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

[2003.10.01]3x^4-2y^2=1の整点

■楕円曲線C: 3x⁴-2y²=1の有理点を既に求めたが、この楕円曲線の整点が(±1,±1),(±3,±11)の8個に限ることを証明する。
これは、参考文献[6](p121, Lemma 4)によると、Bumby(1967)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。
    3x⁴ - 2y² = 1 ------ (1)

(1)より、
    1+2y² = 3x⁴ ------- (2)
となる。(2)の両辺を類数１の２次体Q(sqrt{-2})上で分解すると、
    (1+y*sqrt{-2})(1+y*sqrt{-2}) = (1+sqrt{-2})(1-sqrt{-2})x⁴ -------- (3)
を得る。
体Q(sqrt{-2})の単数は±1であり、その整数環はZ[sqrt{-2}]である。
以下では、Z[sqrt{-2}]の代数的整数について、考察する。
Z[sqrt{-2}]は一意分解整域(UFD)であり、sqrt{-2},1+sqrt{-2},1-sqrt{-2}はいずれも素数であることに注意する。

gp> nf=bnfinit(x^2+2)
time = 513 ms.
%1 = [[;], matrix(0,9), [;], Mat([0.E-38 + 10.65573737811015439752285932*I, 0.E-38 + 11.68534928834977599472928796*I, 0.E-38 + 1.511938820847815389018899756*I, 0.E-38 + 3.141592653589793238462643383*I, 0.E-38 + 1.910633236249018556327714205*I, 0.E-38 + 0.8810213260093969591212855716*I, 0.E-38 + 11.05443179351135756483167377*I, 0.E-38 + 9.887732688709736440884538067*I, 0.E-38 + 2.678637925649436512966035465*I]), [[2, [0, 1]~, 2, 1, [0, 1]~], [3, [-1, 1]~, 1, 1, [1, 1]~], [3, [1, 1]~, 1, 1, [-1, 1]~], [11, [-3, 1]~, 1, 1, [3, 1]~], [11, [3, 1]~, 1, 1, [-3, 1]~], [17, [-7, 1]~, 1, 1, [7, 1]~], [17, [7, 1]~, 1, 1, [-7, 1]~], [19, [-6, 1]~, 1, 1, [6, 1]~], [19, [6, 1]~, 1, 1, [-6, 1]~]]~, [2, 4, 6, 1, 3, 5, 7, 9, 8], [x^2 + 2, [0, 1], -8, 1, [Mat([1, 0.E-96 + 1.414213562373095048801688724*I]), [2; 0.E-96 - 2.828427124746190097603377448*I], [2, 0.E-96; 0.E-96, 4.000000000000000000000000000], [2, 0; 0, -4], [4, 0; 0, 2], [-4, 0; 0, 2], [4, [0, 2]~]], [0.E-96 + 1.414213562373095048801688724*I], [1, x], [1, 0; 0, 1], [1, 0, 0, -2; 0, 1, 1, 0]], [[1, [], []], 1, 0.9482651922610266649, [2, -1], [], 32767], [[;], [], []], 0]
gp> nf.clgp
time = 0 ms.
%2 = [1, [], []]
gp> nf.zk
time = 0 ms.
%3 = [1, x]
gp> nf.fu
time = 0 ms.
%4 = []

■1+y*sqrt{-2},1-y*sqrt{-2}は、どちらも3で割れない。

1+y*sqrt{-2}が3で割り切れると仮定すると、ある有理整数m,nに対して、
    1+y*sqrt{-2}=3(m+n*sqrt{-2})
つまり、
    1+y*sqrt{-2}=3m+3n*sqrt{-2}
となる。これより、
    1=3m, y=3n
となるが、これはmが有理整数であることに反する。
よって、1+y*sqrt{-2}は、3で割れない。

1-y*sqrt{-2}についても同様にして、3で割れないことが分かる。

■1+y*sqrt{-2},1-y*sqrt{-2}は、どちらもsqrt{-2}で割れない。

1+y*sqrt{-2}がsqrt{-2}で割り切れると仮定すると、ある有理整数m,nに対して、
    1+y*sqrt{-2}=sqrt{-2}(m+n*sqrt{-2})
つまり、
    1+y*sqrt{-2}=-2n+m*sqrt{-2}
となる。これより、
    1=-2n, y=m
となるが、これはnが有理整数であることに反する。
よって、1+y*sqrt{-2}は、sqrt{-2}で割れない。

