Integer Points on A^4+B^4+C^4=D^4
[2024.11.20]A^4+B^4+C^4=D^4の整点
■Euler予想(n=4のとき)の反例、つまり、
A^4+B^4+C^4=D^4 ----------(1)
を満たす自明でない整数の組(A,B,C,D) (ただし A*B*C*D!=0かつgcd(A,B,C,D)=1)を探す。
以下では、Elkiesの論文(参考文献[1])の方法によって、(1)を満たす整数の組(A,B,C,D)を探す。
■(1)およびD!=0より、r=A/D,s=B/D.t=C/Dとすると、
r^4+s^4+t^4=1 ----------(2)
つまり、(2)を満たす有理数の組(r,s,t)を見つければ良い。
(3)で、r=x+y,s=x-yとすると、
2*(x^4+6*x^2*y^2+y^4)+t^4=1 ----------(3)
となる。
ここで、ある有理数uに対して、
r=x+y, s=x-y ----------(4)
(u^2+2)*y^2=-(3*u^2-8*u+6)*x^2-2*(u^2-2)*x-2*u ----------(5a)
±(u^2+2)*t^2=4*(u^2-2)*x^2+8*u*x+(2-u^2) ----------(5b)
を満たす有理数の組(x,y,t,r,s)が存在すれば、(r,s,t)が(2)を満たすことが分かる。
[pari/gpによる計算]
gp > Y2(u)
%1 = ((-3*u^2 + 8*u - 6)/(u^2 + 2))*x^2 + ((-2*u^2 + 4)/(u^2 + 2))*x - 2*u/(u^2 + 2)
gp > T2(u)
%2 = ((4*u^2 - 8)/(u^2 + 2))*x^2 + 8*u/(u^2 + 2)*x + ((-u^2 + 2)/(u^2 + 2))
gp > 2*(x^4+6*x^2*Y2(u)+Y2(u)^2)+T2(u)^2
%4 = 1
■方程式系(4),(5a),(5b)は、変換(u,x,y,t)→(2/u,-x,y,-t)で不変であることが分かる。
有理数u=n/m(m.nは整数,m!=0,gcd(m,n)=1)とすると、(5a),(5b)は
(2*m^2+n^2)*y^2=-(5*m-2-8*m*n+3*n^2)*x^2-2*(2*m^2-n^2)*x-2*m*n ----------(6a)
±(2*m^2+n^2)*t^2=4*(2*m^2-n^2)*x^2+8*m*n*x+(n^2-2*m^2) ----------(6b)
となる。
u→2/uの変換で方程式系(4a),(4b)の有理数解(x,y,t)の集合は不変であるので、mは0でない偶数,nは正の奇数として良い。
■正整数kに対して、S(k)=max{d ∈N: d^2|k}, R(k)=k/(S(k))^2とすると、以下の補題1,2が成立する。
[補題1] (6a)が無限個の有理数解(x,y)を持つための必要十分条件は、
R(2*m^2+n^2), R(2*m^2-4*m*n+n^2)
が両方ともに、全ての素因数pがp≡1 (mod 8)を満たすことである。
で表される楕円曲線C(e)の有理点[x,y]を求める。
[補題2] (6b)が無限個の有理数解(x,t)を持つための必要十分条件は、
R(2*m^2-2*m*n+n^2), R(2*m^2+n^2), R(2*m^2+2*m*n+n^2)
が、それぞれの全ての素因数pがp≡1 (mod 8)を満たすことである。
■max{m/4,n}<=100の範囲で、(6a),(6b)が共に、無限後の有理数解をもつようなu=2*m/nを求めると、以下のようになる。
[pari/gpによる計算]
gp > ppu(ssu(100))
time = 247 ms.
%13 = [-8/7, -16/5, 24/5, -16/15, -40/9, -40, 8/25, -16/27, -24/29, -32/35, -48/35, -16/39, 16/39, -80/39, -80/29, -80/11, 80/3, -72/41, -88/15, -88/5, -8/45, -16/45, -64/45, 96/5, 96/13, 96/25, -8/51, -24/55, 24/55, -112/39, -112/27, 112/27, -72/59, -96/59, -120/49, -120/31, -104/63, -128/63, -24/65, -64/65, -96/65, -48/67, 136/33, 32/71, -144/35, -144, 144, 144/25, 144/35, -112/73, -88/75, -128/75, -152/65, -152/45, -152/15, -24/79, -120/79, -160/63, -160/23, -160/3, 160/3, -168/53, -40/87, 40/87, -176/3, -120/89, -40/91, 40/91, -120/91, -184/73, -80/93, -160/93, -72/95, -168/95, -8/99, 8/99, -200/99, -200/59, -200/33, -160/103, -208/75, -208/43, 208/45, 8/105, -64/105, -88/105, -184/105, -192/107, -216/85, -40/111, 40/111, -224/75, -224/47, -224/3, 224/3, 224/47, 224/65, -96/113, 24/115, -72/115, -232/75, -232/21, 232/3, 232/21, -64/117, 64/117, -48/119, 48/119, -240/89, -240/67, -240/49, -240/19, 240/19, 240/49, 240/67, -160/121, -200/123, -248/105, 248/19, 248/33, -184/125, -248/125, -256/75, -256/15, 256/15, -200/131, -264/119, -264/115, -264/35, 264/35, -120/133, -136/133, 16/135, -208/135, -272/105, -72/137, -96/137, -280/117, -280/87, -280/69, -280/43, -280/27, 280/27, 280/69, -80/141, 80/141, -240/143, -256/143, -288/125, -288/55, 288/5, -24/145, -176/145, -176/147, -296/63, 296/65, 304/87, -176/153, -232/153, -24/155, -48/155, -192/155, -312/73, -312/61, -312/23, 312/23, -200/157, -104/159, -160/159, -280/159, -320/39, 320/19, 320/53, -40/161, 40/161, -216/161, -40/163, 72/163, -320/163, -328/141, -328/109, 32/165, -56/165, 56/165, -136/165, -152/165, -208/165, 48/167, -336/167, -336/115, -336/55, 336/83, -136/169, -336/169, -32/171, 80/171, -232/171, -344/93, 344/93, -280/173, 88/175, -304/175, -352/117, -352/105, -352/25, 352/25, -320/177, -360/179, -360/127, -360/83, -360/19, 360/13, 360/19, 360/29, 360/59, 360/89, -368/45, -368/3, 368/65, -232/185, 80/187, -296/187, -376/175, -200/189, -360/191, -384/115, -384/23, 384/23, 384/55, -32/195, -232/195, -256/195, -352/195, -392/165, -392/143, -392/107, -392/99, -392/75, -392/57, -392/51, -392, 392, 392/57, 392/75, 24/197, -32/197, 32/197, -40/199, 40/199, -400/153, -400/99, -400/81, -400/63, -400/51, -400/37, 400/37, 400/51, 400/63, 400/99, -408/157, -408/151, -408/143, -408/35, -408/25, -408/5, 408, 408/35, -416/105, -416/77, -416/15, 416/15, 416/77, 416/105, 416/111, -424/147, -424/105, 