Integral Points on Elliptic Curves: v^4-5u^2=\pm{4}

[2004.02.08]v⁴-5u²=\pm{4}の整点

ax^4+bx^3y+cx^2y^2+dxy^3+ey^4≡D(mod n)の解を探すプログラム(pari/gp)

■Lucas数列に現れる完全平方数は、1,4に限ることを証明する。
Lucas数列を{v_n : n >= 0 }, Fibonacci数列を{u_n : n >= 0 }とする。
Fibonacci数列とLucas数列の関係式
     v_n²-5u_n² = (-1)ⁿ・4
において、v_n=v², u_n=uとすると、
     v⁴-5u² = ±4 -------- (C±)
     ただし、u,vは非負整数
を得る。
よって、２つの楕円曲線
     C₁: v⁴-5u² = 4, ------ (C1)
     C₂: v⁴-5u² = -4 ----- (C2)
の整点を求めることに帰着できる。

■楕円曲線C₁は整点を持たない。
u,vは整数なので、
     v⁴≡ 0, 1 (mod 5)
である。
よって、(C1)はZ/5Z上で解を持たないので、整数解を持たない。
つまり、楕円曲線C₁は整点を持たない。

■楕円曲線C₂の整点は、(±1,±1), (±2,±2)に限る。
(C2)より、
     (sqrt{5}u+2)(sqrt{5}u-2) = v⁴
を得る。
α=(sqrt{5}+1)/2とする。αは２次の代数的整数である。
sqrt{5}=2α-1であるので、
     (2uα-u+2)(2uα-u-2) = v⁴ ------ (1)
を得る。
類数1の２次体Q(α)上では、その代数的整数Z[α]の元は一意に素因数分解できる。

gp>  nf=bnfinit(x^2-x-1)
time = 595 ms.
%1 = [[;], matrix(0,9), [-0.4812118250596034474977589134 + 15.70796326794896619231321691*I; 0.4812118250596034474977589134 + 12.56637061435917295385057353*I], [-0.1501903904951261035222637571 + 0.E-36*I, -3.114720486363412061520809826 + 3.141592653589793238462643383*I, -7.012026945597204152812156849 + 12.56637061435917295385057353*I, 21.17332030262255168990139219 + 9.424777960769379715387930149*I, 68.48226954895881564820402946 + 3.141592653589793238462643383*I, 97.91345002310529121857931577 + 9.424777960769379715387930149*I, 118.1719585343656005247944658 + 6.283185307179586476925286766*I, -0.08968238446746670411458790438 + 3.141592653589793238462643383*I, 44.84238211501058732140616685 + 6.283185307179586476925286766*I; 0.1501903904951261035222637571 + 9.424777960769379715387930149*I, 3.114720486363412061520809826 + 12.56637061435917295385057353*I, 7.012026945597204152812156849 + 3.141592653589793238462643383*I, -21.17332030262255168990139219 + 6.283185307179586476925286766*I, -68.48226954895881564820402946 + 12.56637061435917295385057353*I, -97.91345002310529121857931577 + 9.424777960769379715387930149*I, -118.1719585343656005247944658 + 6.283185307179586476925286766*I, 0.08968238446746670411458790438 + 0.E-38*I, -44.84238211501058732140616685 + 6.283185307179586476925286766*I], [[5, [2, 1]~, 2, 1, [2, 1]~], [11, [-4, 1]~, 1, 1, [3, 1]~], [11, [3, 1]~, 1, 1, [-4, 1]~], [19, [-5, 1]~, 1, 1, [4, 1]~], [19, [4, 1]~, 1, 1, [-5, 1]~], [29, [-6, 1]~, 1, 1, [5, 1]~], [29, [5, 1]~, 1, 1, [-6, 1]~], [31, [-13, 1]~, 1, 1, [12, 1]~], [31, [12, 1]~, 1, 1, [-13, 1]~]]~, [2, 4, 6, 1, 3, 5, 7, 9, 8], [x^2 - x - 1, [2, 0], 5, 1, [[1, -0.6180339887498948482045868343; 1, 1.618033988749894848204586834], [1, 1; -0.6180339887498948482045868343, 1.618033988749894848204586834], [2, 1.000000000000000000000000000; 1.000000000000000000000000000, 3.000000000000000000000000000], [2, 1; 1, 3], [5, 2; 0, 1], [3, -1; -1, 2], [5, [2, 1]~]], [-0.6180339887498948482045868343, 1.618033988749894848204586834], [1, x], [1, 0; 0, 1], [1, 0, 0, 1; 0, 1, 1, 1]], [[1, [], []], 0.4812118250596034474977589134, 0.9725234526003131839, [2, -1], [x], 123], [[;], [], []], 0]
gp>  nf.clgp
time = 0 ms.
%2 = [1, [], []]
gp>  nf.fu
time = 0 ms.
%3 = [x]

有理素数2は、Z[α]の素元であるので、
gcd(2uα-u+2,2uα-u-2)=gcd(2uα-u+2,4)=1,2,4である。

----------------------------------------------------------
[case i] gcd(2uα-u+2,2uα-u-2)=1の場合
(1)より、ある有理整数i(0 <= i <= 3), a, bが存在して、
     2uα-u+2 = ±αⁱ(aα+b)⁴
となる。
[case i-1] i=0の場合
     2u = ±(3a⁴ + 8a³b + 6a²b² + 4ab³), ---------- (2)
     -u+2 = ±(2a⁴ + 4a³b + 6a²b² + b⁴) ---------- (3)
(2),(3)より、uを消去すると、
     4 = ±(7a⁴ + 16a³b + 18a²b² + 4ab³+2b⁴) --------- (4)
を得る。

