Integral Points of Curves: y^2=3x^3+1, y^2=3x^4+1
[2004.06.06]y^2=3x^3+1, y^2=3x^4+1の整点
■Fermat商
pを奇素数とする。商
qp(m)=(mp-1-1)/p
を、底mのpのFermat商と呼ぶ。
Fermatの小定理より、mが整数で、pとmが互いに素であるならば、底mのpのFermat商は整数である。
■以下では、底yの3のFermat商q3(y)=(y2-1)/3が完全３乗数x3または完全４乗数x4に等しくなるものを求める。
つまり、Diophantus方程式
(y2-1)/3 = x3,
および、
(y2-1)/3 = x4
の整数解x,yを求めれば良い。
これは、それぞれ楕円曲線
E3: y2 = 3x3+1,
および、
E4: y2 = 3x4+1
の整点を求めることと同値である。
楕円曲線E3, E4は、どちらも、自明な整点(0,±1)を持つ。
■楕円曲線E3: y2 = 3x3+1の整点を求める。
x,yを整数とする。
(y+1)(y-1) = 3x3 --------- (1)
gcd(y+1,y-1) = gcd(y+1,2) = 1 or 2
より、２つの場合に分ける。
[case i]gcd(y+1,y-1)=1つまりy:偶数の場合
(1)より、ある整数X,Yが存在して、
y+ε = 3X3, -------- (2)
y-ε = Y3, --------- (3)
ここで、x = XY, --------- (4)
ε = ±1, gcd(X,Y)=1
となる。
(2),(3)より、
Y3-3X3 = ±2 ----- (5)
を得る。
pari/gpでThue方程式(5)の解を求めると、±(1,1)に限ることが分かる。
(4)より、x=1, (3)より、y=±2を得る。
よって、この場合、(1)の整数解は(1,±2)に限る。
[pari/gpによる計算]
gp> th=thueinit(x^3-3)
time = 721 ms.
%1 = [x^3 - 3, [[;], matrix(0,6), [-2.524681404706315896966265280 + 0.E-56*I; 2.524681404706315896966265280 - 5.213324281535470351304611926*I], [-1.046929977734996797927074523 + 0.E-57*I, 1.404358425295394660588718319 + 0.E-57*I, 0.4368267782472231690496729127 + 0.E-56*I, -17.67276983294421127876385696 + 9.424777960769379715387930149*I, -11.57647704579658268690425187 + 9.424777960769379715387930149*I, 18.13448291799616282838615596 + 3.141592653589793238462643383*I; 1.046929977734996797927074523 + 5.027648138363861692133876439*I, -1.404358425295394660588718319 + 4.352680840545421123747478583*I, -0.4368267782472231690496729127 + 1.547180628968307193363938736*I, 17.67276983294421127876385696 + 11.67781738429520386396158172*I, 11.57647704579658268690425187 + 0.3216569898567189359694977601*I, -18.13448291799616282838615596 + 2.914981175598477285581030805*I], [[2, [3, 1, 0]~, 1, 1, [1, 1, 1]~], [2, [1, 1, 1]~, 1, 2, [1, 1, 0]~], [3, [0, 1, 0]~, 3, 1, [0, 0, 1]~], [5, [-2, 1, 0]~, 1, 1, [-1, 2, 1]~], [11, [2, 1, 0]~, 1, 1, [4, -2, 1]~], [17, [-7, 1, 0]~, 1, 1, [-2, 7, 1]~]]~, [1, 4, 5, 3, 2, 6], [x^3 - 3, [1, 1], -243, 1, [[1, 1.442249570307408382321638310, 2.080083823051904114530056824; 1, -0.7211247851537041911608191554 + 1.249024766483406479413179543*I, -1.040041911525952057265028412 - 1.801405432764004090013198081*I], [1, 2; 1.442249570307408382321638310, -1.442249570307408382321638310 - 2.498049532966812958826359087*I; 2.080083823051904114530056824, -2.080083823051904114530056824 + 3.602810865528008180026396162*I], [3, 0.E-96, 0.E-96; 0.E-96, 6.240251469155712343590170473, 0.E-96; 0.E-96, 0.E-96, 12.98024613276667544089474479], [3, 0, 0; 0, 0, 9; 0, 9, 0], [9, 0, 0; 0, 9, 0; 0, 0, 3], [-81, 0, 0; 0, 0, -27; 0, -27, 0], [81, [0, 27, 0]~]], [1.442249570307408382321638310, -0.7211247851537041911608191554 + 1.249024766483406479413179543*I], [1, x, x^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 3, 0, 3, 0; 0, 1, 0, 1, 0, 0, 0, 0, 3; 0, 0, 1, 0, 1, 0, 1, 0, 0]], [[1, [], []], 2.524681404706315896966265280, 1.018203458815742423, [2, -1], [x^2 - 2], 189], [[;], [], []], 0], [1.442249570307408382321638310 + 0.E-48*I, -0.7211247851537041911608191554 - 1.249024766483406479413179543*I, -0.7211247851537041911608191554 + 1.249024766483406479413179543*I]~, [1.683120936470877264644176853]~, [0.08008382305190411453005682435 + 0.E-47*I; -3.040041911525952057265028412 + 1.801405432764004090013198081*I; -3.040041911525952057265028412 - 1.801405432764004090013198081*I], Mat(-0.3960895810995705469404505321), [0.6409998090896148174899462884, 2.498049532717008005529677791, 0.3662040962227032304650817456, 0.8006246367840018177836435915, 1.29875477 E-48, 7]]
gp> thue(th,2)
time = 25 ms.