424/65, -432/185, -432/179, 432/55, 432/103, 432/115, 432/125, -440/151, -440/133, -440/119, -440/117, -440/81, -440/79, -440/21, 440/21, 440/51, 440/63, 440/79, 440/117, 440/119, -448/167, -448/165, -448/69, -448/33, 448/33, 448/69, 448/85, 456, 456/65, -464/165, -464/45, -464/35, 464/45, -472/163, -472/109, -472/75, 472/45, 472/75, -480/161, -480/113, -480/83, -480/47, -480/31, -480/29, -480/19, 480/7, 480/29, 480/83, 480/97, 480/113, 480/121, -488/175, -488/165, -488/85, -488/23, 488/139, -496/105, 496/105, 496/115, -504/155, -504/5, 504/5, -512/135, -512/105, -512/27, 512/25, 512/135, -520/189, -520/171, -520/147, -520/33, -520/27, 520/33, 520/51, 520/121, 520/147, -528/191, -528/95, -528/85, 528/85, 528/125, -536/181, -536/75, -536/15, -544/105, -544/15, -544, 544/105, -552/145, -552/65, 552/17, 552/65, 552/145, -560/157, -560/123, -560/69, -560/57, 560/33, 560/99, -568/147, -568/3, 568/147, -576/175, -576/121, -576/35, -584/195, -584/155, -584/135, -584/105, 584/25, 584/45, 584/105, 584/155, 592/149, -600/161, -600/143, -600/107, -600/77, -600/67, 600/13, 600/41, 600/47, 600/67, 600/77, 600/103, 600/143, 600/161, -608/105, 608/75, -616/135, -616/123, -616/39, 616/39, -624/55, 624/71, -632/21, -632/3, 632/3, -640/177, -640/159, -640/93, -640/27, -640/21, 640/21, 640/27, 640/117, 640/159, -648/131, -648/83, -648/31, -648/5, 648/5, -656/7, 656/7, 656/165, -664/165, -664/135, -664/131, -664/95, 664/135, -672/185, -672/145, -672/41, 672/41, 672/65, -680/171, -680/143, -680/93, -680/37, -680/9, 680/9, 680/81, 680/93, 680/127, 680/141, 680/171, 680/189, -688/195, -688/135, -688/123, -688/105, -688/39, -688/27, 688/105, -696/163, -696/41, -696/37, -696/13, 696/35, -704/185, -704/63, 704/63, -712/25, -720/173, -720/113, -720/43, 720/17, 720/19, 720/73, 720/173, 720/191, -728/187, 728/45, 728/187, -736/33, -736/5, 736/5, 736/33, 736/67, 736/95, -744/157, 744/157, -752/135, -752/15, 752/15, 752/117, -760/153, -760/111, -760/89, -760/63, 760/9, 760/63, 760/81, 760/93, 760/197, -768/175, -768/65, 768/35, 768/95, 768/175, -776/165, 776/171, 776/195, 784/39, 792/7, 792/25, 792/149, 792/175, -800/189, -800/159, -800/117, -800/97, -800/69, -800/11, -800/3, 800/11]
■これらのuについて、(3),(5a),(5b)を満たす有理数解(x,y,t)を持たないものもあれば、有理数解(x,y,t)を持つものもある。
これらのuを順に調べて、方程式系(5a),(5b)が有理数解(x,y,t)を持つことが確認できたものは、
u=-16/5, -96/113, -184/105, -64/117, -136/133, 752/15, -800/97
の6個である(他にもあることが予想される)。
■以下では、u=-136/133のときに、(5a),(5b)を満たす有理数解(x,y,t)を求める。
u=-136/133のとき、(4a),(4b)は
y^2=-153163/26937*x^2 + 16882/26937*x + 18088/26937 ------- (6)
±t^2=-33764/26937*x^2 - 72352/26937*x + 8441/26937 ------- (7)
(6)より、
(2993*3)^2*y~2=-458416859*x^2 + 50527826*x + 54137384 ------ (8)
よって、ratpointsによって、□=-458416859*x^2 + 50527826*x + 54137384 となる有理数xをいくつか求めると、
-4/591, -212/819, 271/858, 682/2493. -973/3838, ...
となる。
[ratpointsによる計算]
-bash-3.1$ ratpoints '487236456 454750434 -4125751731' 10000
This is ratpoints-1.5 by Michael Stoll (2001-04-23).
Please acknowledge use of the program in published work.
y^2 = - 4125751731 x^2 + 454750434 x + 487236456
max. Height = 10000
Search region:
[-10000.500000, 10000.500000]
Using speed ratios 1000.000000 and 4.000000
10 primes used for first stage of sieving,
49 primes used for both stages of sieving together.
Sieving primes:
First stage: 251, 197, 139, 137, 113, 103, 101, 89, 83, 67
Second stage: 43, 241, 229, 199, 167, 157, 149, 131, 127, 239, 223, 211, 179, 233, 227, 193, 191, 61, 181, 173, 109, 163, 151, 97, 71, 107, 31, 29, 53, 79, 23, 59, 47, 37, 19, 13, 11, 17, 7
Probabilities: Min(251) = 0.500008, Cut1(67) = 0.500111, Cut2(7) = 0.632653, Max(3) = 1.000000
Forbidden divisors of the denominator:
4, 17, 23, 29, 31, 37, 47, 59, 61, 79, 107, 127, 131, 149, 151, 157, 163, 167, 173, 181, 191, 193, 199, 227, 229, 233, 241
(-4 : 591)
(-212 : 819)
(271 : 858)
(682 : 2493)
(-973 : 3838)
(154 : 3929)
(-1253 : 4290)
(475 : 4986)
(1451 : 5590)
(2651 : 6706)
(3131 : 8190)
(-1438 : 9801)
82376 candidates survived the first stage,
12 candidates survived the second stage.
12 rational point pairs found.
