When we last left our heroes (in 2003) there were 7 elliptic curves in the family x^3 + y^3 + z^3 = n x y z which were apparently of rank 1 but on which no rational points had been found; these were the only examples left in the range n < 1000 for which the number of known independent points was less than the rank. In order of expected heights of the generators, they were: n = 901, 609, 802, 748, 907, 697, and 769. Several correspondents have continued the search, and notified me when they found something new. The search for the points is complete as of May, 2005. More significantly, as the last email from Mark Watkins shows, recent developments in the theory and corresponding software have made spectacular improvements in the kind of computations that can be effectively carried out. Here are excerpts of the email. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Thu, 06 May 2004 17:58:18 +0100 From: "Prof. J. E. Cremona" So far I have found for 901 the following point on the minimal model: n=901 E: y^2 + x*y + y = x^3 - 13729605380*x + 619204181234842 POINT = [257427463274884457842403544888118702511974889478774701840194654064521834405591644196224624947405497630430045100777340168175869525348516060:42846550018041241573271979664009343270823661201847151217355924327259691224153184676413936351826522851290220316890138716760107511932244335279:2503478708751002857018548800447375205923107994952941352941304402800308745537914556750502960647652681002516363838764304110010515017408] HEIGHT = 205.14199076889173056770 where elkqi took an hour or so (I did not measure exactly) to find the point (-118284454761:-48546457015:-43233122682:6981358683) on the quadric intersection [1,1,8,1;1,-9,-11,20;8,-11,-19,-15;1,20,-15,36] [9,-2,-1,7;-2,-39,4,21;-1,4,7,6;7,21,6,-26] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Wed, 24 Nov 2004 12:06:49 +0000 From: "Allan MacLeod" It is a simple matter to show that this point corresponds to the point on Y^2 = X^3 + 901^2 X^2 - 72*901*X - 64*901^3 -432 with X = 648583460084781852830853310223 284340197111505365839745672470 730751535911764628052950171223805 over 460931027407283186401522365020 392298080348389282922230733024 2252295465413564879782007076 From this one can easily find the integer solutions to the problem. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: "Randall L. Rathbun" Date: Tue, 1 Mar 2005 20:41:31 -0800 901, 102,6 , 731432674, [1, 0, 1, -13729605380, 619204181234842] 1155354261148416*x^4 + 679447196288*x^3 - 459316592*x^2 + 14 [SymmetricMatrix([ 4, 1, -4, 17, -18, 7, 7, -10, -8, -10 ]), SymmetricMatrix([ 4, -39, 8, -3, 2, 7, 11, -17, -8, 0 ])] p=[924610497625152926289609230674087039971493915281315106810214588681845654459092553807401262/10103218575606878774712615689460294650914958612656558875056608040193102391757081398441, 363547604215507421525825345494176369530000345247759222483204769508079904419740729030497972859529991407491319523821077633966254751141763/32113646548016808332459642524038323681960379654513715946133670344097354101086438501331246413993719607042885560703541290405859989] height:205.1419907688917385306339557 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Thu, 24 Mar 2005 13:10:29 +0000 From: "Allan MacLeod" I have managed to find the solution to your ABCN N=609, which was one of your remaining unsolved values. The underlying version of your (***) formula is y^2 = x^3 + 370881x^2 - 43848x - 14455458288 on which is the point with x-coordinate -1826843426987651703064905348967723957420211543012028718 015201716003872663044484720530473090955156221944209073 / 1450168540540529706542652716082310713679738480179146621 4759538839707587100959642045056468658757727492649 from which the y-coordinate can be easily computed. From the point we get the following solution to d^3 + e^3 + f^3 = 609 d e f d = 294862252674304954123771140539307835473668857600049426 405079075836511512420626827510295128263320537212336389 9825467960127768773511653705609296540406216597746 e = 185436615869089266089702956453425148040625320413197730 573785651767583018783905247919058936164534220905090842 279604088423597124984446387574830586425949377755529 f = 