1-y*sqrt{-2}についても同様にして、sqrt{-2}で割れないことが分かる。

■1+y*sqrt{-2},1-y*sqrt{-2}は互いに素である。

2をZ[sqrt{-2}]上で素因数分解すると、2=(-1)*sqrt{-2}²となるが、1+y*sqrt{-2}は(Z[sqrt{-2}]の)素数sqrt{-2}で割れないので、
    gcd(1+y*sqrt{-2},1-y*sqrt{-2}) = gcd(1+y*sqrt{-2},2) = 1
となる。

■(3)より、ある有理整数a,b,c,dに対して、
    x = (a+b*sqrt{-2})(c+d*sqrt{-2}),
    gcd(a+b*sqrt{-2},c+d*sqrt{-2}) = 1
であり、さらに、
(case i)
    1+y*sqrt{-2}=±(1+sqrt{-2})(a+b*sqrt{-2})⁴, ------ (4)
    1-y*sqrt{-2}=±(1-sqrt{-2})(c+d*sqrt{-2})⁴.
または、
(case ii)
    1+y*sqrt{-2}=±(1-sqrt{-2})(a+b*sqrt{-2})⁴, ------ (5)
    1-y*sqrt{-2}=±(1+sqrt{-2})(c+d*sqrt{-2})⁴.
のどちらかが成立する。

■[case i]
(4)より、
    1=±(a⁴-8a³b-12a²b²+16ab³+4b⁴), ------ (6)
    y=±(a⁴+4a³b-12a²b²-8ab³+4b⁴). ------ (7)
を得る。
pari/gpでThue方程式(6)を解くと、(a,b)=(±1,0),±(1,1)となる。
よって、(7)より、y=±1,±11を得る。(2)より、x=±1,±3を得る。