例えば、(x,y)=(-4/591,1448/1773)は(5a)の有理点なので、任意の有理数kに対して、直線
y-1448/1773=k*(x+4/591)
つまり、
y=k*(x+4/591)+1448/1773 ------------- (9)
と２次曲線(6)の交点は高々2個の有理点であり、(-4/591,1448/1773)と異なる交点(x(k),y(k))は、以下のようになる。
x(k)=(107748*k^2 + 26003184*k - 10589914)/(-15919767*k^2 - 90519333) ------- (10)
y(k)=(39004776*k^2 - 33607698*k - 221780024)/(-47759301*k^2 - 271557999) ----- - (11)
さらに、(10)を(7)に代入して、
±t^2=(-756095391747225*k^4 - 9943858871147520*k^3 + 3313508734978218*k^2 - 63112850850203904*k + 1329472461417583)/(-2280950832008601*k^4 - 25938840426397398*k^2 - 73743746820884001) -------- (12)
を得る。
両辺に適当な有理数の平方数を掛けて、
±t^2*(26937*k^2 + 153163)^2*1773^2=756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583
つまり、
±□=756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583
を満たす有理数kを求めれば良い。
[pari/gpによる計算]
gp > T2(u0)
%3 = -33764/26937*x^2 - 72352/26937*x + 8441/26937
gp > h2(26937)
%4 = [2993, 3]
gp > Y2(u0)*(2993*3)^2
%5 = -458416859*x^2 + 50527826*x + 54137384
gp > gg(u0)
ratpoints '487236456 454750434 -4125751731' 10000
ratpoints '227375217 -1948945824 -909500868' 10000
ratpoints '-227375217 1948945824 909500868' 10000
%6 = [-4125751731*x^2 + 454750434*x + 487236456, -909500868*x^2 - 1948945824*x + 227375217]
gp > a=YY(u0,-4/591)
%7 = 1448/1773
gp > xx(u0,-4/591,a)
%8 = (107748*k^2 + 26003184*k - 10589914)/(-15919767*k^2 - 90519333)
gp > xk(k)=(107748*k^2 + 26003184*k - 10589914)/(-15919767*k^2 - 90519333)
%9 = (k)->(107748*k^2+26003184*k-10589914)/(-15919767*k^2-90519333)
gp > factor(xk(k))
%10 =
[ 26937*k^2 + 153163 -1]
[53874*k^2 + 13001592*k - 5294957 1]
gp > k*(xk(k)+4/591)+1448/1773
%11 = (39004776*k^2 - 33607698*k - 221780024)/(-47759301*k^2 - 271557999)
gp > TT2(u0,xk(k))
%12 = (-756095391747225*k^4 - 9943858871147520*k^3 + 3313508734978218*k^2 - 63112850850203904*k + 1329472461417583)/(-2280950832008601*k^4 - 25938840426397398*k^2 - 73743746820884001)
gp > TT2(u0,xk(k))*(26937*k^2 + 153163)^2
%13 = 9334511009225/38809*k^4 + 368291069301760/116427*k^3 - 122722545739934/116427*k^2 + 21037616950067968/1047843*k - 1329472461417583/3143529
gp > h2(3143529)
%14 = [1, 1773]
gp > TT2(u0,xk(k))*(26937*k^2 + 153163)^2*1773^2
%15 = 756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583
gp > factor(%15)
%16 =
[756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583 1]
■楕円曲線E+またはE-の有理点(k,w)をMAGMAを使って求める。ここで、
E+: w^2=756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583
E-: w^2=-(756095391747225*k^4 + 9943858871147520*k^3 - 3313508734978218*k^2 + 63112850850203904*k - 1329472461417583)
である。
楕円曲線E+の有理点はMAGMA 4-descentに失敗するので、見つからなかった。
[MAGMA 4-descemntによる計算]
> SetClassGroupBounds("GRH");
> P := PolynomialRing(Rationals());
> C := HyperellipticCurve((756095391747225*k^4 + 9943858871147520*k^3 - 331350\
8734978218*k^2 + 63112850850203904*k - 1329472461417583));
> fd := FourDescent(C : RemoveTorsion);
> #fd;
0
楕円曲線E-の有理点はMAGMA 4-descentを使って、見つかった。
[MAGMA 4-descentによる計算]
> function RP4(fd,M)
function> T0:=Realtime();
function> for J:=1 to #fd do
function|for> FD:=fd[J];
function|for> printf "J="; J;
function|for> pts:=PointsQI(FD,M);
function|for> F,m:=AssociatedEllipticCurve(FD); F;
function|for> for K:=1 to #pts do
function|for|for> P:=m(pts[K]); P; printf "height "; Height(P);
function|for|for> IsPoint(F,P[1]);
function|for|for> end for; //K
function|for> end for; //J
function> T1:=Realtime(T0);
function> printf "realtime="; T1;
function> return #fd;
function> end function;
>
> SetClassGroupBounds("GRH");
> P := PolynomialRing(Rationals());
> C := HyperellipticCurve(-(756095391747225*k^4 + 9943858871147520*k^3 - 33135\
08734978218*k^2 + 63112850850203904*k - 1329472461417583));
> fd := FourDescent(C : RemoveTorsion);
> #fd;
32
> RP4(fd,10^10);
J=1
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=2
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=3
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=4
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=5
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=6
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=7
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=8
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=9
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=10
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=11
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=12
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=13
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=14
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=15
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=16
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=17
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=18
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=19
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=20
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=21
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=22
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=23
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
(8260877400000179849849/6302308993600 : -1190969441972391858917789983730643/158\
21568589893184000 : 1)
height 39.3965170140096596689036629566
true (8260877400000179849849/6302308993600 :
1190969441972391858917789983730643/15821568589893184000 : 1)
(8260877400000179849849/6302308993600 : -1190969441972391858917789983730643/158\
21568589893184000 : 1)
height 39.3965170140096596689036629566
true (8260877400000179849849/6302308993600 :
1190969441972391858917789983730643/15821568589893184000 : 1)
(533623704685573455739882286646972692742064637617047590586281/11599413197260540\
61744311607582851710260521101434944 : 45538938693047332040103080728905584162578\
7091519889509891213024141692487015602095997442699/39505178539691178233772393767\
112171314592204905355424784157883591755864048128 : 1)
height 127.253344089485755159750160533
true (533623704685573455739882286646972692742064637617047590586281/115994131972\
6054061744311607582851710260521101434944 :
4553893869304733204010308072890558416257870915198895098912130241416924870156020\
95997442699/3950517853969117823377239376711217131459220490535542478415788359175\
5864048128 : 1)
(533623704685573455739882286646972692742064637617047590586281/11599413197260540\
61744311607582851710260521101434944 : 45538938693047332040103080728905584162578\
7091519889509891213024141692487015602095997442699/39505178539691178233772393767\
112171314592204905355424784157883591755864048128 : 1)
height 127.253344089485755159750160533
true (533623704685573455739882286646972692742064637617047590586281/115994131972\
6054061744311607582851710260521101434944 :
4553893869304733204010308072890558416257870915198895098912130241416924870156020\
95997442699/3950517853969117823377239376711217131459220490535542478415788359175\
5864048128 : 1)
J=24
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=25
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=26
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=27
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=28
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=29
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
(6176859732099143695357148145088807157111914912587541241/3159529783121008593400\
529004994269402288984064 : 2059263728185733327290328139163631980242224265861839\
9325183997885647364273695632781/17759620464120442856372034466207127675752817982\
2216433517394479808512 : 1)
height 114.867773549861329024805353459
true (6176859732099143695357148145088807157111914912587541241/31595297831210085\
93400529004994269402288984064 : 20592637281857333272903281391636319802422242658\
618399325183997885647364273695632781/177596204641204428563720344662071276757528\
179822216433517394479808512 : 1)
(1051502703981043542278921/30140451361024 :
1079975500089924718756938074614354971/165472042466465312768 : 1)
height 43.5250405272178017138852653147
true (1051502703981043542278921/30140451361024 :
1079975500089924718756938074614354971/165472042466465312768 : 1)
(6176859732099143695357148145088807157111914912587541241/3159529783121008593400\
529004994269402288984064 : 2059263728185733327290328139163631980242224265861839\
9325183997885647364273695632781/17759620464120442856372034466207127675752817982\
2216433517394479808512 : 1)
height 114.867773549861329024805353459
true (6176859732099143695357148145088807157111914912587541241/31595297831210085\
93400529004994269402288984064 : 20592637281857333272903281391636319802422242658\
618399325183997885647364273695632781/177596204641204428563720344662071276757528\
179822216433517394479808512 : 1)
(1051502703981043542278921/30140451361024 :
1079975500089924718756938074614354971/165472042466465312768 : 1)
height 43.5250405272178017138852653147
true (1051502703981043542278921/30140451361024 :
1079975500089924718756938074614354971/165472042466465312768 : 1)
J=30
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=31
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
J=32
Elliptic Curve defined by y^2 = x^3 - x^2 + 3971619223592226604*x -
1791602749600376016515086080 over Rational Field
realtime=51718.714
32
>
楕円曲線E2: y^2 = x^3 - x^2+3971619223592226604*x-1791602749600376016515086080の有理点
Q1(8260877400000179849849/6302308993600, 1190969441972391858917789983730643/15821568589893184000)
height(Q1)=39.396517014009659668903662956579790265
Q2(1051502703981043542278921/30140451361024, 1079975500089924718756938074614354971/165472042466465312768)
height(Q2)=43.525040527217801713885265314659623039
は１次独立であるので、rank(E2)は2以上である。
これより、楕円曲線E0: y^2=x^3+50863653035430650648132587239190032*x-2596579570917378368973512525646015766366106324875392の有理点を求めると、
P1(233714768991672015003916550889/1575577248400, 179221698729508418377839623024366696263826037/197769607373664800)
height(P1)=39.396517014009659668903662956579790265
P2(29748863191601190279969223028793/7535112840256, 162518900058271624795006804896461339908243018989/20684005308308164096)
height(P2)=43.525040527217801713885265314659623039
となる。
楕円曲線E0の有理点P1,P2から、楕円曲線E-の有理点P(x,y)をいくつか求めると、そのx座標kは以下のようになる。
-33911/44895,
-528613/41475,
-8461/571461,
-1764801439/390505689,
-7120040532103/1277846701103,
-18462026010773/33730649253075,
-71338389553919/42257187855015,
-13829171930102373919/5909680725265070169,
-144299282565611338439299750477/12929206500636822398629544645,
-6367375348949905801835214005701/2198213930842716226274514448701,
-289427327430500418566364978685741/28031392606194750943890326256741,
-50857619740851149060224133043757/441945990273105933948748068815355,
...