191706060808040346368834241692091759563783860373218147 559146130835383172170300634585089506737802463043237192 52284175017348206491850754496659371605138841041825 I found the point as part of a project to develop a Pari Heegner-point code. I estimate the program took about 8 days using the Windows version of Pari on a 3.4GHz PC. I cannot be more precise as there was a bug which caused the program to take longer than it should have. I did not directly find the point on the elliptic curve but rather used the related quartic and found the point which makes this a square, then lifted up to the original curve - this approach needs much less accuracy and is much faster. I have now started on the N=907 value which has a height of 162, but the smallest conductor of the remaining curves. It is the value of the conductor which is the dominant factor in the time taken by a Heegner-pt computation. I estimate between 11 and 13 days should be sufficient. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Wed, 13 Apr 2005 09:15:18 +0100 From: "Allan MacLeod" I have managed to find a solution to another of your ABCN unsolveds, namely N=907 which has a height of 162. I used the Heegner-point method with my own code to find a solution in about 14-15 days on a single PC using Pari. For the curve Y^2 = X^3 + n^2 X^2 - 72 n X - 16(4n^3+27) with N = 907, we have Y^2 = X^3 + 822649 X^2 - 65304X - 47753129584 The curve has the point (X,Y) with X = 13596 98418 03391 85071 97885 00883 71955 66716 11333 86555 18466 83181 91305 31731 96242 78902 65180 80861 81510 93411 17672 02428 43768 51126 27772 71874 59393 82406 81498 4 OVER 15354 41252 38159 66796 79405 91056 01786 82556 16184 19863 93824 30840 99900 94286 53436 59837 67443 67144 39131 50110 63396 12464 41994 19068 40497 84545 88193 75602 5 and Y = 16082 93092 15239 37912 82727 90014 81397 23103 61683 50797 63762 34009 49890 60611 78699 83256 34450 61595 93364 77847 22521 43920 14416 49521 80811 47255 22009 14882 44110 65928 07205 86291 47460 81563 73628 45914 94285 17413 09980 11957 54802 70678 59279 0252 OVER 19026 10139 09176 10434 91133 84517 62319 12403 81800 78374 40953 04830 79844 46249 19491 83242 55800 95021 02468 65341 16693 26730 47452 91028 96880 63213 99122 78914 14769 58261 19954 90817 51897 34703 57330 16036 01308 33386 21897 79645 97445 21387 5 This leads to the following solution to x^3 + y^3 + z^3 = 907 x y z x = 12136 94899 26899 64099 43216 51292 41272 43751 11041 24228 20806 90212 47288 73316 92248 55174 52425 59770 77191 54846 43226 74112 95781 68646 84201 52810 69062 46776 11355 94712 96632 36307 67445 53287 03579 79099 20156 87163 43482 06861 09046 09314 1 y = 20494 14247 87938 42257 06824 95305 28636 38395 28154 70281 81599 85343 62192 61262 87277 24935 12044 27661 14452 18204 61523 78527 21240 92489 32394 11557 67691 45388 15988 37892 28038 03442 42569 20811 42053 25438 48249 91735 01794 39652 03104 35193 5 z = -47500 08059 51521 66209 97829 53061 35350 98090 74051 02260 85064 40342 02372 39203 47219 81013 51245 88965 46934 70257 01367 99405 71847 76785 65132 53193 23211 15633 32794 16143 76038 57738 86580 22126 30885 87114 84680 39466 98745 19500 48410 40471 56 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Wed, 25 May 2005 10:07 AM From: "Prof. J. E. Cremona" Here is a solution for your curve for n=907 which I don't think we had before. The method alluded to by Mark Watkins is (1) compute the point as a real point to many decimal places using Heegner techniques (which we have much improved over the last 6 months); (2) find a 4-covering by doing a 2-descent followed by a second descent; (3) pull back the real point on E to a real point on the 4-covering, which has smaller height so is easier to recognise; (4) recognise that as a rational point using what is known as "p-adic Elkies"; (5) map back to the original curve. Most of theese steps are prettya utomatic in the new version of Mgagma due to be released in July. > Subject: Re: test cases for Heegner/4-descent > Date: Wed, 25 May 2005 10:33:34 +1000 (EST) > From: Mark Watkins > To: John.Cremona@nottingham.ac.uk (Prof. J. E. Cremona) > > Back on Oct 22 you sent me a list of rank 1 curves for which some