Thue方程式(4)はZ/8Z上で解を持たないので、整数解を持たない。

gp>  z=Mod(x,x^2-x-1)
time = 5 ms.
%1 = Mod(x, x^2 - x - 1)
gp>  w=(z-1)/2        
time = 1 ms.
%2 = Mod(1/2*x - 1/2, x^2 - x - 1)
gp>  p=lift((a*z+b)^4)
time = 31 ms.
%3 = (3*a^4 + 8*b*a^3 + 6*b^2*a^2 + 4*b^3*a)*x + (2*a^4 + 4*b*a^3 + 6*b^2*a^2 + b^4)
gp>  p1=polcoeff(p,1,x)
time = 4 ms.
%4 = 3*a^4 + 8*b*a^3 + 6*b^2*a^2 + 4*b^3*a
gp>  p0=polcoeff(p,0,x)
time = 3 ms.
%5 = 2*a^4 + 4*b*a^3 + 6*b^2*a^2 + b^4
gp>  p1+2*p0
time = 3 ms.
%6 = 7*a^4 + 16*b*a^3 + 18*b^2*a^2 + 4*b^3*a + 2*b^4
gp>  read("ide7.gp")
time = 4 ms.
gp> g([7,16,18,4,2],4,8)
time = 4 ms.
gp> g([7,16,18,4,2],-4,8)
time = 4 ms.

[case i-2] i=1の場合
     2u = ±(5a⁴ + 12a³b + 12a²b² + 4ab³+b⁴), --------- (5)
     -u+2 = ±(3a⁴ + 6a³b + 6a²b² + 4ab³) ----------- (6)
uを消去すると、
     4 = ±(11a⁴ + 28a³b + 24a²b² + 12ab³+b⁴) --------- (7)
を得る。

Thue方程式(7)の整数解(a,b)をpari/gpで計算すると、±(1,-1)に限ることが分かる。
よって、(5-)より、u=-1を得る。このとき、v=±1である。

gp> p=lift((a*z+b)^4*z)
time = 2 ms.
%11 = (5*a^4 + 12*b*a^3 + 12*b^2*a^2 + 4*b^3*a + b^4)*x + (3*a^4 + 8*b*a^3 + 6*b^2*a^2 + 4*b^3*a)
gp> p1=polcoeff(p,1,x)
time = 1 ms.
%12 = 5*a^4 + 12*b*a^3 + 12*b^2*a^2 + 4*b^3*a + b^4
gp> p0=polcoeff(p,0,x)
time = 2 ms.
%13 = 3*a^4 + 8*b*a^3 + 6*b^2*a^2 + 4*b^3*a
gp>  p1+p0*2
time = 0 ms.
%14 = 11*a^4 + 28*b*a^3 + 24*b^2*a^2 + 12*b^3*a + b^4
gp>  th1=thueinit(11 + 28*b + 24*b^2 + 12*b^3 + b^4)
time = 708 ms.
%15 = [b^4 + 12*b^3 + 24*b^2 + 28*b + 11, [[;], matrix(0,5), [-0.2406059125298017237488794567 - 9.424777960769379715387930150*I, 1.061275061905035652033018916 - 18.84955592153875943077586030*I; -0.2406059125298017237488794567 - 6.283185307179586476925286766*I, -1.061275061905035652033018916 - 6.283185307179586476925286766*I; 0.4812118250596034474977589134 - 9.424777960769379715387930150*I, -1.83086773 E-75 - 20.65866971014352215903049389*I], [0.3849259339947735406604092067 + 0.E-77*I, -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.2783553960709223679498188577 + 3.141592653589793238462643383*I, -16.06740304625427791605153537 + 3.141592653589793238462643383*I, -6.122001996137858632501172741 + 9.424777960769379715387930149*I; -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.3849259339947735406604092067 + 0.E-77*I, -1.013329510184337886411080338 + 0.E-77*I, 18.95467399661189860103808885 + 3.141592653589793238462643383*I, 1.537333281547615534057839952 + 0.E-76*I; -0.1501903904951261035222637571 + 6.866618316202791859756644622*I, -0.1501903904951261035222637571 + 5.699752298156381094093928910*I, 0.7349741141134155184612614809 + 8.748618865266467509624187051*I, -2.887270950357620684986553480 + 7.859228936850032346962950573*I, 4.584668714590243098443332788 + 9.245093114906993628990287255*I], [[2, [1, 1, 0, 0]~, 2, 2, [1, 1, 0, 0]~], [11, [-2, 1, 0, 0]~, 1, 1, [-4, -5, -3, -1]~], [11, [0, 1, 0, 0]~, 1, 1, [0, 3, -1, -1]~], [19, [-1, 1, 0, 0]~, 1, 1, [6, -9, -3, -9]~], [19, [6, 1, 0, 0]~, 1, 1, [-4, 9, 3, -9]~]]~, [3, 2, 5, 1, 4], [b^4 + 12*b^3 + 24*b^2 + 28*b + 11, [2, 1], -400, 100, [[1, -9.838278303799419875235999640, 13.51448331982080697376256752, -98.57787136076781352275123524; 1, -0.6338576512001595175823476976, 0.1937206126785621154649534844, -0.6156241442275597982505854680; 1, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I, 0.1458980337503154553862394969 + 0.6005662120015552157733894199*I, -0.4032522475023133394990896439 - 0.3879045354686796176564585540*I], [1, 1, 2; -9.838278303799419875235999640, -0.6338576512001595175823476976, -1.527864045000420607181652662 + 2.172868967516401787912506591*I; 13.51448331982080697376256752, 0.1937206126785621154649534844, 0.2917960675006309107724789938 - 1.201132424003110431546778839*I; -98.57787136076781352275123524, -0.6156241442275597982505854680, -0.8065044950046266789981792879 + 0.7758090709373592353129171081*I], [4, -12.00000000000000000000000000, 14.00000000000000000000000000, -100.0000000000000000000000000; -12.00000000000000000000000000, 100.7213595499957939281834733, -134.6099033699941114994568627, 971.6857314549575186746530810; 14.00000000000000000000000000, -134.6099033699941114994568627, 183.4427190999915878563669467, -1332.931849429938591351478711; -100.0000000000000000000000000, 971.6857314549575186746530810, -1332.931849429938591351478711, 9718.601879714548688494086693], [4, -12, 14, -100; -12, 96, -132, 970; 14, -132, 182, -1332; -100, 970, -1332, 9718], [-10, -6, 0, -2; 0, -2, 0, 0; 0, 0, -10, -8; 0, 0, 0, -2], [-4280, -8480, -16360, -1440; -8480, -17320, -33280, -2920; -16360, -33280, -63280, -5520; -1440, -2920, -5520, -480], [200, [-80, 40, 0, 0]~]], [-9.838278303799419875235999640, -0.6338576512001595175823476976, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I], [1, b, 1/10*b^2 - 2/5*b - 1/10, 1/10*b^3 + 3/10*b - 2/5], [1, 0, 1, 4; 0, 1, 4, -3; 0, 0, 10, 0; 0, 0, 0, 10], [1, 0, 0, 0, 0, 1, 0, -8, 0, 0, -1, 12, 0, -8, 12, -91; 0, 1, 0, 0, 1, 4, -2, -8, 0, -2, 0, 13, 0, -8, 13, -104; 0, 0, 1, 0, 0, 10, -4, -21, 1, -4, -1, 34, 0, -21, 34, -269; 0, 0, 0, 1, 0, 0, 1, -12, 0, 1, -2, 17, 1, -12, 17, -126]], [[1, [], []], 0.5106981094295658352378116261, 1.043760129382343073, [2, -1], [1/10*b^3 + b^2 + 3/10*b + 3/5, 1/10*b^3 + 11/10*b^2 + 9/10*b + 1/2], 243], [[;], [], []], 0], [-9.838278303799419875235999640 + 0.E-67*I, -0.6338576512001595175823476976 + 0.E-67*I, -0.7639320225002103035908263312 + 1.086434483758200893956253295*I, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I]~, [0.2406059125298017237488794567, 0.5306375309525178260165094581]~, [-0.7861513777574232860695585858 + 0.E-66*I, 2.890053638263963812457009296 + 0.E-66*I; 0.7861513777574232860695585858 + 0.E-67*I, 0.3460143392358258839521643726 + 0.E-68*I; 1.48368246 E-67 - 1.272019649514068964252422461*I, -0.6180339887498948482045868343 - 0.7861513777574232860695585858*I; 1.48368246 E-67 + 1.272019649514068964252422461*I, -0.6180339887498948482045868343 + 0.7861513777574232860695585858*I], [-2.078086921235027537601322606, -2.078086921235027537601322606; 0.4711313946287194353179767784, -0.4711313946287194353179767784], [0.7259477025765315359820075412, 1.094193414953955986267911274, 0.7134565368684148183671856481, 0.7629801282406652058906211739, 5.64610170 E-66, 9]]
gp>  thue(th1,4)
time = 64 ms.
%16 = []
gp>  thue(th1,-4)
time = 61 ms.
%17 = [[1, -1], [-1, 1]]
gp> p=lift((1*z-1)^4*z)
time = 0 ms.
%18 = 2*x - 3