[pari/gpによる計算]
gp > v=-v
%72 = [-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
gp > e0=E0(v)
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
%73 = [0, 0, 0, 50863653035430650648132587239190032, -2596579570917378368973512525646015766366106324875392, 0, 101727306070861301296265174478380064, -10386318283669513475894050102584063065464425299501568, -2587111200108673641517013036002797701622059255927877506757439408161024, -2441455345700671231110364187481121536, 2243444749272614910793114822158157622140315864692338688, -11334396694791432938128847004105111102854085759061688459412582125435987411068857311294216892497510400000000, 27063661893511868646330032528630353036510581794248833419696/21078447061438971412035064201047596877693981451233984375, Vecsmall([1]), [Vecsmall([128, -1])], [0, 0, 0, 0, 0, 0, 0, 0]]
gp > e2=E2(v)
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
rr=[10638, -37722348, 0, 0]
%74 = [0, -1, 0, 3971619223592226604, -1791602749600376016515086080, -4, 7943238447184453208, -7166410998401504066060344320, -15773759250040909860586045668425028496, -190637722732426876976, 1547944774510898541874473111232, -5396082447728376681480976435468184800689659251515900000000, 27063661893511868646330032528630353036510581794248833419696/21078447061438971412035064201047596877693981451233984375, Vecsmall([1]), [Vecsmall([128, -1])], [0, 0, 0, 0, 0, [1819287834348575465767591440, 512, [2, 4; 3, 1; 5, 1; 7, 1; 41, 1; 73, 1; 1201, 1; 1801, 1; 8849, 1; 63113, 1; 299513, 1], [[4, -1, 0, 2], [1, 10, 0, 2], [1, 12, 0, 2], [1, 5, 0, 1], [1, 6, 0, 2], [1, 6, 0, 2], [1, 5, 0, 1], [1, 8, 0, 4], [1, 6, 0, 2], [1, 6, 0, 2], [1, 5, 0, 1]]], 0, [[2]~]]]
gp > Q1=QQ(e2,8260877400000179849849/6302308993600)
%75 = [8260877400000179849849/6302308993600, 1190969441972391858917789983730643/15821568589893184000]
gp > Q2=QQ(e2,1051502703981043542278921/30140451361024)
%76 = [1051502703981043542278921/30140451361024, 1079975500089924718756938074614354971/165472042466465312768]
gp > matdet(ellheightmatrix(e2,[Q1,Q2]))
time = 1 ms.
%77 = 1249.8654473108438081171110057696349606
gp > Q3=QQ(e2,6176859732099143695357148145088807157111914912587541241/3159529783121008593400529004994269402288984064)
%78 = [6176859732099143695357148145088807157111914912587541241/3159529783121008593400529004994269402288984064, 20592637281857333272903281391636319802422242658618399325183997885647364273695632781/177596204641204428563720344662071276757528179822216433517394479808512]
gp > matdet(ellheightmatrix(e2,[Q1,Q2,Q3]))
time = 1 ms.
%79 = -9.402942544651957338 E-34
gp > Q4=QQ(e2,533623704685573455739882286646972692742064637617047590586281/1159941319726054061744311607582851710260521101434944)
%80 = [533623704685573455739882286646972692742064637617047590586281/1159941319726054061744311607582851710260521101434944, 455389386930473320401030807289055841625787091519889509891213024141692487015602095997442699/39505178539691178233772393767112171314592204905355424784157883591755864048128]
gp > matdet(ellheightmatrix(e2,[Q1,Q2,Q4]))
time = 1 ms.
%81 = -9.402942544651957338 E-34
gp > matdet(ellheightmatrix(e2,[Q1,Q2]))
%82 = 1249.8654473108438081171110057696349606
gp > P1=ellchangepointinv(Q1,[10638, -37722348, 0, 0])
%83 = [233714768991672015003916550889/1575577248400, 179221698729508418377839623024366696263826037/1977696073736648000]
gp > P2=ellchangepointinv(Q2,[10638, -37722348, 0, 0])
%84 = [29748863191601190279969223028793/7535112840256, 162518900058271624795006804896461339908243018989/20684005308308164096]
gp > elltors(e0)
%85 = [2, [2], [[48769295475937284, 0]]]
gp > T0=[48769295475937284, 0]
%86 = [48769295475937284, 0]
gp > L1=cc2(10,v,P1,P2);
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
time = 718 ms.
gp > length(L1)
%88 = 230
gp > L2=cc2t(10,v,P1,P2,T0);
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
time = 530 ms.
gp > length(L2)
%90 = 0
gp > L1=cc2(20,v,P1,P2);
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
time = 18,159 ms.
gp > length(L1)
%92 = 860
gp > L1=cc2(30,v,P1,P2);
v=[-756095391747225, -9943858871147520, 3313508734978218, -63112850850203904, 1329472461417583]
I=-1883839001312246320301206934784816
J=96169613737680680332352315764667250606152086106496
time = 2min, 12,127 ms.
gp > length(L1)
%94 = 1890
gp > L10=hsort(L1,10^500);
time = 18 ms.
