Integral Points on Elliptic Curve: x^2-3y^4=1

[2003.09.01]x²-3y⁴=1の整点

■楕円曲線C: x²-3y⁴=1の有理点を既に求めたが、この楕円曲線の整点が(±1,0),(±2,±1),(±7,±2)の10個に限ることを証明する。
これは、参考文献[2](p43)によると、Ljunggren(1936)によって証明されたとある。

x,yを以下を満たす有理整数(以下では単に整数と呼ぶ)とする。
    x² - 3y⁴ = 1 ------ (1)

xy=0となる(1)の解を先に求める。
(1)でy=0とすると、x=±1を得る。
(1)でx=0とすると、3y⁴=-1を得るが、これを満たす整数yは存在しない。
よって、xy=0なる解は、(±1,0)のみである。

xy!=0かつ(x,y)が(1)の解であれば、(±x,±y)も(1)の解であるので、x,yはどちらも正整数として良い。
(1)を変形すると、
    (x-1)(x+1) = 3y⁴ --------- (2)
となる。
(2)およびy > 0, x+1 > 0より、x-1 > 0として良い。
    gcd(x-1,x+1)=gcd(2,x+1)=1,2
であるので、gcd(2,x+1)=1の場合とgcd(2,x+1)=2の場合に分ける。

■gcd(2,x+1)=1の場合
x+1,x-1は正整数なので、(2)より、互いに素な正整数a,bに対して、
    y = ab --------- (3)
であり、かつ、(case i)
    x-1 = 3a⁴, ---------- (4)
    x+1 = b⁴, --------- (5)
または、(case ii)
    x-1 = a⁴, --------- (6)
    x+1 = 3b⁴, --------- (7)
となる。

[case i]
(4),(5)より、
    b⁴-3a⁴ = 2 ------- (8)
となる。
Thue方程式(8)はZ/3Zで解を持たないので、整数解を持たない。

gp>  f(a,b)=a^4-3*b^4;
time = 0 ms.
gp>  g(n,d)=
{
   for(i=0,n-1,
     for(j=0,n-1,
       if((f(i,j)-d)%n==0,print([i,j]))
     )
  )
}
time = 0 ms.
gp>  g(3,2)
time = 0 ms.

[case ii]
(6),(7)より、
    a⁴-3b⁴ = -2 ------- (9)
となる。
pari/gpでThue方程式(9)を解くと、
    (a,b) = (1, 1), (1, -1), (-1, 1), (-1, -1)
を得る。a,bは正整数なので、a=b=1であり、(6),(7),(3)より、
    (x,y) = (2, 1)
を得る。

gp> th1=thueinit(x^4-3)
time = 991 ms.
%1 = [x^4 - 3, [[;], matrix(0,10), [1.316957896924816708625046347 + 12.56637061435917295385057353*I, 1.991652391049436824068996675 + 9.424777960769379715387930149*I; 1.316957896924816708625046347 + 0.E-76*I, -1.991652391049436824068996675 + 3.141592653589793238462643383*I; -2.633915793849633417250092694 + 0.E-76*I, 0.E-76 + 3.684120161041834835318491296*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], 252], [[;], [], []], 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(th1,-2)
time = 82 ms.
%2 = [[-1, 1], [1, -1], [-1, -1], [1, 1]]

■gcd(2,x+1)=2の場合
x+1,x-1は正整数なので、(2)より、互いに素な正整数a,bに対して、
    y = 2ab --------- (10)
であり、かつ、(case iii)
    x-1 = 3・2a⁴, ---------- (11)
    x+1 = 2³b⁴ --------- (12)
または、(case iv)
    x-1 = 3・2³a⁴, ---------- (13)
    x+1 = 2b⁴ --------- (14)
または、(case v)
    x-1 = 2a⁴, --------- (15)
    x+1 = 3・2³b⁴ --------- (16)
または、(case vi)
    x-1 = 2³a⁴, --------- (17)
    x+1 = 3・2b⁴ --------- (18)
となる。

[case iii]
(11),(12)より、
    4b⁴-3a⁴ = 1 ------- (19)
となる。
ここで、
    u = 2a-b, v = -a+b ------- (20)
とすると、
    a = u+v, b = u+2v ------- (21)
であり、a,b:整数⇔u,v:整数となる。
(19)に(21)を代入すると、
    u⁴ + 20u³v + 78u²v² + 116uv³ + 61v⁴ = 1 ------- (22)
を得る。
Thue方程式(22)をpari/gpで解くと、
    (u,v) = (±1,0),(3,-2),(-3,2)
を得る。(20)より、
    (a,b) = (1,1),(-1,-1),(-1,1),(1,-1)
a,bは正整数なので、a=b=1であり、(11),(12),(10)より、
    (x,y) = (7,2)
を得る。