[case i-3] i=2の場合
     2u = ±(8a⁴ + 20a³b + 18a²b² + 8ab³+b⁴), --------- (8)
     -u+2 = ±(5a⁴ + 12a³b + 12a²b² + 4ab³) --------- (9)
(8),(9)よりuを消去すると、
     4 = ±(18a⁴ + 44a³b + 42a²b² + 16ab³+3b⁴) --------- (10)
を得る。

Thue方程式(10)はZ/3Z上で解を持たないので、整数解を持たない。

gp>  p=lift((a*z+b)^4*z^2)
time = 1 ms.
%23 = (8*a^4 + 20*b*a^3 + 18*b^2*a^2 + 8*b^3*a + b^4)*x + (5*a^4 + 12*b*a^3 + 12*b^2*a^2 + 4*b^3*a + b^4)
gp> p1=polcoeff(p,1,x)
time = 0 ms.
%24 = 8*a^4 + 20*b*a^3 + 18*b^2*a^2 + 8*b^3*a + b^4
gp> p0=polcoeff(p,0,x)
time = 0 ms.
%25 = 5*a^4 + 12*b*a^3 + 12*b^2*a^2 + 4*b^3*a + b^4
gp>  p1+p0*2
time = 1 ms.
%26 = 18*a^4 + 44*b*a^3 + 42*b^2*a^2 + 16*b^3*a + 3*b^4
gp> g([18,44,42,16,3],4,3)
time = 1 ms.
gp> g([18,44,42,16,3],-4,3)
time = 1 ms.

[case i-4] i=3の場合
     2u = ±(13a⁴ + 32a³b + 30a²b² + 12ab³+2b⁴), -------- (11)
     -u+2 = ±(8a⁴ + 72a³b + 66a²b² + 28ab³+b⁴) -------- (12)
(11),(12)より、uを消去すると、
     4 = ±(29a⁴ + 72a³b + 66a²b² + 28ab³+4b⁴) --------- (13)
を得る。

(13)で、a=A-B,b=B[逆変換は、A=a+b,B=b]と置換すると、A,Bは整数であり、
     4 = ±(29A⁴ - 44A³B + 24A²B² - 4AB³+B⁴) --------- (14)
を得る。
Thue方程式(14)の整数解(A,B)をpari/gpで計算すると、±(1,1)に限ることが分かる。
a=A-B,b=Bより、Thue方程式(13)の整数解(a,b)は、(0,±1)に限る。
よって、(11+)より、u=1を得る。このとき、v=±1である。