gp > length(L10)
%102 = 207
gp > L10[1..30]
%103 = [-33911/44895, -528613/41475, -8461/571461, -1764801439/390505689, -7120040532103/1277846701103, -18462026010773/33730649253075, -71338389553919/42257187855015, -13829171930102373919/5909680725265070169, -144299282565611338439299750477/12929206500636822398629544645, -6367375348949905801835214005701/2198213930842716226274514448701, -289427327430500418566364978685741/28031392606194750943890326256741, -50857619740851149060224133043757/441945990273105933948748068815355, -953872343045322853243312366953111/715163261658934278969354351190625, -2246986664829060726040148760020231/29673206229232970184322157560226481, -8692109990438974967719362198755591/39456951033990775076748759839457135, -299413176643300877397136041395566333/34868224257745759888645436328095083, -29089813350866365793906681953016145251119/2117156214297585441149869182896405146455, -82234532747137738385542844575173492343469/270230843354622576214720237819632489482469, 18111647850719132066415845915535998018921/1099424954708204288220280970620553767015329, -3962771320058204922220664484172732093026017813/524430309045042527890209864272260630710810195, -405554291438259330166085341035994354552609869853555781/110660828796349064264427297512752549224817560411758781, -35081950231385307135502095326423249142820538518966790317/16619936836506512248599713865465879600345797952832261755, -83015448003147487499515446468611722493411441962192871189/44172978684849860757999072782173576437811937128845330189, -1199829996053041216784445712132050624323973004048223950957/1196486101569998269389437150786797776523047683704546086075, -48135627465571034658495485036176639592430585706894037250799/7824784253231353661895504488548706417738636990260580161815, -58063359279611664317387239031946358679977679527923392465905437/125901225791998705768185874666404488031866721093962326151194187, -957638725638877183607181975540121488527979496387505361327126871/288675492927039076941556516327177641054439184916379904164054335, -45030859155500597064571092348488435312246634934815765597595423871/39650225693017316102949294453361448329866944414416677098520080121, -427924590060929211275517284891399643547577623031991858670784212678566226508855359/1142090533290770058787676235992350940965506694499784049000956349246998109450455015, -25690442880130936360212799480893372046647158295224871000299292327466244415311659951/3745764233670390277345601643654661344692296057019019950093346785804345249551056201]
■これらの有理数kを(10),(11),(12)に代入すると、方程式系(5a),(5b)の有理数解x,y,tが求まる。
(4)より、(2)の有理数解(r,s,t)が求まる。
(2)の両辺に適当な整数を掛けることにより、(1)の整数解(A,B,C,D)をいくつか求めつことができる。A,B,C,Dを正整数として良い。
ここで、0 <= A <=B <= C <= Dを満たすように、A,B,Cを交換して、Dの小さい順に(1)の等式を並べ替えると、以下のようになる。
■(2)に(10),(11),(12)を代入して、変数kをxに置き換えると、以下の整数係数多項式の等式を得る。
(-38681532*x^2+111617250*x+190010282)^4+(39328020*x^2+44401854*x-253549766)^4+(756095391747225*x^4+9943858871147520*x^3-3313508734978218*x^2+63112850850203904*x-1329472461417583)^2=(-47759301*x^2-271557999)^4 -------(13)
ここで、楕円曲線
E(-136/133): y^2=-(756095391747225*x^4 + 9943858871147520*x^3 - 3313508734978218*x^2 + 63112850850203904*x - 1329472461417583)
の有理点P(x,y)を求めることができれば、そのx座標を(13)に代入することにより、A^4+B^4+C^4=D^4の整数解が得られる。
■同様に、他の有理数u=-16/5, -96/113, -184/105, -64/117についても、以下の整数係数多項式の等式を得ることができた。
各楕円曲線は有理点を持つので、有理点のx座標から、A~4+B^4+C^4=D^4の整数解を得ることができる。
- u=-16/5のとき
(765*x^2 + 11184*x - 8221)^4+(612*x^2 - 11490*x - 7442)^4+16*(-2574990*x^4 + 115209*x^3 - 17668620*x^2 + 2922009*x + 63237532)^2=(3213*x^2 + 16359)^4
楕円曲線 E(-16/5): y^2=-2574990*x^4 + 115209*x^3 - 17668620*x^2 + 2922009*x + 63237532
2682440^4+15365639^4+18796760^4=20615673^4
186668000^4+260052385^4+582665296^4=589845921^4
219076465^4+275156240^4+630662624^4=638523249^4
664793200^4+2448718655^4+3134081336^4=3393603777^4
502038853976^4+2480452675600^4+4987588419655^4=5062297699257^4
3579087147375440^4+14890026433468471^4+18565945114216720^4=20249506709579721^4
8813425670440240^4+47886740272114976^4+56827813308111785^4=62940516903410601^4
38751631463616255521^4+46196947347028916440^4+107238802094189542120^4=108593344076382641697^4
5967420362778572362681840^4+11389900458885552539102735^4+19270755733101284410120384^4=19874054816411213708481009^4
41328162329293632574512440^4+74522041242387759937530799^4+129410861225043592041256520^4=133140691304639620846181457^4
169218021322170204480680305^4+1288056982586427591062203384^4+1507524066882038472584786800^4=1677479490238223823661446513^4
8069533957326324451238272976^4+8825302955667506173409411815^4+17155429148630710395388779200^4=17644444348539480178025528601^4
52942590681106640544280040360^4+249130710645573788837883103720^4+303237986735307793676413527001^4=333129191568549251867199876057^4
66039836886902596560454161520^4+379334234716115837893323648560^4+760582529580318898681536071471^4=772094261800702773645712832721^4
14274433969472603571500233839424^4+93833840730275457894522336118855^4+112194490587542543850845554026640^4=123946497886751603284917774006201^4
27240579159414963163833435822329^4+248781160855666365777930155696720^4+284600416391998689341341493583440^4=319280388704872568697808021788921^4
175424290963444842846031401841759^4+679990789464791197416063781213640^4+1376543993038340844176274474873560^4=1396680473961341619647008005444513^4
31058706146444468337573470216746671304^4+45723546709411652690271893833764888905^4+101347936969652305104608658289319870960^4=102597776485948701261490206779151828297^4
1046738335539830238434807071109719218424^4+1381599778333161751111622711637871247945^4+3131108479640288781452874540720691424240^4=3169834808503132719068553468475898078793^4
1037354240745377010703026223898481485504^4+1739791427884216975168428630991483380185^4+3128294165330114744727797686810591297040^4=3209362476706814620541224955667144860121^4
464024162889108246965771239428843936418960^4+2351269049425087935841789662825005390318455^4+2942894144172212096782850563589047020019936^4=3205776133114443285786355494410044071015561^4
10301089588007086166857838935541176607862672400^4+16701371834728694354127181708107566951768007656^4+30542792711349255243090311315337919056425054695^4=31295958743027261374800168000606313319708062233^4
56848660621859333145298924209778474490790862960^4+57537351751926091192521443554403022567670623695^4+116155990279165125223764999207442763432438641384^4=119429377346393956325881294067788995970216498737^4
...