%2 = [[-1, -1]]
gp> thue(th,-2)
time = 16 ms.
%3 = [[1, 1]]
[case ii]gcd(y+1,y-1)=2つまりy:奇数の場合
(1)より、ある整数X,Yが存在して、(case ii-1)
y+ε = 3・2X3, -------- (6)
y-ε = 22Y3, --------- (7)
または、(case ii-2)
y+ε = 3・22X3, -------- (8)
y-ε = 2Y3, --------- (9)
ここで、x = 2XY, --------- (10)
ε = ±1, gcd(X,Y)=1
となる。
[case ii-1](6)&(7)の場合
3X3-2Y3 = ±1 ----- (11)
を得る。
ここで、X=X', Y=X'+Y'[逆変換は、X'=X, Y'=-X+Y]とおくと、X',Y'も整数であり、(11)より、
X'3-6X'2Y'-6X'Y'2-2Y'3 = ±1 ----- (12)
を得る。
pari/gpで、Thue方程式(12)の解を求めると、(±1,0)に限ることが分かる。
よって、Thue方程式(11)の解は、±(1,1)に限る。
(10)より、x=2XY=2, (6)より、y=±5を得る。
この場合、(1)の整数解は(2,±5)に限る。
[pari/gpによる計算]
gp> f(x,y)=3*x^3-2*y^3
time = 1 ms.
gp> f(x,x+y)
time = 2 ms.
%5 = x^3 - 6*y*x^2 - 6*y^2*x - 2*y^3
gp> th=thueinit(x^3-6*x^2-6*x-2)
time = 249 ms.
%6 = [x^3 - 6*x^2 - 6*x - 2, [[;], matrix(0,11), [5.105835485554791888818939596 - 6.283185307179586476925286766*I; -5.105835485554791888818939596 - 2.907971098933654523958540339*I], [-2.946314244410729270671828165 + 6.283185307179586476925286766*I, -0.5450993388624919968924532840 + 0.E-57*I, 0.9216914562360849233216501214 + 0.E-57*I, -3.403890323703194592545959730 + 9.424777960769379715387930149*I, 6.807780647406389185091919461 + 9.424777960769379715387930149*I, 2.703398747911805023347663956 + 6.283185307179586476925286766*I, -6.363257659590970449694668901 + 3.141592653589793238462643383*I, 5.441566203354885526373018779 + 9.424777960769379715387930149*I, -0.4615207527183764735340035451 + 3.141592653589793238462643383*I, 0.3051991501759277131975005998 + 3.141592653589793238462643383*I, -3.008597898087732736545164556 + 3.141592653589793238462643383*I; 2.946314244410729270671828165 + 0.07693644086591002180081936094*I, 0.5450993388624919968924532840 + 4.790853485662575629126220195*I, -0.9216914562360849233216501214 + 6.672565963180626031401452855*I, 3.403890323703194592545959730 + 4.033042501682298508280789148*I, -6.807780647406389185091919461 + 2.405890508601380444980566314*I, -2.703398747911805023347663956 + 4.449636633852909697394726275*I, 6.363257659590970449694668900 + 11.79765009339360143003995784*I, -5.441566203354885526373018779 + 6.662525172144118446259736363*I, 0.4615207527183764735340035451 + 1.553005495186321112500037956*I, -0.3051991501759277131975005998 + 2.869779524145674578870723671*I, 3.008597898087732736545164555 + 5.246954456360588677585123586*I], [[2, [0, 1, 0]~, 3, 1, [0, 0, 1]~], [3, [1, 1, 0]~, 3, 1, [1, -1, 1]~], [5, [-2, 1, 0]~, 1, 1, [1, 1, 1]~], [11, [2, 1, 0]~, 1, 1, [-1, 3, 1]~], [13, [5, 1, 0]~, 1, 1, [-3, 2, 1]~], [13, [-1, 1, 0]~, 1, 1, [2, -5, 1]~], [13, [3, 1, 0]~, 1, 1, [-5, 4, 1]~], [17, [8, 1, 0]~, 1, 1, [4, 3, 1]~], [19, [-6, 1, 0]~, 1, 1, [-6, 0, 1]~], [19, [-5, 1, 0]~, 1, 1, [8, -1, 1]~], [19, [5, 1, 0]~, 1, 1, [-8, 8, 1]~]]~, [3, 4, 6, 1, 2, 10, 7, 5, 8, 11, 9], [x^3 - 6*x^2 - 6*x - 2, [1, 1], -972, 1, [[1, 6.910169879315560342757745704, 47.75044776100002579216255865; 1, -0.4550849396577801713788728521 - 0.2869253958842582408342928967*I, 0.1247761194999871039187206750 + 0.2611508529445446973360396340*I], [1, 2; 6.910169879315560342757745704, -0.9101698793155603427577457043 + 0.5738507917685164816685857935*I; 47.75044776100002579216255865, 0.2495522389999742078374413501 - 0.5223017058890893946720792680*I], [3, 6.000000000000000000000000000, 48.00000000000000000000000000; 6.000000000000000000000000000, 48.32930473121335326339472749, 329.7002767525334992232183326; 48.00000000000000000000000000, 329.7002767525334992232183326, 2280.272799071974652839259769], [3, 6, 48; 6, 48, 330; 48, 330, 2280], [18, 0, 12; 0, 18, 6; 0, 0, 3], [540, 2160, -324; 2160, 4536, -702; -324, -702, 108], [324, [-108, 54, 0]~]], [6.910169879315560342757745704, -0.4550849396577801713788728521 - 0.2869253958842582408342928967*I], [1, x, x^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 2, 0, 2, 12; 0, 1, 0, 1, 0, 6, 0, 6, 38; 0, 0, 1, 0, 1, 6, 1, 6, 42]], [[1, [], []], 5.105835485554791888818939596, 0.9180660735476202720, [2, -1], [3*x^2 + 3*x + 1], 181], [[;], [], []], 0], [6.910169879315560342757745704 + 0.E-48*I, -0.4550849396577801713788728521 + 0.2869253958842582408342928967*I, -0.4550849396577801713788728521 - 0.2869253958842582408342928967*I]~, [3.403890323703194592545959730]~, [164.9818529209467584047609130 + 0.E-46*I; 0.009073539526620797619543468806 + 0.07732362881914063049475978834*I; 0.009073539526620797619543468806 - 0.07732362881914063049475978834*I], Mat(0.1958543323280110747701402746), [0.07362509091896981387462225415, 0.5738507917111314024917341453, 0.6443314073460819109151301908, 1.488189768304959669214639253, 5.88528364 E-49, 7]]