gp> p=lift((a*z+b)^4*z^3)
time = 1 ms.
%31 = (13*a^4 + 32*b*a^3 + 30*b^2*a^2 + 12*b^3*a + 2*b^4)*x + (8*a^4 + 20*b*a^3 + 18*b^2*a^2 + 8*b^3*a + b^4)
gp> p1=polcoeff(p,1,x)
time = 1 ms.
%32 = 13*a^4 + 32*b*a^3 + 30*b^2*a^2 + 12*b^3*a + 2*b^4
gp> p0=polcoeff(p,0,x)
time = 0 ms.
%33 = 8*a^4 + 20*b*a^3 + 18*b^2*a^2 + 8*b^3*a + b^4
gp>  p1+p0*2
time = 0 ms.
%34 = 29*a^4 + 72*b*a^3 + 66*b^2*a^2 + 28*b^3*a + 4*b^4
gp>  f(a,b)=29*a^4 + 72*b*a^3 + 66*b^2*a^2 + 28*b^3*a + 4*b^4
time = 0 ms.
gp>  f(A-B,B)
time = 2 ms.
%35 = 29*A^4 - 44*B*A^3 + 24*B^2*A^2 - 4*B^3*A - B^4
gp>  th3=thueinit(-(29-44*B+24*B^2-4*B^3-B^4))
time = 374 ms.
%52 = [B^4 + 4*B^3 - 24*B^2 + 44*B - 29, [[;], matrix(0,5), [0.2406059125298017237488794567 + 3.141592653589793238462643383*I, 1.061275061905035652033018916 - 6.283185307179586476925286766*I; 0.2406059125298017237488794567 - 1.03634022 E-75*I, -1.061275061905035652033018916 + 6.283185307179586476925286766*I; -0.4812118250596034474977589134 - 3.141592653589793238462643383*I, 5.18170113 E-76 + 17.04044213293399670252122670*I], [-0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.3849259339947735406604092067 + 0.E-77*I, 0.2783553960709223679498188577 + 3.141592653589793238462643383*I, -8.681380701877172561494280241 + 6.283185307179586476925286766*I, -13.84748166483059446784480842 + 0.E-75*I; 0.3849259339947735406604092067 + 0.E-77*I, -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, -1.013329510184337886411080338 + 0.E-77*I, 7.237745226698362219001003501 + 3.141592653589793238462643383*I, 15.03735485095559273937458259 + 9.424777960769379715387930149*I; -0.1501903904951261035222637571 + 5.699752298156381094093928910*I, -0.1501903904951261035222637571 + 6.866618316202791859756644622*I, 0.7349741141134155184612614809 + 8.748618865266467509624187051*I, 1.443635475178810342493276740 + 1.001982800176547508059178401*I, -1.189873186124998271529774172 + 8.486675079415861480760762869*I], [[2, [0, 0, 1, 1]~, 2, 2, [1, 1, 0, 0]~], [11, [-2, 1, 0, 0]~, 1, 1, [5, 1, 5, -1]~], [11, [-4, 1, 0, 0]~, 1, 1, [4, -5, 3, -1]~], [19, [-3, 1, 0, 0]~, 1, 1, [-8, 5, -6, -9]~], [19, [4, 1, 0, 0]~, 1, 1, [9, -2, 0, -9]~]]~, [3, 2, 5, 1, 4], [B^4 + 4*B^3 - 24*B^2 + 44*B - 29, [2, 1], -400, 100, [[1, -7.838278303799419875235999639, 4.676205016021387098526567882, -46.00580635324123130588382903; 1, 1.366142348799840482417652302, 0.5598629614784025978826057867, -0.3548734217566656582079076543; 1, 1.236067977499789696409173668 - 1.086434483758200893956253295*I, 0.3819660112501051517954131656 - 0.4858682717566456781828638759*I, -0.8196601125010515179541316563 - 0.04381071473395100488488192205*I], [1, 1, 2; -7.838278303799419875235999639, 1.366142348799840482417652302, 2.472135954999579392818347337 + 2.172868967516401787912506591*I; 4.676205016021387098526567882, 0.5598629614784025978826057867, 0.7639320225002103035908263312 + 0.9717365435132913563657277518*I; -46.00580635324123130588382903, -0.3548734217566656582079076543, -1.639320225002103035908263312 + 0.08762142946790200976976384411*I], [4, -4.000000000000000000000000000, 6.000000000000000000000000000, -48.00000000000000000000000000; -4.000000000000000000000000000, 68.72135954999579392818347337, -33.88854381999831757127338935, 358.1903898849802314624623248; 6.000000000000000000000000000, -33.88854381999831757127338935, 22.94427190999915878563669467, -215.9148550549911672491852940; -48.00000000000000000000000000, 358.1903898849802314624623248, -215.9148550549911672491852940, 2118.007677514901998526674929], [4, -4, 6, -48; -4, 64, -36, 358; 6, -36, 22, -216; -48, 358, -216, 2118], [-10, -2, 0, 0; 0, -2, 0, 0; 0, 0, -10, -6; 0, 0, 0, -2], [-760, 1440, -3720, -640; 1440, -2760, 6480, 1160; -3720, 6480, -14320, -2640; -640, 1160, -2640, -480], [200, [40, 40, 0, 0]~]], [-7.838278303799419875235999639, 1.366142348799840482417652302, 1.236067977499789696409173668 - 1.086434483758200893956253295*I], [1, B, 1/10*B^2 + 1/5*B + 1/10, 1/10*B^3 - 3/10*B - 1/5], [1, 0, -1, 2; 0, 1, -2, 3; 0, 0, 10, 0; 0, 0, 0, 10], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, -6; 0, 1, 0, 0, 1, -2, 0, -10, 0, 0, -1, 4, 0, -10, 4, -45; 0, 0, 1, 0, 0, 10, 2, 21, 1, 2, 3, -10, 0, 21, -10, 103; 0, 0, 0, 1, 0, 0, 1, -4, 0, 1, 0, 3, 1, -4, 3, -28]], [[1, [], []], 0.5106981094295658352378116261, 1.043760129382343073, [2, -1], [1/10*B^3 + 3/5*B^2 - 11/10*B + 7/5, 1/10*B^3 + 1/2*B^2 - 23/10*B + 23/10], 246], [[;], [], []], 0], [-7.838278303799419875235999639 + 0.E-67*I, 1.366142348799840482417652302 + 0.E-67*I, 1.236067977499789696409173668 + 1.086434483758200893956253295*I, 1.236067977499789696409173668 - 1.086434483758200893956253295*I]~, [0.2406059125298017237488794567, 0.5306375309525178260165094581]~, [-1.272019649514068964252422461 + 0.E-66*I, 2.890053638263963812457009296 + 0.E-66*I; 1.272019649514068964252422461 + 0.E-67*I, 0.3460143392358258839521643726 + 0.E-68*I; 1.85460307 E-67 + 0.7861513777574232860695585858*I, -0.6180339887498948482045868343 - 0.7861513777574232860695585858*I; 1.85460307 E-67 - 0.7861513777574232860695585858*I, -0.6180339887498948482045868343 + 0.7861513777574232860695585858*I], [2.078086921235027537601322606, 2.078086921235027537601322606; 0.4711313946287194353179767784, -0.4711313946287194353179767784], [0.7259477025765315359820075412, 1.094193414953955986267911274, 0.8418239574966185067958180081, 0.7629801282406652058906211739, 5.029457483314110202 E-66, 9]]
gp>  thue(th3,4)
time = 36 ms.
%36 = []
gp>  thue(th3,-4)
time = 40 ms.
%37 = [[-1, -1], [1, 1]]
gp> p=lift((0*z+1)^4*z^3)
time = 0 ms.
%38 = 2*x + 1