- u=-95/113のとき
(-3093106*x^2 + 40474650*x + 7552656)^4+(6916046*x^2 - 20456346*x - 47468592)^4+(88701693774137*x^4 + 179587087819776*x^3 + 240380808658614*x^2 + 2132889183387648*x - 2068027156905327)^2=(-10061283*x^2 - 55307817)^4
楕円曲線 E(-95/113): y^2=88701693774137*x^4 + 179587087819776*x^3 + 240380808658614*x^2 + 2132889183387648*x - 2068027156905327
1809372297612384971225^4+2064212400686193381720^4+2076053953455407507832^4=2624574574278836761049^4
212221446638922320619625747320^4+248727846969321093756176028288^4+253491817712488283032983319985^4=316108046580586691194170309041^4
1841986851833863637127588778913^4+2058103859391620832225294037080^4+2129164668675024740160566040960^4=2659214384653731859473704802209^4
29078274618068115351387455624821602600^4+156918444932542019867985838782320361295^4+237691119090729820468665272460435043968^4=248265151574081866714040701170722358449^4
2410618091093688819988876382159461878643752^4+34070710958420771416270817319870503451391560^4+64359558289160776669400535229561015175148295^4=65587636843593652805329955352041063767872761^4
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885247296899104370743913414404096629805713977011197929705^4+1167470211067621438573854199078729562238653761689552572760^4+1332530172515393071716062217880037889110777556037092981048^4=1540017732907487720557021038584775261174488579215319494889^4
5083869287493935383449336992528640709518044599644877640095^4+23126600429316662933542540959852842906101959960460592736840^4+34842549399059828307775771804818840103001676290404809191208^4=36425923399060563823975962857530549017281796545228025820769^4
1014813496510646689358729137454825472485381202081010517593318622280^4+2637725437530271482565211005487183015571677934154093673639856168223^4+3815275825578777751335993776901753954447175222905408281881918880640^4=4020757590678649469497930430058494241016165510486433463394944820961^4
39152308198323726963935739763379871164733198818152171716666825295^4+6857034385456255978939299764367172825916650698380527179282066627432^4+10521592669878021062117975066576441523615987923743162417499608793160^4=10967007213528679662308570034732342606394451281648540793528204963249^4
1151125269807508663513807251006210606315454281373174380558184225271245664089080^4+38025741317036672785042112695055833991829669857492145935476535594135549855054143^4+58328496092745481730926843514436757393166793009418914143491201795503684878355840^4=60800781414619069911392399465265002823503645237351673288477463732012582988958401^4
10177548493344689855304235337413546100877987375312383778618552401826291793982592^4+75381882024379247160695878608520840669032304067950497802882583216756007396834535^4+134896638507103594978443740451359333036471537256988935644704117144546334947658280^4=138072367193812630154711361703285259408517870170577357369164954892236778052953881^4
...
- u=-184/105のとき
(-65242302*x^2 + 205510490*x + 413771372)^4+(81902290*x^2 + 88778694*x - 434464324)^4+(432617336765985*x^4 + 38356656367961344*x^3 + 8062466049405030*x^2 + 206603328950201344*x - 70451814984618887)^2=(89198023*x^2 + 514194549)^4
楕円曲線 E(-164/105): y^2=432617336765985*x^4 + 38356656367961344*x^3 + 8062466049405030*x^2 + 206603328950201344*x - 70451814984618887
8428029040133597878014994881016481495^4+30844087392158518233066378628867203000^4+50959085080541127211225264739779330824^4=52597738518268114502668246372803215657^4
496105156458016145849469689975353074801^4+1072580163735569048387870467616930443560^4+1410475105777829777510125492390667750240^4=1520283515690949145570133876920610580593^4
396415520963823574069525431920468445294905^4+786986053669731843497495429254978647805160^4+941353475180821259453013079076215345120312^4=1045223861509219185829610470498369350170937^4
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654913039713792679000878724387789442639324188320503048^4+908817999572155789057773873279142680513451282605194465^4+1261104431130396708990191888394488515665932758047613120^4=1357454627376644142063980511644218143776116932911972577^4
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396142040109433215154415820529206530330477597007918207943207199181749182807870851520^4+710748265553559009505343236494327124880114567853552145177669884165348672654468679697^4+968886977582482218700412575466215133936599670563978113303483541505253086495536123480^4=1038037381462651265302121345194758792649119944440544952543760361118026323757256641553^4
16191486463157984493556603696020585750769696323308879434488508707966164682420354855945640^4+218300274740482473036335352520977853936992598746373982347603601197207642615385707298795295^4+269486069645817749293941228652489401898503813398871580320275806975098030123211806820617264^4=294724812942723615334829489746710377417665828607201705271905967721032060862641532181992673^4
180300068721467653289182695300511902813405114288137248269093399328819700264706140516302280691040^4+287095939454387764695101610895952925169562117029585162609377217492688120264337708752561987773809^4+374714163601336269393721196639200673004419367246742686483594069156688878408776140236593070093720^4=407466723076324504462446448899776151102400934061661575101744576425591755459519612906502388640113^4
...
- u=-64/117のとき
(-708165*x^2 + 70440496*x - 24768731)^4+(7616708*x^2 - 53790750*x - 65587958)^4+ 16*(89935248322350*x^4 + 33070932611785*x^3 + 583933936927068*x^2 + 927183285571609*x - 1901784280370312)^2=(-19088981*x^2 - 93598719)^4
楕円曲線 E(-64/117): y^2=-(89935248322350*x^4 + 33070932611785*x^3 + 583933936927068*x^2 + 927183285571609*x - 1901784280370312)
795982072092256232620732814048^4+1313272433331994944019605734745^4+2048922910929014528766490488600^4=2140692291273266871272786028889^4
488651333354248804387699812729543^4+2898150661569318012594653410666840^4+3763674383631279656295881792982240^4=4058317807600886242386280247206201^4
55592434990965403952467546519339118870672160^4+107584196447592312203600076400120718889023960^4+195814457311356847249907285801580736290807897^4=200427560891069506796750606026430459746092121^4
155264563538576725829619843501018854148454541868863640^4+480594478289963813159886307937159688664036643087920135^4+1097265008892750174525734952567934939198970429665114848^4=1107330952754886698971866986834601583386034575500343289^4
111596812628305600750508871422053860693476366902498123346425477903322676840^4+1052803295902991454813648638143142722538856259563344671706049222219997676832^4+2352291885407255047582143281394915918558491052381721438903203403171776900665^4=2375544718922721267516620265402335797602783647230064226451068941039505429049^4
10786221755953392499325563154122435784615348387133555667449373757793016187898471660063342183^4+14338952965510873865029732427040902551543834311844044238122922574703380592333927919061717800^4+31876899259695099263382830685145733596776674898767940527662884545497696172275628277601770720^4=32299176700552564707481773214254737959820971251470421052314145701193317078213147061914166681^4
612853487940237177964943427383347725527655598954682238187321699833877820748453759025647602191769454556321509777839316706080^4+1868858653066342214956078073304554648544697553777569570838622336833685840086134455246774146847645702944689677202000744063400^4+2850814954584949602318326586887848905579599736819004834881907389550842410692038630693906574077199983754162687955728821631737^4=2975536189027491519979552732528021243622421663762228161627459903994762418182002025280311240223307497485124863220779192213241^4
...
