 0 min, 8 sec, 727 millisec Total elapsed time: 0 min, 8 sec, 742 millisec ****** END PROGRAM ******** ? ---- medium size examples ---- ? v=[439,761,1451,2741,3331,4349,4391,5231,6101,6379,7219,8209,9049,9689] %2 = [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],100)) ************ 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 R(439) = 0.6848413406172976205500589562641432789833003765778232254569757262119178545989548929995377666520234187 r(439) = -0.3785680870846511332738924797707621261900712932007989412149406911361612286303217030691356718235450335 log h_1(439) = 270.1566866946350674506088554326011720719479729367888672315753720572177481921117247976462231271013444 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 70 millisec Total elapsed time: 0 min, 0 sec, 70 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 R(761) = 1.469582858131415524913226569841312009167105693745979875895487742512449405710194222484970666391467311 r(761) = 0.3849785905416971348930257146146982531796000884837153780033224185880192273751533946008315658983278917 log h_1(761) = 569.8998118871323534351369990084464657861538351899655956066820527015310821055012791139097648383151527 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 106 millisec Total elapsed time: 0 min, 0 sec, 106 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 R(1451) = 1.489316072080934425611321346752153251112925802956900290040634402655807967015047584928112919576415704 r(1451) = 0.3983170025528813314231091463545884443659620970780983069222156624629822149855265946906186284878454208 log h_1(1451) = 1314.913590959739633280304174458475051730174849655225728115040943264726912600350259557667300698165713 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 0 sec, 365 millisec Total elapsed time: 0 min, 0 sec, 365 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 R(2741) = 1.498121015176665823721124535220681126275264457810035935757209895578753430046456359802102518921683664 r(2741) = 0.4042116663292426669821341985560782898487403326206660145789286980627674932710947874619294622553687136 log h_1(2741) = 2913.635351929796928432167472612271200974702541972086071992801845378552195256634369280557140261024826 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 1 sec, 286 millisec Total elapsed time: 0 min, 1 sec, 286 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 R(3331) = 0.6424292976347195066887411522702535091860164308136846530503923245084279154930131836095028527208646306 r(3331) = -0.4424985108597686750119146368506688388719272425446184888306794304679990907032738872976868720493865016 log h_1(3331) = 3700.727036232571937039498347892554270122091028094027172466294586777915190036095724169998072766942764 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 1 sec, 903 millisec Total elapsed time: 0 min, 1 sec, 903 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 R(4349) = 1.518570512426339397454202981116036423512615333649478776056105247513514423760156552056212687759932816 r(4349) = 0.4177694400043232472325215983909787844424794591317313642447234301990471958313879959425452519904262301 log h_1(4349) = 5120.505093552877767548544472627536999455469258347353430438266163043065315405697762230776943858603984 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 3 sec, 234 millisec Total elapsed time: 0 min, 3 sec, 234 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 R(4391) = 1.507776410131052825600361832032637669138026153232363037151066461540707638889901157797803074416318708 r(4391) = 0.4106359894489440206811727927960384538866256726862301144954329290568552795104618433235264511066036294 log h_1(4391) = 5180.426151613542166468864891843764775854358647874100600958902353834225058234317241929933803779523544 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 3 sec, 357 millisec Total elapsed time: 0 min, 3 sec, 357 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 R(5231) = 1.556562247546690554629305894110505251320095101081599698820908878895451219655926127607265380179629768 r(5231) = 0.4424797020990258532407953334401841546922209643815483895179503362507707783343939457336472621420181313 log h_1(5231) = 6398.932205422665116359605763722252251733047138748820912967196262294815862544585734907654808725570170 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 4 sec, 675 millisec Total elapsed time: 0 min, 4 sec, 675 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 R(6101) = 1.511405291132409881116244836469133082076697844961579765627450761700422515970899394283746027261592536 r(6101) = 0.4130398744137640926775072936011235563729980540246994086774851203441076993971279864804599726390813160 log h_1(6101) = 7696.514498484196814923446293509463839644713139346047599730700163856609162381034803617672712337705130 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 6 sec, 356 millisec Total elapsed time: 0 min, 6 sec, 356 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 R(6379) = 0.6735230262787954049821487359022947355039240336610888832179088947738237816527269985036067406314787700 r(6379) = -0.3952330947350118129557002367176298891357809287595165999709979986053057490015478075153389765614891747 log h_1(6379) = 8117.111098284461879294444954979907779173393589703526250308788773295906368060779424037877904045125372 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 6 sec, 951 millisec Total elapsed time: 0 min, 6 sec, 951 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 R(7219) = 