gp> thue(th,1)
time = 18 ms.
%7 = [[1, 0]]
gp> thue(th,-1)
time = 11 ms.
%8 = [[-1, 0]]
[case ii-2](8)&(9)の場合
Y3-6X3 = ±1 ----- (13)
を得る。
pari/gpで、Thue方程式(13)の解を求めると、(0,±1)に限ることが分かる。
(10)より、x=2XY=0, (6)より、y=±1を得る。
この場合、(1)の整数解は(0,±1)に限る。
[pari/gpによる計算]
(14:13) gp > th=thueinit(x^3-6)
time = 171 ms.
%9 = [x^3 - 6, [[;], matrix(0,8), [-5.789932142319291516526504589 + 6.283185307179586476925286766*I; 5.789932142319291516526504589 + 9.590701812201609056340229381*I], [-0.7384478947135952746909346924 + 0.E-57*I, -2.776866551697951145777598052 + 0.E-56*I, 2.389787971114667873525087653 + 3.141592653589793238462643383*I, 3.859954761546194344351003059 + 6.283185307179586476925286766*I, -3.859954761546194344351003059 + 6.283185307179586476925286766*I, 0.3870785805832832722525103990 + 0.E-57*I, -1.523303539085574946403018715 + 0.E-57*I, 2.964889267884110630293778109 + 6.283185307179586476925286766*I; 0.7384478947135952746909346924 + 4.904140062566477981300029173*I, 2.776866551697951145777598052 + 6.620161354366356610276624400*I, -2.389787971114667873525087653 + 9.203883241537456739673797968*I, -3.859954761546194344351003059 + 1.983779201438375931673562767*I, 3.859954761546194344351003059 + 2.205011003348015052943295076*I, -0.3870785805832832722525103990 + 3.025511325634946080825437930*I, 1.523303539085574946403018715 + 8.675806284035571386712563514*I, -2.964889267884110630293778109 + 12.10075092764215580663374222*I], [[2, [0, 1, 0]~, 3, 1, [0, 0, 1]~], [3, [0, 1, 0]~, 3, 1, [0, 0, 1]~], [5, [-1, 1, 0]~, 1, 1, [1, 1, 1]~], [7, [2, 1, 0]~, 1, 1, [-3, -2, 1]~], [7, [-3, 1, 0]~, 1, 1, [2, 3, 1]~], [7, [1, 1, 0]~, 1, 1, [1, -1, 1]~], [11, [3, 1, 0]~, 1, 1, [-2, -3, 1]~], [17, [-5, 1, 0]~, 1, 1, [8, 5, 1]~]]~, [3, 5, 4, 1, 2, 6, 7, 8], [x^3 - 6, [1, 1], -972, 1, [[1, 1.817120592832139658891211756, 3.301927248894626683874609952; 1, -0.9085602964160698294456058781 + 1.573672595132472278291282234*I, -1.650963624447313341937304976 - 2.859552878990809721272226986*I], [1, 2; 1.817120592832139658891211756, -1.817120592832139658891211756 - 3.147345190264944556582564469*I; 3.301927248894626683874609952, -3.301927248894626683874609952 + 5.719105757981619442544453972*I], [3, 0.E-96, 0.E-96; 0.E-96, 9.905781746683880051623829857, -3.74534108 E-96; 0.E-96, -3.74534108 E-96, 32.70817067097851386004181161], [3, 0, 0; 0, 0, 18; 0, 18, 0], [18, 0, 0; 0, 18, 0; 0, 0, 3], [-324, 0, 0; 0, 0, -54; 0, -54, 0], [324, [0, 54, 0]~]], [1.817120592832139658891211756, -0.9085602964160698294456058781 + 1.573672595132472278291282234*I], [1, x, x^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 6, 0, 6, 0; 0, 1, 0, 1, 0, 0, 0, 0, 6; 0, 0, 1, 0, 1, 0, 1, 0, 0]], [[1, [], []], 5.789932142319291516526504589, 0.9803244459404400962, [2, -1], [3*x^2 - 6*x + 1], 181], [[;], [], []], 0], [1.817120592832139658891211756 + 0.E-48*I, -0.9085602964160698294456058781 - 1.573672595132472278291282234*I, -0.9085602964160698294456058781 + 1.573672595132472278291282234*I]~, [3.859954761546194344351003059]~, [0.003058189691042098276559319263 + 0.E-47*I; 1.498470905154478950861720340 + 18.02069420776726283356437436*I; 1.498470905154478950861720340 - 18.02069420776726283356437436*I], Mat(-0.1727135958452574924940842135), [0.4038045762253003818165390493, 3.147345189950210037556070013, 0.5972531564093516669374924528, 0.6354561953312910491716059969, 5.13522760 E-49, 7]]
gp> thue(th,1)
time = 13 ms.