- u=752/15のとき
(9757046960*x^2 + 7519367358*x - 7454262002)^4+(1358289600*x^2 - 14711305762*x - 24119434078)^4+(-39159013139399690271*x^4 + 354843087677126656000*x^3 + 156153692910200849094*x^2 + 688695624015988023040*x - 590408355413958103463)^2=(10146706289*x^2 + 28822251027)^4
楕円曲線 E(752/15): y^2=-39159013139399690271*x^4 + 354843087677126656000*x^3 + 156153692910200849094*x^2 + 688695624015988023040*x - 590408355413958103463
114862632623427728870024826165344657847113178256^4+177017292014028386678857881623193380518849269735^4+830360780713533802860637029331340002295867415600^4=830865076481620806281128937452391730363875897369^4
16579089914696428063210037221564337382228209740797135352627384864846107916186325315082503543035382037923553797995092678327782367119692154798855248215507684498885161789097175544272147566793626438141357137867194146949804183342774102520328650656867579497324109499485349108092687950101900526433480336609852667300910115717720269072036332300137196798542452384662967338643469511081013201105901104868673619313909581103695752604742233312247480621234750407945^4+34555903302907218523013846079457363014359157177352112703952252429210711093524409960925155086864479358721373468739138393913211035922380730551633693624516204288050309029977133962552874171891350988981520209070842412320211719197966479174394587176670041705354586561223973247930521062434584875210031212793508020597036672553715076042640046775932146863196536876738688199071997667997133932396957436376044339307328119520916309741081238922541770843287267573872^4+53987867925998155568036932969640105607875853568507274040677942747988633650380234120148645348539003359304983780320733182907430502798352943000868201145989977275236803348560426363283680884905373875611376682021699784681380667029996816377681946979663502495780306171311491790306253734955951985118019569065622149787177415448554830932513158718014627476336165394878440417584699507507338573277215783842063679279706078174309494949919432647713882935008937151600^4=56229729332652522086967774006460514705505625942740374713210225318122749350045056651778076025572451691557718309451072737434485464693707339432624914509785651842596259102523998384946477456488439502143369417709927345451027953211929672510547194467353895290955524851707315748889165358253849803180208855272493551112617388391023104190237893744687032374770443130626257139868806383848520713605665176768613831300278215964680436548995660206911430579364545119497^4
58752883416686702572511263192243409196947882199227976822638066507478171255021201002631977347875250272152231331381601661626222785066638219098535783489280809850781304388886841070754399867014339208295022003827273361440372910686900857733818582168265401343409936929411512165596968013568423161669684289350075736438746734940805419919265661737570695363527582696313021948059003889332035300515935542082256913521420668401282667984416067002091905462894035586373374826830108402800768050176231632836353338186549253288272230897171456924777175852019574121809484580871103463187706966831631271201858514130966125981339997849930904311611664616217443179400954856749422643600267547141815575600249921663847424059392298492983347418506048454080936222613625646144893058026930607242309644553384703918762479094041254920241293427416940002888542117939535715686115731927504769155915376664535851557397326850335917575743736933614058535613612186087968036550843933175322994915089335072662477882422835158852980876067398780693280609776243439315641936846681757002577905543137176879227588150820380144585467457180171777031447378363078695022924490309374794998173578527712504554460717857054409680933066377840971991633051761924249682331406492327024481412916575201364523485904075619717310150032^4+205356652573940666691341642527845500959455017779355116255574826789847620483762052480541713752777119081488277031109443967966823748798591424602853155602235286711522707652945531351783041228813843761418052058398554318572030849514399694294852754409239516208567995896216599056933420215915187316073145787939947224194827180723184983459466926547892764734635838649228174488528838877278205442040536953208258858066975091184546486371174961920622601016660125869540953362499639144078511582524966903917429821850085898850745735146052677161017521832827461851033300633416075611578171401034431492427596487516858841942465301345102271818397982363581491621775089372926078082727095708309445127019546933443840582715153376851184737804736191068666392883787746027868203625834762643932928004440315945288481634770449830133128011617917153591381324245640655366041228971759102292955662046808288775843298558587802554603883270677638377975510431016124896270017800546669782388637648027020516674522565386434561153469345832798235841135816333877746436880264341713723994755705713245530630210667091028943583847556885067964813500124995024005390647467851162620206751965336629130084465364664221831952534374390911680966839117111485283006482814684487016228668101366082980223589642808546553166166000^4+235355327834246339703375988184554216659161275506504133799612383454884787064953560220210465601846210246203431664981652098713976583270415975692179841226913664194494000531952971845057589139860709880305989888804053775283810525568358018795518281530782655041417810380856270295176730645561099128376871973852946955258042156816469711851787400490775503503155979619499061746425376842392703490294366282172639693437486595003582513113843690578906856694568779624813469203357410856671133868170135361441824450295594531875040916730416491399049704878207620938371588305406539699259855230539497858104680972401281312755430138894222116758820210460921554398135753824069753454570632428467635420713292217902738505799956957916234412229704993690442403099572161585227154204012181357468172446095204794453052916931759969655485095627235174989619952222116127570650086716409807842049453818783356109460448845276266626526572241424314437670732613089111251107353751646963745529802473074973136982764046473634646420933931733762758695859329239156307477419482482604639536958897601723242155644223390659973781391274562978029143309233638819634216043348812428884705195392050755407593616814278470141316122562086813205996575605679431260197986800224817602255215843607545208376798856055833036882806295^4=264014869497329966804697634169454802601597994528057433680501197739339888983149421533738361005040963787395961673174694086179140671022539065391798093073224365413859446863872212330828815915394798104347079681299075670571566738390490660673150971320187239364210612414543385642614960115292869282674679270932972922360561771656450348584366028626763170847046198168767447611408047475991512591553312339732378851121901213073331572881438802590527655827152799578998309593174197981336056992352438566405656038902968049774790070191703636324689687363681847128343095684073567834489365906157247555423090243445056080536999228960026567581828163414549920699393510552759996066156236253331971064052408730874326782573235027884256180453794922523916020830957765332390599521887855212447743896170139940525952991194091873480756563591254346675605532913116469603178178308572891155371734387015317811842494571531267321053802702229751785715325218238770603395187714420133568302831936143120725586586103040948862256054231452121522340482703816215076619109751602844494123621913364886256577828851863636252358680311680944000961886724963379829227207624420633482268396211673604317188265482187002450997796426235320241113672320963273291224124179452561094634357126507260462058777059881138191525935593^4
...