0.6580840900963173782917427954501610677165051676074859457699979024909345356062724683817079928138142686 r(7219) = -0.4184225593231341881390391865968328228681099237096141378317362239807921335116232075448861831035187392 log h_1(7219) = 9408.290079050164171830738368133724370888673576633927217652921585081925428084948934914143340216832348 ------ Precomputation time: 0 min, 0 sec, 0 millisec Final step Kummer ratio computation time: 0 min, 8 sec, 878 millisec Total elapsed time: 0 min, 8 sec, 878 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 R(8209) = 0.6720450390038575959197342229439232609976262750154917933921909165493734131266509841484378210080177636 r(8209) = -0.3974299183777142869478443959855038282974856500694812347635728463427906339340289957102647941999939874 log h_1(8209) = 10961.30930020921879616344751908167941674845253703093388805424462242932470900767181460686227886344939 ------ Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 11 sec, 511 millisec Total elapsed time: 0 min, 11 sec, 512 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 R(9049) = 0.6676142441711162320155692165757091780734362715451223843405201217765372698141087373945342826962493320 r(9049) = -0.4040447510363401613599741435530353985623629250605273593475581509889944551571660633221918386612050366 log h_1(9049) = 12302.59002796725916356889459683790798659775694001638248193743009911555980091798723524595124433267196 ------ Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 13 sec, 968 millisec Total elapsed time: 0 min, 13 sec, 969 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 R(9689) = 1.524371504087494924535704793958075704772032498943478845835776791349364507084395024449618832401814645 r(9689) = 0.4215821966438812205877909269900967007050037924828259948662040970896255986275870321082174455181326706 log h_1(9689) = 13338.54099012319929661558842691821209687690754083646956370884449712213077724342568058945705663677746 ------ Precomputation time: 0 min, 0 sec, 1 millisec Final step Kummer ratio computation time: 0 min, 15 sec, 965 millisec Total elapsed time: 0 min, 15 sec, 966 millisec ****** END PROGRAM ******** ? ## *** last result: cpu time 1min, 18,626 ms, real time 1min, 19,008 ms. ---- a bit larger examples ---- v=[37189, 42611, 149119, 198221, 305741, 401179] for (i=1,length(v),init_Kummer_Bernoulli_final();global_kummer_Bernoulli(v[i],v[i],100)) ? for (i=1,length(v),init_Kummer_Bernoulli_final();global_kummer_Bernoulli(v[i],v[i],100)) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[37189,37189] Performing precomputations of g^k%q for this interval Precomputation for q = 37189 Starting computation of KUMMER RATIO R(37189) = 0.6252312557876547952334176018598454788919061504315316351837630004347066179262859506534156570641097278 r(37189) = -0.4696336884220332991843240570184268692189244807035367291133131112384916757960954518239657760177741016 log h_1(37189) = 63676.73497184487211564289813423054808495059045178845465897084765303987705626043848098912627227638178 ------ Precomputation time: 0 min, 0 sec, 7 millisec Final step Kummer ratio computation time: 3 min, 56 sec, 897 millisec Total elapsed time: 3 min, 56 sec, 904 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[42611,42611] Performing precomputations of g^k%q for this interval Precomputation for q = 42611 Starting computation of KUMMER RATIO R(42611) = 1.619906571157532399867361172777560352578175436944176426790030204404437466272596893962498628164837124 r(42611) = 0.4823684754562117636750253027478134255954416635260924019475519685992449917188075880665895804742273902 log h_1(42611) = 74410.10531636314534476262079708120362950244647452734233315467772342805221966481739299483410396060951 ------ Precomputation time: 0 min, 0 sec, 3 millisec Final step Kummer ratio computation time: 5 min, 13 sec, 378 millisec Total elapsed time: 5 min, 13 sec, 381 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[149119,149119] Performing precomputations of g^k%q for this interval Precomputation for q = 149119 Starting computation of KUMMER RATIO R(149119) = 0.6241497159784014254093473958479025529053241079277606079498127687690000186179385198233337714430849142 r(149119) = -0.4713650099386109951094263781653659431306377710268248890229476490884583836786037591575866254036444811 log h_1(149119) = 307073.8990002758277646124697205493221623129706168469511910097919065690200805211393476120775263018772 ------ Precomputation time: 0 min, 0 sec, 10 millisec Final step Kummer ratio computation time: 63 min, 47 sec, 641 millisec Total elapsed time: 63 min, 47 sec, 651 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE KUMMER ********** ******* RATIO IN ONE INTERVAL ******** ********* with the the generalised bernoulli numbers ********** Interval=[198221,198221] Performing precomputations of g^k%q for this interval Precomputation for q = 198221 Starting computation of KUMMER RATIO R(198221) = 1.623477270751197661500864242418175767860985763523973784554298824019733332984017347273604704369243755 1.623477270751197661500864242333 r(198221) = 0.4845703123095099379315541083678552897005566258746352635852921196871048962784432021820722427368377283 log h_1(198221) = 422290.5449031533371686012803824014414146554010729521684512963192601556208701469030436161058803059910 ------ Precomputation time: 0 min, 0 sec, 14 millisec Final step Kummer ratio computation time: 112 min, 4 sec, 834 millisec Total elapsed time: 112 min, 4 sec, 848 millisec ****** END PROGRAM ******** *******/