[case i]の結果をまとめると、得られた(C2)の整数解(u,v)は、
     (±1,±1)
である。

----------------------------------------------------------
[case ii] gcd(2uα-u+2,2uα-u-2)=2の場合
2⁴|(2uα-u+2)(2uα-u-2)=v⁴より、2|vとなるので、
(ii-1) 2|(2uα-u+2), 2³|(2uα-u-2) または、
(ii-2) 2³|(2uα-u+2), 2|(2uα-u-2) である。

[case ii-1]
2²|(2uα-u-2), (2uα-u+2)=(2uα-u-2)+4より、
2²|(2uα-u+2)を得る。
よって、2²|gcd(2uα-u-2,2uα-u+2)=2となり、矛盾する。
[case ii-2]
(ii-1)の場合と同様にして、2²|gcd(2uα-u-2,2uα-u+2)=2となり、矛盾する。

よって、gcd(2uα-u+2,2uα-u-2)=2は成立しないことが分かる。

----------------------------------------------------------
[case iii] gcd(2uα-u+2,2uα-u-2)=4の場合
(1)より、ある有理整数i(0 <= i <= 3), A, Bが存在して、
     2uα-u+2 = ±2²αⁱ(Aα+B)⁴
となる。
[case iii-1] i=0の場合
     u = ±(6a⁴ + 16a³b + 12a²b² + 8ab³), --------- (16)
     -u+2 = ±(8a⁴ + 16a³b + 24a²b² + 4b⁴) --------- (17)
(16),(17)より、uを消去すると、
     1 = ±(7a⁴ + 16a³b + 18a²b² + 4ab³+2b⁴) --------- (18)
を得る。

Thue方程式(18)はZ/5Z上で解を持たないので、整数解を持たない。

gp>  p=lift(4*(a*z+b)^4)
time = 0 ms.
%56 = (12*a^4 + 32*b*a^3 + 24*b^2*a^2 + 16*b^3*a)*x + (8*a^4 + 16*b*a^3 + 24*b^2*a^2 + 4*b^4)
gp> p1=polcoeff(p,1,x)
time = 0 ms.
%57 = 12*a^4 + 32*b*a^3 + 24*b^2*a^2 + 16*b^3*a
gp> p0=polcoeff(p,0,x)
time = 0 ms.
%58 = 8*a^4 + 16*b*a^3 + 24*b^2*a^2 + 4*b^4
gp>  p1/2
time = 0 ms.
%59 = 6*a^4 + 16*b*a^3 + 12*b^2*a^2 + 8*b^3*a
gp>  p1/2+p0
time = 0 ms.
%60 = 14*a^4 + 32*b*a^3 + 36*b^2*a^2 + 8*b^3*a + 4*b^4
gp>  (p1/2+p0)/2
time = 0 ms.
%61 = 7*a^4 + 16*b*a^3 + 18*b^2*a^2 + 4*b^3*a + 2*b^4
gp>  g([7,16,18,4,2],1,5)
time = 2 ms.
gp>  g([7,16,18,4,2],-1,5)
time = 2 ms.

[case iii-2] i=1の場合
     u = ±(10a⁴ + 24a³b + 24a²b² + 8ab³ + 2b⁴), -------- (19)
     -u+2 = ±(12a⁴ + 32a³b + 24a²b² + 16ab³) -------- (20)
(19),(20)より、uを消去すると、
     1 = ±(11a⁴ + 28a³b + 24a²b² + 12ab³ + b⁴) --------- (21)
を得る。

Thue方程式(21)の整数解(a,b)をpari/gpで計算すると、(0,±1)に限ることが分かる。
よって、(19+)より、u=2を得る。このとき、v=±2である。