gp> x=Mod(x,x^2+2)
time = 3 ms.
%1 = Mod(x, x^2 + 2)
gp> (1+x)*(a+b*x)^4
time = 17 ms.
%2 = Mod((a^4 + 4*b*a^3 - 12*b^2*a^2 - 8*b^3*a + 4*b^4)*x + (a^4 - 8*b*a^3 - 12*b^2*a^2 + 16*b^3*a + 4*b^4), x^2 + 2)
gp> (1-x)*(a+b*x)^4
time = 5 ms.
%3 = Mod((-a^4 + 4*b*a^3 + 12*b^2*a^2 - 8*b^3*a - 4*b^4)*x + (a^4 + 8*b*a^3 - 12*b^2*a^2 - 16*b^3*a + 4*b^4), x^2 + 2)
gp> th1=thueinit(a^4-8*a^3-12*a^2+16*a+4)
time = 1,030 ms.
%4 = [a^4 - 8*a^3 - 12*a^2 + 16*a + 4, [[;], matrix(0,9), [1.909668421483239999893802577 + 9.424777960769379715387930149*I, -0.8335003601945001014486245809 + 12.56637061435917295385057353*I, 1.458931309366677586352162730 + 12.56637061435917295385057353*I; 0.9158046094693063195753709193 + 6.283185307179586476925286766*I, -0.7174959518732310540818583589 + 15.70796326794896619231321691*I, -3.009927621434408741882645670 + 15.70796326794896619231321691*I; -1.909668421483239999893802577 + 9.424777960769379715387930149*I, -1.458931309366677586352162730 + 9.424777960769379715387930149*I, 0.8335003601945001014486245809 + 9.424777960769379715387930149*I; -0.9158046094693063195753709193 + 12.56637061435917295385057353*I, 3.009927621434408741882645670 + 6.283185307179586476925286766*I, 0.7174959518732310540818583589 + 6.283185307179586476925286766*I], [-2.136853886500424273876644618 + 0.E-94*I, 2.570361249745706825851904753 + 9.424777960769379715387930149*I, 2.051593165816257547073832738 + 0.E-94*I, -12.67531109189848989862870505 + 12.56637061435917295385057353*I, -0.5731079173902944219501968278 + 0.E-96*I, 12.76194983943368343486865241 + 3.141592653589793238462643383*I, -0.4335073632452825519752601347 + 3.141592653589793238462643383*I, 0.9571302395066873965711230090 + 0.E-96*I, -0.1272460482297605844303582373 + 0.E-96*I; -0.5690735667600707870914616443 + 0.E-96*I, 0.1355662035147882351162015095 + 0.E-96*I, -1.146855128557999054533284356 + 3.141592653589793238462643383*I, 1.235159171496846624956114873 + 3.141592653589793238462643383*I, 0.5731079173902944219501968278 + 0.E-96*I, -0.8137994408140254940020238721 + 9.424777960769379715387930149*I, 0.4335073632452825519752601347 + 0.E-96*I, 0.3475789104625023966291437423 + 0.E-96*I, -1.177463101739429208769908514 + 3.141592653589793238462643383*I; 0.2779295801845291380511174423 + 3.141592653589793238462643383*I, 0.1555777830607534139241426924 + 6.283185307179586476925286766*I, -0.2950875458161320417567570967 + 0.E-94*I, 0.06693690931201261572437484085 + 3.141592653589793238462643383*I, -0.5731079173902944219501968278 + 3.141592653589793238462643383*I, 0.4567141283723305094523597813 + 4.19478201 E-94*I, -0.4335073632452825519752601347 + 3.141592653589793238462643383*I, -0.1272460482297605844303582373 + 9.424777960769379715387930149*I, 0.9571302395066873965711230090 + 9.424777960769379715387930149*I; 2.427997873075965922916988820 + 0.E-94*I, -2.861505236321248474892248955 + 0.E-94*I, -0.6096504914421264507837912857 + 9.424777960769379715387930149*I, 11.37321501108963065794821533 + 9.424777960769379715387930149*I, 0.5731079173902944219501968278 + 3.141592653589793238462643383*I, -12.40486452699198845031898832 + 6.283185307179586476925286766*I, 0.4335073632452825519752601347 + 0.E-96*I, -1.177463101739429208769908514 + 0.E-96*I, 0.3475789104625023966291437423 + 9.424777960769379715387930149*I], [[2, [1, 0, 0, 1]~, 4, 1, [2, 1, 1, 0]~], [3, [1, 1, 0, 0]~, 4, 1, [1, 1, 0, -1]~], [5, [-2, 1, 0, 0]~, 1, 1, [0, -1, 1, -2]~], [5, [1, 1, 0, 0]~, 1, 1, [2, 0, -1, -2]~], [5, [-4, -2, 4, 0]~, 1, 2, [1, -1, -1, 0]~], [19, [-4, 1, 0, 0]~, 1, 1, [7, 8, 3, 8]~], [19, [-9, 1, 0, 0]~, 1, 1, [6, -5, 4, 8]~], [29, [9, 1, 0, 0]~, 1, 