%10 = [[1, 0]]
gp> thue(th,-1)
time = 11 ms.
%11 = [[-1, 0]]
以上より、(1)の整数解(x,y)は、
(0,±1), (1,±2), (2,±5)
の6個に限る。
よって、楕円曲線E3の整点は、以下の6個に限る。
E3(Z) = { (0,±1), (1,±2), (2,±5) }
■定理１
底が正整数mである3のFermat商q3(m)が完全３乗数になるのは、mが1,2,5の時(以下の３個)に限る。
(12-1)/3 = 03,
(22-1)/3 = 13,
(52-1)/3 = 23.
■楕円曲線E4: y2 = 3x4+1の整点を求める。
x,yを整数とする。
(y+1)(y-1) = 3x4 --------- (14)
gcd(y+1,y-1) = gcd(y+1,2) = 1 or 2
より、２つの場合に分ける。
[case i]gcd(y+1,y-1)=1つまりy:偶数の場合
(1)より、ある整数X,Yが存在して、
|y|+ε = 3X4, -------- (15)
|y|-ε = Y4, --------- (16)
ここで、x = XY, --------- (17)
ε = ±1, gcd(X,Y)=1
となる。
(15),(16)より、
Y4-3X4 = ±2 ----- (18)
を得る。
(18+)は、Z/3Z上で解を持たないので、整数解を持たないことが分かる。
pari/gpでThue方程式(18-)の解を求めると、(±1,±1)に限ることが分かる。
(17)より、x=±1, (15)より、y=±2を得る。
よって、この場合、(14)の整数解は(±1,±2)に限る。
[pari/gpによる計算]
gp> th=thueinit(x^4-3)
time = 335 ms.
%12 = [x^4 - 3, [[;], matrix(0,10), [1.316957896924816708625046347 + 12.56637061435917295385057353*I, 1.991652391049436824068996675 + 9.424777960769379715387930149*I; 1.316957896924816708625046347 + 0.E-75*I, -1.991652391049436824068996675 + 3.141592653589793238462643383*I; -2.633915793849633417250092694 - 25.13274122871834590770114706*I, 1.38178696 E-76 - 8.882250453317338118532082236*I], [0.1174508346406441433899344293 + 0.E-77*I, -1.137288379897010220493220320 + 3.141592653589793238462643383*I, -2.108802667565993887713690979 + 12.56637061435917295385057353*I, 0.6665867212935142348782367508 + 9.424777960769379715387930149*I, 0.E-77 + 3.141592653589793238462643383*I, -0.2934177322109196956370319221 + 0.E-76*I, -1.695837406775185536218644618 + 12.56637061435917295385057353*I, 4.098057806552099119569367520 + 2.76357393 E-76*I, 0.7412216508286685064214332452 + 9.424777960769379715387930149*I, 0.1699368566088486626401086618 + 0.E-76*I; -1.137288379897010220493220320 + 3.141592653589793238462643383*I, 0.1174508346406441433899344293 + 0.E-77*I, 4.279119766373124935988345407 + 0.E-76*I, -1.325065669755922589190759924 + 0.E-77*I, 0.E-77 + 0.E-77*I, 0.7570164272534220646124701888 + 0.E-77*I, 3.866154505582316584493299046 + 0.E-76*I, -8.902290699208863585094114642 + 0.E-76*I, 0.1699368566088486626401086618 + 0.E-77*I, 0.7412216508286685064214332452 + 3.141592653589793238462643383*I; 1.019837545256366077103285891 + 5.385357040546151906308405904*I, 1.019837545256366077103285891 + 7.181013573813021047542167629*I, -2.170317098807131048274654428 + 6.362139101713159819379718051*I, 0.6584789484624083543125231736 + 8.125245387700503894584532414*I, 0.E-77 + 9.424777960769379715387930149*I, -0.4635986950425023689754382666 + 8.409537773104259896608012097*I, -2.170317098807131048274654428 + 1.435056336700095105440668358*I, 4.804232892656764465524747122 + 8.926008017200831086272748558*I, -0.9111585074375171690615419070 + 4.749632653351470097007422335*I, -0.9111585074375171690615419070 + 7.816737961007702856843151198*I], [[2, [1, 1, 0, 0]~, 4, 1, [1, 1, 1, 1]~], [3, [0, 1, 0, 0]~, 4, 1, [0, 0, 0, 1]~], [11, [-4, 1, 0, 