gp>  p=lift(4*(a*z+b)^4*z)
time = 0 ms.
%154 = (20*a^4 + 48*b*a^3 + 48*b^2*a^2 + 16*b^3*a + 4*b^4)*x + (12*a^4 + 32*b*a^3 + 24*b^2*a^2 + 16*b^3*a)
gp>  p1=polcoeff(p,1,x)
time = 0 ms.
%155 = 20*a^4 + 48*b*a^3 + 48*b^2*a^2 + 16*b^3*a + 4*b^4
gp>  p0=polcoeff(p,0,x)
time = 0 ms.
%156 = 12*a^4 + 32*b*a^3 + 24*b^2*a^2 + 16*b^3*a
gp>  p1/2
time = 0 ms.
%157 = 10*a^4 + 24*b*a^3 + 24*b^2*a^2 + 8*b^3*a + 2*b^4
gp>  p1/2+p0
time = 0 ms.
%158 = 22*a^4 + 56*b*a^3 + 48*b^2*a^2 + 24*b^3*a + 2*b^4
gp>  (p1/2+p0)/2
time = 0 ms.
%159 = 11*a^4 + 28*b*a^3 + 24*b^2*a^2 + 12*b^3*a + b^4
gp>  th1=thueinit(11 + 28*b + 24*b^2 + 12*b^3 + b^4)
time = 272 ms.
%160 = [b^4 + 12*b^3 + 24*b^2 + 28*b + 11, [[;], matrix(0,5), [-0.2406059125298017237488794567 - 3.141592653589793238462643383*I, 1.061275061905035652033018916 - 12.56637061435917295385057353*I; -0.2406059125298017237488794567 - 6.283185307179586476925286766*I, -1.061275061905035652033018916 - 18.84955592153875943077586030*I; 0.4812118250596034474977589134 - 9.424777960769379715387930150*I, 1.27124401 E-74 - 33.22504032450269511288106742*I], [0.3849259339947735406604092067 + 0.E-77*I, -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.2783553960709223679498188577 + 3.141592653589793238462643383*I, -8.681380701877172561494280241 + 3.141592653589793238462643383*I, -16.30948911295629625269722680 + 3.141592653589793238462643383*I; -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.3849259339947735406604092067 + 0.E-77*I, -1.013329510184337886411080338 + 0.E-77*I, 7.237745226698362219001003501 + 6.283185307179586476925286766*I, 18.94299777426010486672027771 + 3.141592653589793238462643383*I; -0.1501903904951261035222637571 + 6.866618316202791859756644622*I, -0.1501903904951261035222637571 + 5.699752298156381094093928910*I, 0.7349741141134155184612614809 + 8.748618865266467509624187051*I, 1.443635475178810342493276740 + 7.285168107356133984984465168*I, -2.633508661303808614023050913 + 10.77242379164035642706202025*I], [[2, [1, 1, 0, 0]~, 2, 2, [1, 1, 0, 0]~], [11, [-2, 1, 0, 0]~, 1, 1, [-4, -5, -3, -1]~], [11, [0, 1, 0, 0]~, 1, 1, [0, 3, -1, -1]~], [19, [-1, 1, 0, 0]~, 1, 1, [6, -9, -3, -9]~], [19, [6, 1, 0, 0]~, 1, 1, [-4, 9, 3, -9]~]]~, [3, 2, 5, 1, 4], [b^4 + 12*b^3 + 24*b^2 + 28*b + 11, [2, 1], -400, 100, [[1, -9.838278303799419875235999640, 13.51448331982080697376256752, -98.57787136076781352275123524; 1, -0.6338576512001595175823476976, 0.1937206126785621154649534844, -0.6156241442275597982505854680; 1, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I, 0.1458980337503154553862394969 + 0.6005662120015552157733894199*I, -0.4032522475023133394990896439 - 0.3879045354686796176564585540*I], [1, 1, 2; -9.838278303799419875235999640, -0.6338576512001595175823476976, -1.527864045000420607181652662 + 2.172868967516401787912506591*I; 13.51448331982080697376256752, 0.1937206126785621154649534844, 0.2917960675006309107724789938 - 1.201132424003110431546778839*I; -98.57787136076781352275123524, -0.6156241442275597982505854680, -0.8065044950046266789981792879 + 0.7758090709373592353129171081*I], [4, -12.00000000000000000000000000, 14.00000000000000000000000000, -100.0000000000000000000000000; -12.00000000000000000000000000, 100.7213595499957939281834733, -134.6099033699941114994568627, 971.6857314549575186746530810; 14.00000000000000000000000000, -134.6099033699941114994568627, 183.4427190999915878563669467, -1332.931849429938591351478711; -100.0000000000000000000000000, 971.6857314549575186746530810, -1332.931849429938591351478711, 9718.601879714548688494086693], [4, -12, 14, -100; -12, 96, -132, 970; 14, -132, 182, -1332; -100, 970, -1332, 9718], [-10, -6, 0, -2; 0, -2, 0, 0; 0, 0, -10, -8; 0, 0, 0, -2], [-4280, -8480, -16360, -1440; -8480, -17320, -33280, -2920; -16360, -33280, -63280, -5520; -1440, -2920, -5520, -480], [200, [-80, 40, 0, 0]~]], [-9.838278303799419875235999640, -0.6338576512001595175823476976, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I], [1, b, 1/10*b^2 - 2/5*b - 1/10, 1/10*b^3 + 3/10*b - 2/5], [1, 0, 1, 4; 0, 1, 4, -3; 0, 0, 10, 0; 0, 0, 0, 10], [1, 0, 0, 0, 0, 1, 0, -8, 0, 0, -1, 12, 0, -8, 12, -91; 0, 1, 0, 0, 1, 4, -2, -8, 0, -2, 0, 13, 0, -8, 13, -104; 0, 0, 1, 0, 0, 10, -4, -21, 1, -4, -1, 34, 0, -21, 34, -269; 0, 0, 0, 1, 0, 0, 1, -12, 0, 1, -2, 17, 1, -12, 17, -126]], [[1, [], []], 0.5106981094295658352378116261, 1.043760129382343073, [2, -1], [1/10*b^3 + b^2 + 3/10*b + 3/5, 1/10*b^3 + 11/10*b^2 + 9/10*b + 1/2], 245], [[;], [], []], 0], [-9.838278303799419875235999640 + 0.E-67*I, -0.6338576512001595175823476976 + 0.E-67*I, -0.7639320225002103035908263312 + 1.086434483758200893956253295*I, -0.7639320225002103035908263312 - 1.086434483758200893956253295*I]~, [0.2406059125298017237488794567, 0.5306375309525178260165094581]~, [-0.7861513777574232860695585858 + 0.E-66*I, 2.890053638263963812457009296 + 0.E-66*I; 0.7861513777574232860695585858 + 0.E-67*I, 0.3460143392358258839521643726 + 0.E-68*I; 1.48368246 E-67 - 1.272019649514068964252422461*I, -0.6180339887498948482045868343 - 0.7861513777574232860695585858*I; 1.48368246 E-67 + 1.272019649514068964252422461*I, -0.6180339887498948482045868343 + 0.7861513777574232860695585858*I], [-2.078086921235027537601322606, -2.078086921235027537601322606; 0.4711313946287194353179767784, -0.4711313946287194353179767784], [0.7259477025765315359820075412, 1.094193414953955986267911274, 0.7134565368684148183671856481, 0.7629801282406652058906211739, 5.64610170 E-66, 9]]
gp>  thue(th1,1)
time = 27 ms.
%161 = [[1, 0], [-1, 0]]
gp>  thue(th1,-1)
time = 39 ms.
%162 = []
gp>  p=lift(4*(0*z+1)^4*z)
time = 0 ms.
%163 = 4*x