1, [-1, -6, -10, 8]~], [29, [3, 1, 0, 0]~, 1, 1, [4, -10, 14, 8]~]]~, [3, 4, 7, 1, 2, 6, 5, 9, 8], [a^4 - 8*a^3 - 12*a^2 + 16*a + 4, [4, 0], 27648, 32, [[1, -1.933417311388629813494311268, 1.434525624994309484778059685, -1.386768338497295830227870195; 1, -0.2193386938947763804377404597, 0.5120273656598666033563985729, -0.05615370681110910813237436061; 1, 1.034437825822273617099743118, 0.7675154038729781224328040152, 0.3969734828337338662821887017; 1, 9.118318179461132576832308609, 21.28593160547284578943273772, 97.04594856247467107207805585], [1, 1, 1, 1; -1.933417311388629813494311268, -0.2193386938947763804377404597, 1.034437825822273617099743118, 9.118318179461132576832308609; 1.434525624994309484778059685, 0.5120273656598666033563985729, 0.7675154038729781224328040152, 21.28593160547284578943273772; -1.386768338497295830227870195, -0.05615370681110910813237436061, 0.3969734828337338662821887017, 97.04594856247467107207805585], [4, 8.000000000000000000000000000, 24.00000000000000000000000000, 96.00000000000000000000000000; 8.000000000000000000000000000, 88.00000000000000000000000000, 192.0000000000000000000000000, 888.0000000000000000000000000; 24.00000000000000000000000000, 192.0000000000000000000000000, 456.0000000000000000000000000, 2064.000000000000000000000000; 96.00000000000000000000000000, 888.0000000000000000000000000, 2064.000000000000000000000000, 9420.000000000000000000000000], [4, 8, 24, 96; 8, 88, 192, 888; 24, 192, 456, 2064; 96, 888, 2064, 9420], [24, 0, 0, 16; 0, 12, 0, 4; 0, 0, 12, 0; 0, 0, 0, 8], [89856, -34560, -94464, 23040; -34560, 24192, 43776, -11520; -94464, 43776, 111744, -27648; 23040, -11520, -27648, 6912], [6912, [-2304, 1152, 0, 0]~]], [-1.933417311388629813494311268, -0.2193386938947763804377404597, 1.034437825822273617099743118, 9.118318179461132576832308609], [1, a, 1/4*a^2 + 1/2, 1/8*a^3 + 1/4*a], [1, 0, -2, 0; 0, 1, 0, -2; 0, 0, 4, 0; 0, 0, 0, 8], [1, 0, 0, 0, 0, -2, 0, -4, 0, 0, -2, -6, 0, -4, -6, -31; 0, 1, 0, 0, 1, 0, 0, -4, 0, 0, -2, -9, 0, -4, -9, -43; 0, 0, 1, 0, 0, 4, 0, 7, 1, 0, 4, 10, 0, 7, 10, 52; 0, 0, 0, 1, 0, 0, 2, 8, 0, 2, 4, 20, 1, 8, 20, 90]], [[1, [], []], 17.63089857554484186671204240, 0.9509306494990348175, [2, -1], [1/4*a^3 - 9/4*a^2 - 1/2*a + 5/2, 1/4*a^2 - 1/2, 1/4*a^2 - 2*a - 1/2], 312], [[;], [], []], 0], [-1.933417311388629813494311268 + 0.E-86*I, -0.2193386938947763804377404597 + 0.E-86*I, 1.034437825822273617099743118 + 0.E-86*I, 9.118318179461132576832308609 + 0.E-86*I]~, [1.412736515476273159734586748, 1.504963810717204370941322835, 1.504963810717204370941322835]~, [-6.750849990554747209963966297 + 0.E-85*I, 0.4345256249943094847780596859 + 0.E-86*I, 4.301360247771569111766682222 + 0.E-86*I; 2.498784989333758733965404582 + 0.E-86*I, -0.4879726343401333966436014271 + 0.E-87*I, -0.04929524655058063576812050767 + 0.E-86*I; -0.1481294950116089864306018524 + 0.E-86*I, -0.2324845961270218775671959847 + 0.E-86*I, -2.301360247771569111766682222 + 0.E-86*I; 0.4001944962325974624291635670 + 0.E-85*I, 20.28593160547284578943273772 + 0.E-86*I, 2.049295246550580635768120507 + 0.E-86*I], [0.2829867552572250084614740354, 0.08132073977186383708508334127, -0.2016660154853611713763906942; -0.2827218497335698900705394750, -0.2483018292579450906152887337, -0.4017979978312947692385463283; 0.1534961685733496786232575946, -0.2483018292579450906152887337, 0.03442002047562479945525074127], [0.3986586634667121444689176563, 1.253776519591672345565778578, 0.7258581393734344655270250492, 1, 5.38859960 E-86, 11]]
gp> thue(th1,1)
time = 64 ms.
%5 = [[-1, -1], [1, 1], [1, 0], [-1, 0]]
gp> thue(th1,-1)
time = 6 ms.
%6 = []