gp> f1(a,b)=4*b^4-3*a^4
time = 4 ms.
gp> f1(u+v,u+2*v)
time = 4 ms.
%3 = u^4 + 20*v*u^3 + 78*v^2*u^2 + 116*v^3*u + 61*v^4
gp> th2=thueinit(x^4+20*x^3+78*x^2+116*x+61)
time = 441 ms.
%4 = [x^4 + 20*x^3 + 78*x^2 + 116*x + 61, [[;], matrix(0,6), [0.8314429455293105378262425195 - 6.283185307179586476925286766*I, -1.316957896924816708625046347 - 47.12388980384689857693965075*I; -0.8314429455293105378262425195 + 0.E-76*I, -1.316957896924816708625046347 - 3.141592653589793238462643383*I; 2.76357393 E-76 - 8.675309095351899565520423531*I, 2.633915793849633417250092694 - 56.54866776461627829232758090*I], [-0.3796722499342086910405680023 + 0.E-77*I, -0.7488910472635524783074349734 + 0.E-75*I, 0.3497966969331724487518152909 + 3.141592653589793238462643383*I, -5.041762631904113938971960580 + 3.141592653589793238462643383*I, 3.359148821922442531047324288 + 0.E-74*I, 0.5613618105043447203097266161 + 3.141592653589793238462643383*I; 0.08255189826575805951880754609 + 3.141592653589793238462643383*I, 0.4517706955951018467856745172 + 9.424777960769379715387930149*I, 0.5613618105043447203097266161 + 0.E-77*I, -0.8845479042575612498407479826 + 0.E-75*I, -9.943938306546526074172556024 + 3.141592653589793238462643383*I, 0.3497966969331724487518152909 + 0.E-77*I; 0.2971203516684506315217604561 + 10.83084413414789562880075952*I, 0.2971203516684506315217604561 + 4.127650268383590413644950772*I, -0.9111585074375171690615419070 + 6.644290338771342858732938217*I, 5.926310536161675188812708562 + 12.26349477761036919841312867*I, 6.584789484624083543125231736 + 2.854158269750047960477409567*I, -0.9111585074375171690615419070 + 12.20526558276741657204292208*I], [[2, [1, 0, 0, 1]~, 4, 1, [1, 1, 1, 1]~], [3, [-1, 1, 0, 0]~, 4, 1, [-1, -1, 0, 1]~], [11, [-2, 1, 0, 0]~, 1, 1, [-3, 0, 0, 4]~], [11, [3, 1, 0, 0]~, 1, 1, [-4, 4, 2, 4]~], [23, [-1, 1, 0, 0]~, 1, 1, [10, 6, -8, 4]~], [23, [6, 1, 0, 0]~, 1, 1, [0, -7, 10, 4]~]]~, [3, 4, 6, 1, 2, 5], [x^4 + 20*x^3 + 78*x^2 + 116*x + 61, [2, 1], -1728, 16, [[1, -15.41023084701677874147978214, 59.61880368958686649201285897, -918.7395276795092711957286981; 1, -1.517972383258730432630003224, 0.8260600390840474979537288355, -1.253936326243211589044582366; 1, -1.535898384862245412945107317 - 0.4987096409377726371048605178*I, 0.7775681356645430050167060972 + 0.3829836660157776690894348362*I, -1.003267997123758607613359750 - 0.9760047198041720736200031265*I], [1, 1, 2; -15.41023084701677874147978214, -1.517972383258730432630003224, -3.071796769724490825890214634 + 0.9974192818755452742097210356*I; 59.61880368958686649201285897, 0.8260600390840474979537288355, 1.555136271329086010033412194 - 0.7659673320315553381788696724*I; -918.7395276795092711957286981, -1.253936326243211589044582366, -2.006535994247517215226719500 + 1.952009439608344147240006253*I], [4, -20.00000000000000000000000000, 62.00000000000000000000000000, -922.0000000000000000000000000; -20.00000000000000000000000000, 244.9948452238571284375369951, -922.7639905862550412724773366, 14163.94697185346844018100253; 62.00000000000000000000000000, -922.7639905862550412724773366, 3556.586705953739538940391448, -54777.49517546255721480939443; -922.0000000000000000000000000, 14163.94697185346844018100253, -54777.49517546255721480939443, 844087.8103408523200817565158], [4, -20, 62, -922; -20, 244, -922, 14162; 62, -922, 3556, -54776; -922, 14162, -54776, 844084], [-12, -6, 0, -4; 0, -6, 0, -4; 0, 0, -12, -6; 0, 0, 0, -2], [-84960, -214272, -331344, -18000; -214272, -516960, -784656, -42480; -331344, -784656, -1182528, -63936; -18000, -42480, -63936, -3456], [864, [-144, 144, 0, 0]~]], [-15.41023084701677874147978214, -1.517972383258730432630003224, -1.535898384862245412945107317 - 0.4987096409377726371048605178*I], [1, x, 1/4*x^2 + 1/4, 1/4*x^3 + 1/4*x], [1, 0, -1, 0; 0, 1, 0, -1; 0, 0, 4, 0; 0, 0, 0, 4], [1, 0, 0, 0, 0, -1, 0, 4, 0, 0, 1, -14, 0, 4, -14, 203; 0, 1, 0, 0, 1, 0, 0, -24, 0, 0, -6, 121, 0, -24, 121, -1958; 0, 0, 1, 0, 0, 4, 0, -77, 1, 0, -19, 361, 0, -77, 361, -5753; 0, 0, 0, 1, 0, 0, 1, -20, 0, 1, -5, 81, 1, -20, 81, -1259]], [[1, [], []], 2.189950705914511481040414854, 1.091651065648911351, [2, -1], [1/4*x^3 + 9/2*x^2 + 41/4*x + 13/2, x^3 + 37/2*x^2 + 50*x + 73/2], 242], [[;], [], []], 0], [-15.41023084701677874147978214 + 0.E-67*I, -1.517972383258730432630003224 + 0.E-67*I, -1.535898384862245412945107317 + 0.4987096409377726371048605178*I, -1.535898384862245412945107317 - 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 + 0.E-66*I; -0.3660254037844386467637231707 + 0.9306048591020995989412187469*I, -3.732050807568877293527446341 + 0.E-66*I], [0.6013641737999131206008777974, -0.6013641737999131206008777974; -0.3796628587501034616094920614, -0.3796628587501034616094920614], [2.312388743923661095791688026, 0.4990317098621737900320063640, 1.027718466043327812187847275, 1.234343291841566074273351776, 7.499629852129697630 E-67, 9]]
gp> thue(th2,1)
time = 36 ms.
%5 = [[1, 0], [-1, 0], [-3, 2], [3, -2]]