people > had asked you for generators. I have started to do this using FourDescent > in combination with HeegnerPoint. For instance, on the n=907 curve, I found > > (1266527566883682569438334389195272127923072642784511295600984538 > 45096307441017846000663129624761355382746187269396463046524637281 > 35754271922715469/18458111834916763305338433752183352460423099849 > 05390978282616158472101522516417576527761879383727177402362377791 > 75541761566691806047182256016 : > -6788463077772309579746321733074448920010568039730943671016676126 > 92174943453010044491124782889995566093049413092889242909377770525 > 40079810385030946308262237709605284373045464812732552959800513921 > 044960171743250130393/2507730773891567862747051916526946458257835 > 19165534691004417529511616044209658856677380120779647660176763708 > 34416981943772991515934740422253015347584572024678954507758282680 > 83252921546029106245972921109987536064 : 1) > > in less than 2 hours on our slow 750Mhz Sparcs. > > > filename="dofs.txt.2" > > > > n = -158, -182, -188, -200, -221, -243 > > > > dofs(n)=[0, n^2, 0, -72*n, -16*(4*n^3+27) ] > > > > Summary: points found for n in {-158,-182,-188,-200,-221,-243,901} > > Have 4-covering for n in {609,802,697,769} > > > > > > 901, 102,6 , 731432674, [1, 0, 1, -13729605380, 619204181234842] > > 609, 121.2 , 225866502, [1,-1, 0, -2865684327, 59046511507397] > > 802, 141.3 , 515849581, [0, 1, 1, -8618990809, 307983862552785] > > 748, 145.6 , 418508965, [0, 1, 1, -6521768491, 202717213102626] > > 907, 162.2 , 186535654, [1, 0, 1, -14098991107, 644360446364158] > > 697, 167.9 , 338608846, [1, 0, 1, -4916886147, 132703046307558] > > 769, 178.2 , 454756582, [1, 0, 1, -7285583301, 239355096410424] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Date: Fri, 27 May 2005 11:16:12 +1000 (EST) From: Mark Watkins n=697 [1, 0, 1, -4916886147, 132703046307558] Quad field -3028, With pairings, 3 terms : a[] = [ 1, 1, 1 ] square of index is 8100.1 Using precision 32 digits Taking 144323240 terms Found a point 5362128491705441036491368662525704166630421462562300348423419364121353816 2737732499618869393866267074615302924865711799471321014611377162374037551/ 1557269171495805403378803174382136715386201925389481567488011193460618857 5753008120381963237827143272612722920384596220726384913784262087920484 2091008643102789413113145124435817230952322451944916005912290097115425031 2318720962456385912279035286201629748302297329985185177758412974513867691 624306823314734889736905169117432569723469284225960134218827015891213520889/ 1943325409610403775687061872445878001020543820722597733655693043719604599 0831043695868392183697669986964419815611804400444426372060383863002006300 48621394842228655586445235227860854428758964735698801404368882620648 of height 335.933775094707993645907881756 ================================================================ n=748 [0, 1, 1, -6521768491, 202717213102626] Quad field -4360, With pairings, 3 terms : a[] = [ 1, 1, 1 ] square of index is 1296.0 Using precision 30 digits Taking 139363390 terms Found a point 78653737409040479130262856220948724245848142231328782584532871148 54937430975119193716102043583854908637298952000660899675927448294/ 83614042542474532088713123899973765986832130880425199086790058295 916654967513701908424505043786473049871386353159506218021809 -4964524455345684549895325040013627691789441821314156508824711268 07898162249178665209869058086582908087833221413807525972536652044 478299520971158876284383273567496763089580007958126938533737495084/ 24177914173098797213066316306033172954025691690578610276823421939 33717577694035899530109619057986236270554738549752702226809192840 9449073552993512650689750200895048231854718156012939503127 of height 291.228324714725933482724836391 ================================================================ n=769 [1, 0, 1, -7285583301, 239355096410424] Quad field -2392, With pairings, 2 terms : a[] = [ 1, 1 ] square of index is 1296.0 Using precision 33 digits Taking 224894190 terms 27459277684449987877760118806857426148212282369872982 77607632364030625754997464382740380309193931494836427 551073675501729792268983035660490129899517912107868/ 