[case iii-3] i=2の場合
     u = ±(16a⁴ + 40a³b + 36a²b² + 16ab³ + 2b⁴), -------- (22)
     -u+2 = ±(20a⁴ + 48a³b + 48a²b² + 16ab³ + 4b⁴) -------- (23)
(22),(23)より、uを消去すると、
     1 = ±(18a⁴ + 44a³b + 42a²b² + 16ab³ + 3b⁴) --------- (24)
を得る。

Thue方程式(24)は、Z/5Z上で解を持たないので、整数解を持たない。

gp>  p=lift(4*(a*z+b)^4*z^2)
time = 9 ms.
%164 = (32*a^4 + 80*b*a^3 + 72*b^2*a^2 + 32*b^3*a + 4*b^4)*x + (20*a^4 + 48*b*a^3 + 48*b^2*a^2 + 16*b^3*a + 4*b^4)
gp>  p1=polcoeff(p,1,x)
time = 0 ms.
%165 = 32*a^4 + 80*b*a^3 + 72*b^2*a^2 + 32*b^3*a + 4*b^4
gp>  p0=polcoeff(p,0,x)
time = 0 ms.
%166 = 20*a^4 + 48*b*a^3 + 48*b^2*a^2 + 16*b^3*a + 4*b^4
gp>  p1/2
time = 0 ms.
%167 = 16*a^4 + 40*b*a^3 + 36*b^2*a^2 + 16*b^3*a + 2*b^4
gp>  (p1/2+p0)/2
time = 0 ms.
%168 = 18*a^4 + 44*b*a^3 + 42*b^2*a^2 + 16*b^3*a + 3*b^4
gp>  g([18,44,42,16,3],1,5)
time = 2 ms.
gp>  g([18,44,42,16,3],-1,5)
time = 2 ms.

[case iii-4] i=3の場合
     u = ±(26a⁴ + 64a³b + 60a²b² + 24ab³ + 4b⁴), -------- (25)
     -u+2 = ±(32a⁴ + 80a³b + 72a²b² + 32ab³ + 4b⁴) -------- (26)
(25),(26)より、uを消去すると、
     1 = ±(29a⁴ + 72a³b + 66a²b² + 28ab³ + 4b⁴) --------- (27)
を得る。
(13)で、a=A-B,b=B[逆変換は、A=a+b,B=b]と置換すると、A,Bは整数であり、
     1 = ±(29A⁴ - 44A³B + 24A²B² - 4AB³ - B⁴) --------- (28)
を得る。
Thue方程式(28)の整数解(A,B)をpari/gpで計算すると、(0,±1)に限ることが分かる。
a=A-B,b=Bより、Thue方程式(13)の整数解(a,b)は、±(1,-1)に限る。
よって、(11-)より、u=-2を得る。このとき、v=±2である。