0]~, 1, 1, [-2, 5, 4, 1]~], [11, [4, 1, 0, 0]~, 1, 1, [2, 5, -4, 1]~], [13, [3, 1, 0, 0]~, 1, 1, [-1, -4, -3, 1]~], [13, [-3, 1, 0, 0]~, 1, 1, [1, -4, 3, 1]~], [13, [-2, 1, 0, 0]~, 1, 1, [-5, 4, 2, 1]~], [13, [2, 1, 0, 0]~, 1, 1, [5, 4, -2, 1]~], [23, [-4, 1, 0, 0]~, 1, 1, [-5, -7, 4, 1]~], [23, [4, 1, 0, 0]~, 1, 1, [5, -7, -4, 1]~]]~, [3, 4, 7, 1, 2, 6, 8, 5, 10, 9], [x^4 - 3, [2, 1], -6912, 1, [[1, -1.316074012952492460819218901, 1.732050807568877293527446341, -2.279507056954777641993563252; 1, 1.316074012952492460819218901, 1.732050807568877293527446341, 2.279507056954777641993563251; 1, 0.E-121 - 1.316074012952492460819218901*I, -1.732050807568877293527446341 + 0.E-121*I, 0.E-121 + 2.279507056954777641993563251*I], [1, 1, 2; -1.316074012952492460819218901, 1.316074012952492460819218901, 0.E-121 + 2.632148025904984921638437803*I; 1.732050807568877293527446341, 1.732050807568877293527446341, -3.464101615137754587054892683 + 0.E-121*I; -2.279507056954777641993563252, 2.279507056954777641993563251, 0.E-121 - 4.559014113909555283987126504*I], [4, 0.E-115, 0.E-115, 0.E-115; 0.E-115, 6.928203230275509174109785366, 0.E-115, 0.E-114; 0.E-115, 0.E-115, 12.00000000000000000000000000, 0.E-115; 0.E-115, 0.E-114, 0.E-115, 20.78460969082652752232935609], [4, 0, 0, 0; 0, 0, 0, 12; 0, 0, 12, 0; 0, 12, 0, 0], [12, 0, 0, 0; 0, 12, 0, 0; 0, 0, 12, 0; 0, 0, 0, 4], [-1728, 0, 0, 0; 0, 0, 0, -576; 0, 0, -576, 0; 0, -576, 0, 0], [1728, [0, 576, 0, 0]~]], [-1.316074012952492460819218901, 1.316074012952492460819218901, 0.E-121 - 1.316074012952492460819218901*I], [1, x, x^2, x^3], [1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0; 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0; 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3; 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0]], [[1, [], []], 5.245844688643497921652743802, 0.8869642302877511490, [2, -1], [x^2 + 2, x^3 - x^2 + x - 2], 251], [[;], [], []], 0], [-1.316074012952492460819218901 + 0.E-67*I, 1.316074012952492460819218901 + 0.E-67*I, 0.E-73 + 1.316074012952492460819218901*I, 0.E-73 - 1.316074012952492460819218901*I]~, [1.316957896924816708625046347, 0.9958261955247184120344983376]~, [3.732050807568877293527446341 + 0.E-67*I, -7.327631877476147396340228495 + 0.E-66*I; 3.732050807568877293527446341 + 0.E-67*I, -0.1364697376616071907146641877 + 0.E-67*I; 0.2679491924311227064725536584 + 0.E-73*I, -0.2679491924311227064725536585 - 0.9634330440022851811743443501*I; 0.2679491924311227064725536584 + 0.E-73*I, -0.2679491924311227064725536585 + 0.9634330440022851811743443501*I], [0.3796628587501034616094920614, 0.3796628587501034616094920614; 0.2510478245335478287614651570, -0.2510478245335478287614651570], [0.8773826753893999080763120985, 1.861209718018078226062017574, 0.2746530721670274228488113092, 0.9036020036098448319622180529, 4.23524976 E-67, 9]]
gp> thue(th,2)
time = 11 ms.
%13 = []
gp> thue(th,-2)
time = 38 ms.