55653971635295660084686145258798186730245642938678187 24928092205987366064341261651146889341076331998424420 5716710464499744954881769325237306687356528729 -689405872326432712109904778685087024134389954584598010182 3950844257061139688178461261452472155490193153456502774218 1898633761185632846960526402139585802252072737675001804174 57624438794592333427637339510014950540866321768058206870245/ 4151874748816970770869483331760882928729662844688131670325 5028459851625770690739516066718111193178854085127194992715 3162985185960790513019885390530579980622525567012750083642 734850905312030203572633142212496750984924854188596317 of height 356.423060274797123343250843383 ================================================================ n=802 [0, 1, 1, -8618990809, 307983862552785] Quad field -7123, With pairings, 3 terms : a[] = [ 1, 1, 1 ] square of index is 144.00 Using precision 29 digits Taking 129913875 terms 344119757878014012759045222881991007813783228475152134913457074 47247705487619882723649412109827739150654395539227159166922871/ 642724285450967220630355361504849899779177801506869278332322553 608059004877287340952704562524164438633794984493480976225 3554993960734614962733842624449024712578443232617270656966127 5696203281915508016256761808452255230431766473761078013276995 88494474176865008639232954367432921205608588912596208345715494/ 5152726190134714843703288578234298671363360167245240455269798 5320450310365448629578891100562816290812529660344891094165296 3628969665233353113798462092615015025021573863156170644625 : of height 282.521375682916670776596558628 ================================================================ Total time: 54410.370 seconds, Total memory usage: 221.62MB This was on a 750Mhz UltraSparcIII. On a more modern computer it would take less than 3 hours total. === Mark Watkins watkins@maths.usyd.edu.au %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For n = 769, Watkins' point 2745927768444998787776011880685742614821228236987298277607632364030625754997464382740380309193931494836427551073675501729792268983035660490129899517912107868/55653971635295660084686145258798186730245642938678187249280922059873660643412616511468893410763319984244205716710464499744954881769325237306687356528729, -6894058723264327121099047786850870241343899545845980101823950844257061139688178461261452472155490193153456502774218189863376118563284696052640213958580225207273767500180417457624438794592333427637339510014950540866321768058206870245/415187474881697077086948333176088292872966284468813167032550284598516257706907395160667181111931788540851271949927153162985185960790513019885390530579980622525567012750083642734850905312030203572633142212496750984924854188596317 on the elliptic curve [1, 0, 1, -7285583301, 239355096410424] (i.e. the curve y^2 + x y + y = -7285583301 x + 239355096410424 ) corresponds to the point (X,Y) = 13200185030514635210714569328671891018891811876948839852274099680207033960362\ 564220772967646060344051492373416735244729443569637773251182625386352705370992 /55653971635295660084686145258798186730245642938678187249280922059873660643412\ 616511468893410763319984244205716710464499744954881769325237306687356528729, 267894214910293525842810839723867343449390449879972847652507873696694251463330\ 337640898100189747558161874865742586426409013025349448462623261877903012298768\ 11273895127117159324189755810807052025216430257960569806111097494597130269564/ 415187474881697077086948333176088292872966284468813167032550284598516257706907\ 395160667181111931788540851271949927153162985185960790513019885390530579980622\ 525567012750083642734850905312030203572633142212496750984924854188596317 on the curve Y^2 = X^3 + n^2*X^2 - 72*n*X - 16*(4*n^3+27) discussed in the paper, and then to the point (x,y,z) = 8741637191790101644325486167156645246415013849447125319619167136536543797910\ 606521386007032092206358778287383044032480728518491991006957107477883928964761\ 60744796580931201761196888219680658095672769632361211629282686075900104652767, 18309287724300584710140159014437621798178958488017584164351591197132767064458\ 118579989510610297476293130961159564976281050412254471309193652544951893689717\ 