■[case ii]
(5)より、
1 = ±(-1)(a⁴+8a³b-12a²b²-16ab³+4b⁴), ------ (8)
-y = ±(a⁴-4a³b-12a²b²+8ab³+4b⁴). ------ (9)
を得る。
pari/gpでThue方程式(6)を解くと、(a,b)=(±1,0),±(1,-1)となる。
よって、(9)より、y=±1,±11を得る。(2)より、x=±1,±3を得る。

gp> th2=thueinit(a^4+8*a^3-12*a^2-16*a+4)
time = 340 ms.
%7 = [a^4 + 8*a^3 - 12*a^2 - 16*a + 4, [[;], matrix(0,9), [-0.9158046094693063195753709193 + 3.141592653589793238462643383*I, 3.009927621434408741882645670 + 6.283185307179586476925286766*I, 0.7174959518732310540818583589 + 6.283185307179586476925286766*I; -1.909668421483239999893802577 + 25.13274122871834590770114706*I, -1.458931309366677586352162730 + 15.70796326794896619231321691*I, 0.8335003601945001014486245809 + 15.70796326794896619231321691*I; 0.9158046094693063195753709193 - 3.141592653589793238462643383*I, -0.7174959518732310540818583589 + 3.141592653589793238462643383*I, -3.009927621434408741882645670 + 3.141592653589793238462643383*I; 1.909668421483239999893802577 + 2.34083817 E-94*I, -0.8335003601945001014486245809 + 6.283185307179586476925286766*I, 1.458931309366677586352162730 + 6.283185307179586476925286766*I], [-0.5690735667600707870914616443 + 9.424777960769379715387930149*I, 0.1355662035147882351162015095 + 0.E-95*I, -1.146855128557999054533284356 + 3.141592653589793238462643383*I, -14.58449943942719510075255213 + 3.141592653589793238462643383*I, 0.5731079173902944219501968278 + 0.E-96*I, 15.00585917011001623170664313 + 12.56637061435917295385057353*I, 0.4335073632452825519752601347 + 3.141592653589793238462643383*I, 0.3475789104625023966291437423 + 3.141592653589793238462643383*I, -1.177463101739429208769908514 + 3.141592653589793238462643383*I; -2.136853886500424273876644618 + 3.141592653589793238462643383*I, 2.570361249745706825851904753 + 9.424777960769379715387930149*I, 2.051593165816257547073832738 + 0.E-95*I, -26.21646116323157533397894254 + 12.56637061435917295385057353*I, -0.5731079173902944219501968278 + 0.E-96*I, 26.30309991076676887021888990 + 1.79776372 E-94*I, -0.4335073632452825519752601347 + 0.E-96*I, 0.9571302395066873965711230090 + 3.141592653589793238462643383*I, -0.1272460482297605844303582373 + 0.E-96*I; 2.427997873075965922916988820 + 3.141592653589793238462643383*I, -2.861505236321248474892248955 + 6.283185307179586476925286766*I, -0.6096504914421264507837912857 + 9.424777960769379715387930149*I, 36.36260030025838313486003159 + 3.141592653589793238462643383*I, 0.5731079173902944219501968278 + 3.141592653589793238462643383*I, -37.39424981616074092723080458 + 9.424777960769379715387930149*I, 0.4335073632452825519752601347 + 3.141592653589793238462643383*I, -1.177463101739429208769908514 + 9.424777960769379715387930149*I, 0.3475789104625023966291437423 + 9.424777960769379715387930149*I; 0.2779295801845291380511174423 + 6.283185307179586476925286766*I, 0.1555777830607534139241426924 + 6.283185307179586476925286766*I, -0.2950875458161320417567570967 + 0.E-95*I, 4.438360302400387299871463086 + 9.424777960769379715387930149*I, -0.5731079173902944219501968278 + 3.141592653589793238462643383*I, -3.914709264716044174694728464 + 3.141592653589793238462643383*I, -0.4335073632452825519752601347 + 0.E-96*I, -0.1272460482297605844303582373 + 0.E-96*I, 0.9571302395066873965711230090 + 9.424777960769379715387930149*I], [[2, [1, 0, 0, 1]~, 4, 1, [2, 1, 1, 0]~], [3, [-1, 1, 0, 0]~, 4, 1, [-1, 1, 0, -1]~], [5, [-1, 1, 0, 0]~, 1, 1, [-2, 0, 1, -2]~], [5, [2, 1, 0, 0]~, 1, 1, [0, -1, -1, -2]~], [5, [-4, 2, 4, 0]~, 1, 2, [1, 1, -1, 0]~], [19, [9, 1, 0, 0]~, 1, 1, [-6, -5, -4, 8]~], [19, [4, 1, 0, 0]~, 1, 1, [-7, 8, -3, 8]~], [29, [-3, 1, 0, 0]~, 1, 1, [-4, -10, -14, 8]~], [29, [-9, 1, 0, 0]~, 1, 1, [1, -6, 10, 8]~]]~, [3, 4, 7, 1, 2, 6, 5, 9, 8], [a^4 + 8*a^3 - 12*a^2 - 16*a + 4, [4, 0], 27648, 32, [[1, -9.118318179461132576832308609, 21.28593160547284578943273772, -97.04594856247467107207805585; 1, -1.034437825822273617099743118, 0.7675154038729781224328040152, -0.3969734828337338662821887017; 