[case iv]
(13),(14)より、
b⁴-12a⁴ = 1 ------- (23)
となる。
Thue方程式(23)をpari/gpで解くと、
(a,b)=(0,±1)
を得る。
a,bは正整数なので、(23)は正整数解を持たない。よって、(13),(14)も正整数解を持たない。

gp> th3=thueinit(x^4-12)
time = 284 ms.
%6 = [x^4 - 12, [[;], matrix(0,6), [-0.8314429455293105378262425195 + 6.283185307179586476925286766*I, -1.316957896924816708625046347 - 9.424777960769379715387930150*I; 0.8314429455293105378262425195 + 0.E-76*I, -1.316957896924816708625046347 + 3.141592653589793238462643383*I; 0.E-76 + 8.675309095351899565520423531*I, 2.633915793849633417250092694 - 18.84955592153875943077586030*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], 252], [[;], [], []], 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, -0.2679491924311227064725536585 + 0.E-67*I; 2.296630262886538245704941917 + 0.E-67*I, -0.2679491924311227064725536585 + 0.E-67*I; -0.3660254037844386467637231707 + 0.9306048591020995989412187469*I, -3.732050807568877293527446341 + 0.E-73*I; -0.3660254037844386467637231707 - 0.9306048591020995989412187470*I, -3.732050807568877293527446341 + 0.E-73*I], [-0.6013641737999131206008777974, 0.6013641737999131206008777974; -0.3796628587501034616094920614, -0.3796628587501034616094920614], [0.3102016197317200282838095689, 2.632148025641770119047939311, 0.6212266624470000775574273699, 0.6389431042462724758553493051, 9.28409679 E-67, 9]]
gp> thue(th3,1)
time = 29 ms.
%7 = [[1, 0], [-1, 0]]

[case v]
(15),(16)より、
a⁴-12b⁴ = -1 ------- (24)
となる。
Thue方程式(24)はZ/3Zで解を持たないので、整数解を持たない。
よって、(15),(16)も正整数解を持たない。

gp>  f(a,b)=a^4-12*b^4
time = 0 ms.
gp>  g(3,-1)
time = 7 ms.

[case vi]
(17),(18)より、
3b⁴-4a⁴ = 1 ------- (25)
となる。
Thue方程式(25)はZ/3Zで解を持たないので、整数解を持たない。
よって、(17),(18)も正整数解を持たない。

gp>  f(a,b)=3*b^4-4*a^4
time = 0 ms.
gp>  g(3,1)
time = 1 ms.

■case i～viの結果を整理すると、Diophantus方程式(2)の正整数解(x,y)は、
    (2, 1), (7, 2)
に限る。
これに、xy=0の場合の解が
    (±1, 0)
に限ること、および、
xy!=0かつx,yのどちらか(あるいは両方)が負整数の場合は、(|x|,|y|)が(2)の正整数解になること
を付け加えると、以下が証明できたことになる。
--------------------------------------------------------
Diophantus方程式
    x²-3y⁴ = 1
の有理整数解(x,y)は、
    (±1, 0), (±2, ±1), (±7, ±2)
の10個に限る。
--------------------------------------------------------

[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]加川貴章, "Elliptic curves with everywhere good reduction over real quardratic fields", March, 1998.

Last Update: 2005.06.12

H.Nakao