61147435692613957451346522355780909953716213570145517693786653843564287876680, -35085648134634071628607750076283942812171420224444703200117776102479881326218\ 714311072122739937967287303232625718154605199585331363747089516287251974167943\ 513341779529615995899623507635405757344694780817337466371873307887044992892407 on the curve x^3+y^3+z^3 = 769 x y z . This point does not (I think) come from either of the 3-isogenous curves; in particular the simplest solution to a/b + b/c + c/a = 769 seems to use a, b, c = -1176176241961977645951554607573345691035517205967359050391277279461108894919\ 069946451753171365300604963039819091688026816866877066930231604169829700236927\ 899383584360363115887976845542777947440146354796674236800203327726336323656599\ 925302217876415429980835277281312366024569902975672468208298543125449756522464\ 140439242633205268814748226245446515311034065694937129501022653470775730701945\ 688278320653674164272250117743903773082414392947893031948852276956879790037097\ 995615486616852585565998500042187074735804974999631210782813914259067347919945\ 432043437784630130378446559749010130408794120609022000532289303225914053197831\ 44648267226455499751707858923871959212194234948161954465165066673724516800, 10760979029461314145676763879162724839887727372994282767649010320007148079388\ 551816009727768587822591825425279977928318176448782979385370949981826562673837\ 749579896371588063339055478453185609454082690851208181909001172046233253629932\ 100314293844433425112163821505837955312836762281988149497062225844597162252604\ 938937185563534551023399455046765921905906310197029982133219843773293084321589\ 511835667340549298075808008000225572594140134046580569001302971047754530042218\ 091854721387074275374619645900666717588761933761867095883544499788824701938201\ 865589598419471811895588316082457833875359481613209918065646757833460026788503\ 80277819816022768840462053196440373698301007325809346386053122523495696783, 13991265733006661499122827344086214851315829660491176163315988068192176434499\ 607594054583107193132721145332268611236209040905882475496561728768524904019663\ 462738940081099989613101874230690140029399952120135989072056670164799797933923\ 338866371648808033873768305287146057564433589221321611300372445001133288011364\ 698024114722390470511881896934270790488885398947556849443557493848670612798869\ 927274751963363226284698762854227134079495498038066733518304417661705780737692\ 017160178403720669988527767491442563896432498532842438241950448864080754457053\ 759658850060809396615619628461884196069100468829654407436792422046439110106613\ 22431678629791380186653561234846802030775819156582867648613469166440520 Note that this case (n=769) was hardest to solve because of the great height of the generator; but our method of approach allows us to search the 3-isogenous curves and it is only in this sense that the case n=917 was easier -- the 3-isogenous curve in that case has a generator of somewhat more modest height. Therefore when we show the generators on the curves actually of interest to us, the case n=917 becomes more impressive than the case n=769. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Recall that when the point (X,Y) of a point on our curve Y^2 = X^3 + n^2*X^2 - 72*n*X - 16*(4*n^3+27) is not a double (e.g. if it's a generator), then X < 0 iff the equations a/b+b/c+c/a=n and x^3+y^3+z^3=nxyz have positive solutions. Examining Watkins' points we find that this holds for n=609, 697, 802 but not for n= 748, 769, 901, 907. So our conclusion is that for the first 1000 positive values of n, we have the following dispositions of the curves x^3+y^3+z^3=nxyz : ** 1 (n=3) has genus 0; the solution set is the line x+y+z=0. All others are elliptic curves. ** 1 (n=5) has rank 0 and torsion Z/6; (there is the unique nontrivial solution {x,y,z}={1,1,2} .) All others have torsion Z/3 (three trivial solutions with xyz = 0). ** 449 more have rank 0, i.e. there are no nontrivial solutions All others have infinitely many solutions, possibly of mixed signs. ** 456 have rank 1 (190 have all-positive solutions x,y,z) ** 92 have rank 2 (58 have an all-positive solution) ** 1 has rank 3 (n=484) but only mixed-sign solutions.