gp>  p=lift(4*(a*z+b)^4*z^3)
time = 6 ms.
%169 = (52*a^4 + 128*b*a^3 + 120*b^2*a^2 + 48*b^3*a + 8*b^4)*x + (32*a^4 + 80*b*a^3 + 72*b^2*a^2 + 32*b^3*a + 4*b^4)
gp> p1=polcoeff(p,1,x)
time = 0 ms.
%170 = 52*a^4 + 128*b*a^3 + 120*b^2*a^2 + 48*b^3*a + 8*b^4
gp> p0=polcoeff(p,0,x)
time = -8 ms.
%171 = 32*a^4 + 80*b*a^3 + 72*b^2*a^2 + 32*b^3*a + 4*b^4
gp>  p1/2
time = 0 ms.
%172 = 26*a^4 + 64*b*a^3 + 60*b^2*a^2 + 24*b^3*a + 4*b^4
gp>  (p1/2+p0)/2
time = 0 ms.
%173 = 29*a^4 + 72*b*a^3 + 66*b^2*a^2 + 28*b^3*a + 4*b^4
gp>  f(a,b)=29*a^4 + 72*b*a^3 + 66*b^2*a^2 + 28*b^3*a + 4*b^4
time = 1 ms.
gp>  f(A-B,B)
time = 5 ms.
%174 = 29*A^4 - 44*B*A^3 + 24*B^2*A^2 - 4*B^3*A - B^4
gp>  th3=thueinit(-(29 - 44*B + 24*B^2 - 4*B^3 - B^4))
time = 288 ms.
%175 = [B^4 + 4*B^3 - 24*B^2 + 44*B - 29, [[;], matrix(0,5), [0.2406059125298017237488794567 - 3.141592653589793238462643383*I, 1.061275061905035652033018916 + 12.56637061435917295385057353*I; 0.2406059125298017237488794567 - 6.283185307179586476925286766*I, -1.061275061905035652033018916 + 12.56637061435917295385057353*I; -0.4812118250596034474977589134 - 3.141592653589793238462643383*I, 4.83625438 E-75 + 17.04044213293399670252122670*I], [-0.2347355434996474371381454495 + 3.141592653589793238462643383*I, 0.3849259339947735406604092067 + 0.E-77*I, 0.2783553960709223679498188577 + 3.141592653589793238462643383*I, -5.879915929435840295855481317 + 9.424777960769379715387930149*I, -11.38547421670489268299239004 + 3.141592653589793238462643383*I; 0.3849259339947735406604092067 + 0.E-77*I, -0.2347355434996474371381454495 + 3.141592653589793238462643383*I, -1.013329510184337886411080338 + 0.E-77*I, 1.549009503899409268375651096 + 0.E-75*I, 11.13171192765108061202888747 + 9.424777960769379715387930149*I; -0.1501903904951261035222637571 + 5.699752298156381094093928910*I, -0.1501903904951261035222637571 + 6.866618316202791859756644622*I, 0.7349741141134155184612614809 + 8.748618865266467509624187051*I, 4.330906425536431027479830220 + 6.331898260116669548891217569*I, 0.2537622890538120709635025675 + 6.200926367191366534459505478*I], [[2, [0, 0, 1, 1]~, 2, 2, [1, 1, 0, 0]~], [11, [-2, 1, 0, 0]~, 1, 1, [5, 1, 5, -1]~], [11, [-4, 1, 0, 0]~, 1, 1, [4, -5, 3, -1]~], [19, [-3, 1, 0, 0]~, 1, 1, [-8, 5, -6, -9]~], [19, [4, 1, 0, 0]~, 1, 1, [9, -2, 0, -9]~]]~, [3, 2, 5, 1, 4], [B^4 + 4*B^3 - 24*B^2 + 44*B - 29, [2, 1], -400, 100, [[1, -7.838278303799419875235999639, 4.676205016021387098526567882, -46.00580635324123130588382903; 1, 1.366142348799840482417652302, 0.5598629614784025978826057867, -0.3548734217566656582079076543; 1, 1.236067977499789696409173668 - 1.086434483758200893956253295*I, 0.3819660112501051517954131656 - 0.4858682717566456781828638759*I, -0.8196601125010515179541316563 - 0.04381071473395100488488192205*I], [1, 1, 2; -7.838278303799419875235999639, 1.366142348799840482417652302, 2.472135954999579392818347337 + 2.172868967516401787912506591*I; 4.676205016021387098526567882, 0.5598629614784025978826057867, 0.7639320225002103035908263312 + 0.9717365435132913563657277518*I; -46.00580635324123130588382903, -0.3548734217566656582079076543, -1.639320225002103035908263312 + 0.08762142946790200976976384411*I], [4, -4.000000000000000000000000000, 6.000000000000000000000000000, -48.00000000000000000000000000; -4.000000000000000000000000000, 68.72135954999579392818347337, -33.88854381999831757127338935, 358.1903898849802314624623248; 6.000000000000000000000000000, -33.88854381999831757127338935, 22.94427190999915878563669467, -215.9148550549911672491852940; -48.00000000000000000000000000, 358.1903898849802314624623248, -215.9148550549911672491852940, 2118.007677514901998526674929], [4, -4, 6, -48; -4, 64, -36, 358; 6, -36, 22, -216; -48, 358, -216, 2118], [-10, -2, 0, 0; 0, -2, 0, 0; 0, 0, -10, -6; 0, 0, 0, -2], [-760, 1440, -3720, -640; 1440, -2760, 6480, 1160; -3720, 6480, -14320, -2640; -640, 1160, -2640, -480], [200, [40, 40, 0, 0]~]], [-7.838278303799419875235999639, 1.366142348799840482417652302, 1.236067977499789696409173668 - 1.086434483758200893956253295*I], [1, B, 1/10*B^2 + 1/5*B + 1/10, 1/10*B^3 - 3/10*B - 1/5], [1, 0, -1, 2; 0, 1, -2, 3; 0, 0, 10, 0; 0, 0, 0, 10], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, -6; 0, 1, 0, 0, 1, -2, 0, -10, 0, 0, -1, 4, 0, -10, 4, -45; 0, 0, 1, 0, 0, 10, 2, 21, 1, 2, 3, -10, 0, 21, -10, 103; 0, 0, 0, 1, 0, 0, 1, -4, 0, 1, 0, 3, 1, -4, 3, -28]], [[1, [], []], 0.5106981094295658352378116261, 1.043760129382343073, [2, -1], [1/10*B^3 + 3/5*B^2 - 11/10*B + 7/5, 1/10*B^3 + 1/2*B^2 - 23/10*B + 23/10], 246], [[;], [], []], 0], [-7.838278303799419875235999639 + 0.E-67*I, 1.366142348799840482417652302 + 0.E-67*I, 1.236067977499789696409173668 + 1.086434483758200893956253295*I, 1.236067977499789696409173668 - 1.086434483758200893956253295*I]~, [0.2406059125298017237488794567, 0.5306375309525178260165094581]~, [-1.272019649514068964252422461 + 0.E-66*I, 2.890053638263963812457009296 + 0.E-66*I; 1.272019649514068964252422461 + 0.E-67*I, 0.3460143392358258839521643726 + 0.E-68*I; 1.85460307 E-67 + 0.7861513777574232860695585858*I, -0.6180339887498948482045868343 - 0.7861513777574232860695585858*I; 1.85460307 E-67 - 0.7861513777574232860695585858*I, -0.6180339887498948482045868343 + 0.7861513777574232860695585858*I], [2.078086921235027537601322606, 2.078086921235027537601322606; 0.4711313946287194353179767784, -0.4711313946287194353179767784], [0.7259477025765315359820075412, 1.094193414953955986267911274, 0.8418239574966185067958180081, 0.7629801282406652058906211739, 5.029457483314110202 E-66, 9]]
gp>  thue(th3,1)
time = 29 ms.
%176 = [[1, 0], [-1, 0]]
gp>  thue(th3,-1)
time = 27 ms.
%177 = []
gp> p=lift(4*(1*z-1)^4*z^3)
time = 0 ms.
%178 = 4*x - 4

[case iii]の結果をまとめると、得られた(C2)の整数解(u,v)は、
     (±2,±2)
である。

-------------------------------------------------

以上の結果より、(C2)の有理整数解(u,v)は、
     (±1,±1), (±2,±2)
に限る。

つまり、楕円曲線C₂の整点は、
     (±1,±1), (±2,±2)
の８個に限る。

従って、Lucas数列に現れる完全平方数v²は、
    1, 4
の２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.
[7]J. Mc Laughlin, "Small Prime Powers in the Fibonacci Sequence", Dec 13, 2000, p1-22.

Last Update: 2005.06.12

H.Nakao