%14 = [[-1, 1], [1, -1], [-1, -1], [1, 1]]
[case ii]gcd(y+1,y-1)=2つまりy:奇数の場合
(1)より、ある整数X,Yが存在して、(case ii-1)
|y|+ε = 3・2X4, -------- (19)
|y|-ε = 23Y4, --------- (20)
または、(case ii-2)
|y|+ε = 3・23X4, -------- (21)
|y|-ε = 2Y4, --------- (22)
ここで、x = 2XY, --------- (23)
ε = ±1, gcd(X,Y)=1
となる。
[case ii-1](19)&(20)の場合
3X4-4Y4 = ±1 ----- (24)
を得る。
ここで、X=X', Y=X'+Y'[逆変換は、X'=X, Y'=-X+Y]とおくと、X',Y'も整数であり、(24)より、
X'4+16X'3Y'+24X'2Y'2+16X'Y'3+4Y'4 = ±1 ----- (25)
を得る。
pari/gpで、Thue方程式(25+)の解を求めると、(±1,0),±(1,-2)に限ることが分かる。また、Thue方程式(25-)は整数解を持たない。
よって、Thue方程式(24-)の解は、±(1,1), ±(1,-1)に限る。Thue方程式(24+)は整数解を持たない。
(23)より、x=2XY=±2, (19)より、y=±7を得る。
この場合、(14)の整数解は(±2,±7)に限る。
[pari/gpによる計算]
gp> g(x,y)=3*x^4-4*y^4
time = 0 ms.
gp> g(x,x+y)
time = 2 ms.
%15 = -x^4 - 16*y*x^3 - 24*y^2*x^2 - 16*y^3*x - 4*y^4
gp> th=thueinit(x^4+16*x^3+24*x^2+16*x+4)
time = 374 ms.
%16 = [x^4 + 16*x^3 + 24*x^2 + 16*x + 4, [[;], matrix(0,6), [0.8314429455293105378262425195 - 6.283185307179586476925286766*I, -1.316957896924816708625046347 - 28.27433388230813914616379045*I; -0.8314429455293105378262425195 - 6.283185307179586476925286766*I, -1.316957896924816708625046347 - 21.99114857512855266923850368*I; 3.45446742 E-76 + 3.891061519007273388330150002*I, 2.633915793849633417250092694 - 6.283185307179586476925286766*I], [-0.3796722499342086910405680023 + 0.E-77*I, -0.7488910472635524783074349734 + 0.E-75*I, 0.3497966969331724487518152909 + 0.E-77*I, -3.378876740845492863319475541 + 0.E-75*I, 4.190591767451753068873566807 + 9.424777960769379715387930149*I, 0.5613618105043447203097266161 + 3.141592653589793238462643383*I; 0.08255189826575805951880754609 + 3.141592653589793238462643383*I, 0.4517706955951018467856745172 + 9.424777960769379715387930149*I, 0.5613618105043447203097266161 + 3.141592653589793238462643383*I, -2.547433795316182325493233021 + 3.141592653589793238462643383*I, -10.77538125207583661199879854 + 12.56637061435917295385057353*I, 0.3497966969331724487518152909 + 0.E-77*I; 0.2971203516684506315217604561 + 10.83084413414789562880075952*I, 0.2971203516684506315217604561 + 4.127650268383590413644950772*I, -0.9111585074375171690615419070 + 0.3611050315917563818076514505*I, 5.926310536161675188812708562 + 1.196061894086156544297568382*I, 6.584789484624083543125231736 + 0.4620344815777348718822728035*I, -0.9111585074375171690615419070 + 12.20526558276741657204292208*I], [[2, [0, 0, 1, 1]~, 4, 1, [1, 2, 2, 1]~], [3, [1, 1, 0, 0]~, 4, 1, [1, 1, 0, 1]~], [11, [-3, 1, 0, 0]~, 1, 1, [1, -3, -1, 4]~], [11, [2, 1, 0, 0]~, 1, 1, [-4, -1, 1, 4]~], [23, [-2, 1, 0, 0]~, 1, 1, [8, -1, 3, 4]~], [23, [5, 1, 0, 0]~, 1, 1, [11, -9, -2, 4]~]]~, [3, 4, 6, 1, 2, 5], [x^4 + 16*x^3 + 24*x^2 + 16*x + 4, [2, 1], -1728, 16, [[1, -14.41023084701677874147978214, 45.20857284257008775053307682, -755.2933474577654504611699033; 1, -0.5179723832587304326300032249, 0.3080876558253170653237256105, -0.2937285922497995278133990846; 1, -0.5358983848622454129451073170 - 0.4987096409377726371048605178*I, 0.2416697508022975920715987802 - 0.1157259749219949680154256816*I, -0.2064619749923750055083487758 - 0.3257633626946117034565591357*I], [1, 1, 2; -14.41023084701677874147978214, -0.5179723832587304326300032249, -1.071796769724490825890214634 + 0.9974192818755452742097210356*I; 45.20857284257008775053307682, 0.3080876558253170653237256105, 0.4833395016045951841431975604 + 0.2314519498439899360308513632*I; -755.2933474577654504611699033, -0.2937285922497995278133990846, -0.4129239499847500110166975517 + 0.6515267253892234069131182714*I], [4, -16.00000000000000000000000000, 46.00000000000000000000000000, -756.0000000000000000000000000; -16.00000000000000000000000000, 208.9948452238571284375369951, -651.7691453623979128349403414, 10884.64984531856044480110752; 46.00000000000000000000000000, -651.7691453623979128349403414, 