1, 0.2193386938947763804377404597, 0.5120273656598666033563985729, 0.05615370681110910813237436061; 1, 1.933417311388629813494311268, 1.434525624994309484778059685, 1.386768338497295830227870195], [1, 1, 1, 1; -9.118318179461132576832308609, -1.034437825822273617099743118, 0.2193386938947763804377404597, 1.933417311388629813494311268; 21.28593160547284578943273772, 0.7675154038729781224328040152, 0.5120273656598666033563985729, 1.434525624994309484778059685; -97.04594856247467107207805585, -0.3969734828337338662821887017, 0.05615370681110910813237436061, 1.386768338497295830227870195], [4, -8.000000000000000000000000000, 24.00000000000000000000000000, -96.00000000000000000000000000; -8.000000000000000000000000000, 88.00000000000000000000000000, -192.0000000000000000000000000, 888.0000000000000000000000000; 24.00000000000000000000000000, -192.0000000000000000000000000, 456.0000000000000000000000000, -2064.000000000000000000000000; -96.00000000000000000000000000, 888.0000000000000000000000000, -2064.000000000000000000000000, 9420.000000000000000000000000], [4, -8, 24, -96; -8, 88, -192, 888; 24, -192, 456, -2064; -96, 888, -2064, 9420], [24, 0, 0, 8; 0, 12, 0, 4; 0, 0, 12, 0; 0, 0, 0, 8], [89856, 34560, -94464, -23040; 34560, 24192, -43776, -11520; -94464, -43776, 111744, 27648; -23040, -11520, 27648, 6912], [6912, [2304, 1152, 0, 0]~]], [-9.118318179461132576832308609, -1.034437825822273617099743118, 0.2193386938947763804377404597, 1.933417311388629813494311268], [1, a, 1/4*a^2 + 1/2, 1/8*a^3 + 1/4*a], [1, 0, -2, 0; 0, 1, 0, -2; 0, 0, 4, 0; 0, 0, 0, 8], [1, 0, 0, 0, 0, -2, 0, -4, 0, 0, -2, 6, 0, -4, 6, -31; 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 2, -9, 0, 4, -9, 43; 0, 0, 1, 0, 0, 4, 0, 7, 1, 0, 4, -10, 0, 7, -10, 52; 0, 0, 0, 1, 0, 0, 2, -8, 0, 2, -4, 20, 1, -8, 20, -90]], [[1, [], []], 17.63089857554484186671204240, 0.9509306494990348175, [2, -1], [1/4*a^3 + 9/4*a^2 - 1/2*a - 5/2, 1/4*a^2 - 1/2, 1/4*a^2 + 2*a - 1/2], 311], [[;], [], []], 0], [-9.118318179461132576832308609 + 0.E-86*I, -1.034437825822273617099743118 + 0.E-86*I, 0.2193386938947763804377404597 + 0.E-86*I, 1.933417311388629813494311268 + 0.E-86*I]~, [1.412736515476273159734586748, 1.504963810717204370941322835, 1.504963810717204370941322835]~, [-0.4001944962325974624291635670 + 0.E-85*I, 20.28593160547284578943273772 + 0.E-86*I, 2.049295246550580635768120507 + 0.E-86*I; 0.1481294950116089864306018524 + 0.E-86*I, -0.2324845961270218775671959847 + 0.E-86*I, -2.301360247771569111766682222 + 0.E-86*I; -2.498784989333758733965404582 + 0.E-86*I, -0.4879726343401333966436014271 + 0.E-87*I, -0.04929524655058063576812050767 + 0.E-86*I; 6.750849990554747209963966297 + 0.E-85*I, 0.4345256249943094847780596859 + 0.E-86*I, 4.301360247771569111766682222 + 0.E-86*I], [-0.2829867552572250084614740355, -0.4846527707425861798378647297, -0.2016660154853611713763906942; 0.2827218497335698900705394750, -0.1190761480977248791680068533, 0.03442002047562479945525074127; -0.1534961685733496786232575946, -0.1190761480977248791680068533, -0.4017979978312947692385463283], [0.3986586634667121444689176563, 1.253776519591672345565778578, 0.7258581393734344655270250492, 1, 7.812870064703895905 E-86, 11]]
gp> thue(th2,1)
time = 31 ms.
%8 = [[1, -1], [-1, 1], [1, 0], [-1, 0]]
gp> thue(th2,-1)
time = 3 ms.
%9 = []

■case i～iiの結果を整理すると、Diophantus方程式3x⁴-2y²=1の整数解(x,y)は、
(±1, ±1), (±3, ±11)
の8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.
[3]Michael A. Bennett, Gary Walsh, "The Diophantine Equation b^2X^4-dY^2=1", 1991, p1-10.
[4]Michael A. Bennett, Gary Walsh, "Simultaneous quadratic quations with few or no solutions", 1999, p1-10.
[5]J.H.E.Cohn, "The Diophantine Equation x^4+1=Dy^2",Math. of Comp, Vol.66(1997), No.219, p1347-1351.
[6]Gary Walsh, "A note on a theorem of Ljunggren and the Diophantine equations x^2-kxy^2+y^4=1,4", Arch. Math. 73(1999), p119-125.

Last Update: 2005.08.21

H.Nakao