2044.053570005086584832973770, -34145.84920286903319445958989; -756.0000000000000000000000000, 10884.64984531856044480110752, -34145.84920286903319445958989, 570468.4244870738964045283503], [4, -16, 46, -756; -16, 208, -652, 10884; 46, -652, 2044, -34146; -756, 10884, -34146, 570468], [-12, 0, 0, -2; 0, -6, 0, -2; 0, 0, -12, 0; 0, 0, 0, -2], [-127872, -87120, 326304, 21024; -87120, -58176, 226368, 14544; 326304, 226368, -830016, -53568; 21024, 14544, -53568, -3456], [864, [-288, 144, 0, 0]~]], [-14.41023084701677874147978214, -0.5179723832587304326300032249, -0.5358983848622454129451073170 - 0.4987096409377726371048605178*I], [1, x, 1/4*x^2 + 1/2*x + 1/2, 1/4*x^3 + 1/2*x], [1, 0, -2, 0; 0, 1, -2, -2; 0, 0, 4, 0; 0, 0, 0, 4], [1, 0, 0, 0, 0, -2, -1, 10, 0, -1, 2, -37, 0, 10, -37, 617; 0, 1, 0, 0, 1, -2, -1, 15, 0, -1, 3, -52, 0, 15, -52, 870; 0, 0, 1, 0, 0, 4, 2, -22, 1, 2, -4, 81, 0, -22, 81, -1352; 0, 0, 0, 1, 0, 0, 1, -16, 0, 1, -3, 51, 1, -16, 51, -852]], [[1, [], []], 2.189950705914511481040414854, 1.091651065648911351, [2, -1], [1/4*x^3 + 15/4*x^2 + 2*x + 1/2, x^3 + 31/2*x^2 + 16*x + 4], 244], [[;], [], []], 0], [-14.41023084701677874147978214 + 0.E-67*I, -0.5179723832587304326300032249 + 0.E-67*I, -0.5358983848622454129451073170 + 0.4987096409377726371048605178*I, -0.5358983848622454129451073170 - 0.4987096409377726371048605178*I]~, [0.4157214727646552689131212597, 1.316957896924816708625046347]~, [2.296630262886538245704941917 + 0.E-65*I, -0.2679491924311227064725536585 + 0.E-65*I; 0.4354205446823390478225044237 + 0.E-67*I, -0.2679491924311227064725536585 + 0.E-67*I; -0.3660254037844386467637231707 - 0.9306048591020995989412187470*I, -3.732050807568877293527446341 + 2.95970698 E-67*I; -0.3660254037844386467637231707 + 0.9306048591020995989412187469*I, -3.732050807568877293527446341 - 2.95970698 E-67*I], [0.6013641737999131206008777974, -0.6013641737999131206008777974; -0.3796628587501034616094920614, -0.3796628587501034616094920614], [2.312388743923661095791688026, 0.4990317098621737900320063640, 0.6669846074502505211684060683, 1.234343291841566074273351776, 5.71516289 E-67, 9]]
gp> thue(th,1)
time = 27 ms.
%17 = [[1, 0], [-1, 0], [-1, 2], [1, -2]]
gp> thue(th,-1)
time = 5 ms.
%18 = []
[case ii-2](21)&(22)の場合
Y4-12X4 = ±1 ----- (26)
を得る。
pari/gpで、Thue方程式(26)の解を求めると、(0,±1)に限ることが分かる。
(23)より、x=2XY=0, (19)より、y=±1を得る。
この場合、(14)の整数解は(0,±1)に限る。
[pari/gpによる計算]
gp> th=thueinit(x^4-12)
time = 253 ms.
%21 = [x^4 - 12, [[;], matrix(0,6), [-0.8314429455293105378262425195 + 6.283185307179586476925286766*I, 1.316957896924816708625046347 + 12.56637061435917295385057353*I; 0.8314429455293105378262425195 + 0.E-76*I, 1.316957896924816708625046347 + 0.E-76*I; 6.90893484 E-77 + 8.675309095351899565520423531*I, -2.633915793849633417250092694 + 25.13274122871834590770114706*I], [0.4517706955951018467856745172 + 0.E-76*I, 0.08255189826575805951880754609 + 9.424777960769379715387930149*I, 0.5613618105043447203097266161 + 3.141592653589793238462643383*I, -0.7449609469958594460693828466 + 3.141592653589793238462643383*I, 1.489921893991718892138765693 + 9.424777960769379715387930149*I, 0.3497966969331724487518152909 + 3.141592653589793238462643383*I; -0.7488910472635524783074349734 + 3.141592653589793238462643383*I, -0.3796722499342086910405680023 + 0.E-77*I, 0.3497966969331724487518152909 + 0.E-77*I, 0.08648199853345109175685967294 + 0.E-77*I, -0.1729639970669021835137193458 + 0.E-76*I, 0.5613618105043447203097266161 + 0.E-77*I; 0.2971203516684506315217604561 + 2.155535038795996063280335994*I, 0.2971203516684506315217604561 + 1.735526480211277325049814007*I, -0.9111585074375171690615419070 + 12.20526558276741657204292208*I, 0.6584789484624083543125231736 + 7.479247201265743021222855148*I, -1.316957896924816708625046347 + 7.032654172597066626792793385*I, -0.9111585074375171690615419070 + 6.644290338771342858732938217*I], [[2, [1, 1, 1, 0]~, 4, 1, [1, 2, 2, 1]~], [3, [0, 1, 0, 0]~, 4, 1, [0, 1, 0, 1]~], [11, [-1, 1, 0, 0]~, 1, 1, [-1, -3, 4, 4]~], [11, [1, 1, 0, 0]~, 1, 1, [1, 1, -4, 4]~], [23, [-3, 1, 0, 0]~, 1, 1, [-2, 1, -11, 4]~], [23, [3, 1, 0, 0]~, 1, 1, [2, -10, 11, 4]~]]~, [3, 4, 6, 1, 2, 5], [x^4 - 12, [2, 1], -1728, 16, [[1, -1.861209718204199197882437494, 0.4354205446823390478225044237, -2.542459756837412478271225982; 1, 1.861209718204199197882437493, 2.296630262886538245704941917, 2.542459756837412478271225982; 1, 0.E-121 - 1.861209718204199197882437494*I, -0.3660254037844386467637231707 - 0.9306048591020995989412187470*I, 0.E-121 + 0.6812500386332132803887884880*I], [1, 1, 2; -1.861209718204199197882437494, 1.861209718204199197882437493, 0.E-121 + 3.722419436408398395764874987*I; 0.4354205446823390478225044237, 2.296630262886538245704941917, -0.7320508075688772935274463415 + 1.861209718204199197882437493*I; -2.542459756837412478271225982, 2.542459756837412478271225982, 0.E-121 - 1.362500077266426560777576976*I], [4, 0.E-115, 2.000000000000000000000000000, 0.E-115; 0.E-115, 13.85640646055101834821957073, 6.928203230275509174109785366, 6.928203230275509174109785366; 2.000000000000000000000000000, 6.928203230275509174109785366, 7.464101615137754587054892683, 3.464101615137754587054892683; 0.E-115, 6.928203230275509174109785366, 3.464101615137754587054892683, 13.85640646055101834821957073], [4, 0, 2, 0; 0, 0, 0, 12; 2, 0, 4, 6; 0, 12, 6, 12], [-12, 0, 0, -6; 0, -6, 0, -2; 0, 0, -12, 0; 0, 0, 0, -2], [-576, -144, 288, 0; -144, 0, 288, -144; 288, 288, -576, 0; 0, -144, 0, 0], [864, [0, 144, 0, 0]~]], [-1.861209718204199197882437494, 1.861209718204199197882437493, 0.E-121 - 1.861209718204199197882437494*I], [1, x, 1/4*x^2 + 1/2*x + 1/2, 1/4*x^3 + 1/2*x], [1, 0, -2, 0; 0, 1, -2, -2; 0, 0, 4, 0; 0, 0, 0, 4], [1, 0, 0, 0, 0, -2, -1, 2, 0, -1, 0, 1, 0, 2, 1, 1; 0, 1, 0, 0, 1, -2, -1, -1, 0, -1, -1, 0, 0, -1, 0, -2; 0, 0, 1, 0, 0, 4, 2, 2, 1, 2, 2, 1, 0, 2, 1, 4; 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0]], [[1, [], []], 2.189950705914511481040414854, 1.091651065648911351, [2, -1], [1/4*x^2 + 1/2*x + 1/2, 1/2*x^2 + 2], 253], [[;], [], []], 0], [-1.861209718204199197882437494 + 0.E-67*I, 1.861209718204199197882437493 + 0.E-67*I, 0.E-73 + 1.861209718204199197882437493*I, 0.E-73 - 1.861209718204199197882437494*I]~, [0.4157214727646552689131212597, 1.316957896924816708625046347]~, [0.4354205446823390478225044237 + 0.E-68*I, 3.732050807568877293527446341 + 0.E-67*I; 2.296630262886538245704941917 + 0.E-67*I, 3.732050807568877293527446341 + 0.E-67*I; -0.3660254037844386467637231707 + 0.9306048591020995989412187469*I, 0.2679491924311227064725536584 + 0.E-73*I; -0.3660254037844386467637231707 - 0.9306048591020995989412187470*I, 0.2679491924311227064725536584 + 0.E-73*I], [-0.6013641737999131206008777974, 0.6013641737999131206008777974; 0.3796628587501034616094920614, 0.3796628587501034616094920614], [0.3102016197317200282838095689, 2.632148025641770119047939311, 0.6212266624470000775574273699, 0.6389431042462724758553493051, 5.71516289 E-67, 9]]
gp> thue(th,1)
time = 19 ms.
%22 = [[1, 0], [-1, 0]]
gp> thue(th,-1)
time = 4 ms.
%23 = []
以上より、(14)の整数解(x,y)は、
(0,±1), (±1,±2), (±2,±7)
の10個に限る。
よって、E4の整点は、以下の10個に限る。
E4(Z) = { (0,±1), (±1,±2), (±2,±7) }
■定理２
底が正整数mである3のFermat商q3(m)が完全４乗数になるのは、mが1,2,7の時(以下の３個)に限る。
(12-1)/3 = 04,
(22-1)/3 = 14,
(72-1)/3 = 24.
[参考文献]
- [1]Nigel P. Smart, "The Algorithmic Resolution of Diophantine Equations", LMSST 41, Cambridge University Press, 1998, ISBN0-521-64633-2.
- [2]Paulo Ribenboim, "The Little Book of Big Primes", Springer-Verlag, 1991, ISBN0-367-97508-X.
| Last Update: 2005.06.12 |
| H.Nakao |