- u=-800/97のとき
(-4045142520*x^2+3568938018*x+8069930618)^4+(92234520*x^2-11843692098*x-8240824502)^4+(6060056701806149841*x^4+77967010154082816000*x^3-32880985420958691354*x^2+66868751094884194560*x-254345673531010056887)^2=(4177235529*x^2+16467888987)^4
楕円曲線 E(-800/97): y^2=6060056701806149841*x^4+77967010154082816000*x^3-32880985420958691354*x^2+66868751094884194560*x-254345673531010056887
16306696482461560^4+21794572772239369^4+87375622888246360^4=87486470529871881^4
140703840741824401472415321160^4+211846693725917814575333200001^4+779297054673311472994724152360^4=780564938636045965056301701249^4
999473823197475882886367591972007214130994383772809937995102187892357207726086203419640^4+6082019744073345700478988043855300837425714492006934574531115419823104074090059112505359^4+7455233192553681474101071025153048080363092134788491775804921546012434630736927978115560^4=8171423456128565238242271071836894174975585475772593732718357705981103159997839790106609^4
552245125243368972777413573288657778142862530322787354569559764231700682947879270380578428147380029717771109298902226920^4+652572386970659817949016039039228900852673822541837360703347330987824914037595843487623328249714722084858282999614053401^4+2873944370365812669072077676341536744667500161019442866281711608569791515664172925266412035287988286486191024749057246920^4=2876829526059988937840544104529117500319990007237076684552432419919500977390519569906943200751499344372395255192993997849^4
837352585512780124942219866765551514954494911931682473356272035396910027160937786402271290146925078709923787725310795266980040^4+4447954658142508118916442458556205930257335891638434495272552834722442251037001149548129432634987807505764394050698696886000551^4+5266234392710618891126472672934499667750237585584815597179530716141602156515003396626590769268879338878901389939702370847796760^4=5837299488569617041772110252191764260877432583239846443374664140099847052913196975260009063905052865404643647323332687370611801^4
[2024.11.29追記]
参考文献[4][5][6][7]に、Tom Womackの文書"The quartic surfaces x^4+y^4+z^4=N"を追加した。
2007/04/15に、この元々のWebページ(現在は参照できなくなっているようなので、Internet Archive Wayback Machineから探し出した)を見て、いつか、「ここに掲載されていない整数解を見つけよう」と決心したことを思い出した。
実際に、N=1の時の整数解を多数見つけることができた。
[2024.12.02追記]
u=752/15のとき、方程式系(5a),(5b)が有理数解(x,y,t)を持つことを確認できた。
楕円曲線
y^2=-2574990*x^4 + 115209*x^3 - 17668620*x^2 + 2922009*x + 63237532
の有理点から、A^4+B^4+C^4=D^4の整数解を見つけることができた。
[2024.12.09追記]
参考文献[5]のMAGMAプログラム"elk18.mag"は、2つの円錐
Cone1: 17*x^2 + 7*y^2 - 26*y*z + 7*z^2=0
Cone2: 13*y^2 - 7*y*z + 13*z^2 - 51*t^2=0
の共通点からx^4+y^4+z^4=18*t^4の整数解(x,y,z,t)を求めるものであるが、
MAGMA V2.25-3で実行するためには、下から3行目の
print "Expected rank was ",Rank(AssociatedEC(-crv));
を
print "Expected rank was ",Rank(AssociatedEllipticCurve(-crv));
に修正する必要がある。
その実行結果は、以下の通りである。
Point on cone1 = (-7/12 : 7/12 : 1)
Known point arises from k= 3925/2329
true 7440191712/107839375
Hunting points on 147579*k^4 - 262500*k^3 + 273546*k^2 - 69972*k + 97971
before cputime call
cputime was 0.000
Time: 0.000
[ 1, 1, 2, 1 ]
[ 1, 1, 2, 1 ]
[ 1, 1, 2, 1 ]
[ 1, 1, 2, 1 ]
[ 1, 2, 1, 1 ]
[ 1, 2, 1, 1 ]
[ 1, 2, 1, 1 ]
[ 1, 2, 1, 1 ]
[ 191, 278, 199, 149 ]
[ 191, 278, 199, 149 ]
[ 191, 278, 199, 149 ]
[ 191, 278, 199, 149 ]
[ 463, 1433, 538, 701 ]
[ 463, 1433, 538, 701 ]
[ 359, 1214, 439, 593 ]
[ 359, 1214, 439, 593 ]
Elliptic curve underlying it all is Elliptic Curve defined by y^2 = x^3 - 194133*x - 16912532 over Rational Field
of rank 2 true
Lift of <1, <-3, -3>> is [ 380847727, 391808458, 573695561, 303601733 ]
Lift of <1, <-3, -2>> is [ 359, 1214, 439, 593 ]
Lift of <1, <-3, -1>> is [ 25018077593, 67580333026, 223815462953, 108889884353 ]
Lift of <1, <-3, 0>> is [ 1835722393106619138500955324013201, 3646989810030598874532679932938582, 1834484169241706842691587875323671, 1824795736290816467165977786779221 ]
Lift of <1, <-3, 1>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 4957852588672482902866271904146679167109963229805184612141323926079002, 5028724473680793958525465456720590515991642590439820323981026319061577, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <1, <-3, 2>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <1, <-3, 3>> is [ 2604371813533642029120177880897647722051828558434955284902239822560336282641843752529860581502866220959121051767579501077312442113081319218314534927632265448240104129479850075053257353, 20187408102718866119271381799424342231001736248548196340542291947627757088884931666380899338403055980139794314793937206566797692864936196164701342331832386608789309733301513159951745138, 6160982233387947673812937129173690684648294889828579489895058864373219750124337295626227629312347039700049885860869551934716084169519730912474806357625479982101236489675388401153566121, 9822681920026626580588678610032546061187161443863531376743622216001282289240218569210545237190376989706691392851765576735937551767767123401540291185813842905529815706186334357021109913 ]
Lift of <1, <-2, -3>> is [ 53419645724359129, 190923427496780263, 641520316904741774, 312065608178758649 ]
Lift of <1, <-2, -2>> is [ 463, 1433, 538, 701 ]
Lift of <1, <-2, -1>> is [ 191, 199, 278, 149 ]
Lift of <1, <-2, 0>> is [ 4259434036158583, 4430033064864377, 6227038270223074, 3331706515089377 ]
Lift of <1, <-2, 1>> is [ 5116229387274642538565233425637571354179511, 16012611855028525659365480679495496163831911, 5981493490547537409678213927870161419673726, 7831448293825266049231711312969657728319697 ]
Lift of <1, <-2, 2>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <1, <-2, 3>> is [ 34020930773066529360747451299085973403020342233104113631044346873266437213433803817229907863126663405827456097965585906263621891786030342129, 70715084707455271696407680041830258148978097990099430510965457134575934790640960489474770166450309305634838712157837879459008742105905464263, 34191317102346158645981269995435731421627869496844501442086911977565083952506996803860237656711802187193000170341596288946856962797226296774, 35224993527274155682589066980541733238289334114500499380412061893291848645637541434466065642353487979486869191409487227231335360021097711149 ]
Lift of <1, <-1, -3>> is [ 1245744907257334112362374121044601, 1241605322843810001611154407013866, 2386180687038887250859493065400737, 1199034612820213430969683790335141 ]
Lift of <1, <-1, -2>> is [ 4598174177, 5418167038, 5529566303, 3341373121 ]
Lift of <1, <-1, -1>> is [ 1, 2, 1, 1 ]
Lift of <1, <-1, 0>> is [ 6617, 18454, 61247, 29797 ]
Lift of <1, <-1, 1>> is [ 49628196412110951312599, 165799267838517450086714, 60244793925280254852769, 81001697183174090229373 ]
Lift of <1, <-1, 2>> is [ 26068327452197351755586417776003396036278658054695315343, 26860551671620357628493544883373042361731871675526624782, 39087678176477024213173181197080641024770665878794825439, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <1, <-1, 3>> is [ 721125922502601138479303582069821317661661495686735600023189156986722193151457750249819291403496383759, 759184059387327461307263952030989015347125881703200036948010565220279153261631638743232889337140553106, 1024448694126496258974105333988441592817148236862052707632628378764757071318953798781775674946380748417, 554693105227654025023841912817201048428239714476218223918997192743853116695875684021868900944287977881 ]
Lift of <1, <0, -3>> is [ 26068327452197351755586417776003396036278658054695315343, 39087678176477024213173181197080641024770665878794825439, 26860551671620357628493544883373042361731871675526624782, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <1, <0, -2>> is [ 49628196412110951312599, 60244793925280254852769, 165799267838517450086714, 81001697183174090229373 ]
Lift of <1, <0, -1>> is [ 6617, 61247, 18454, 29797 ]
Lift of <1, <0, 1>> is [ 4598174177, 5529566303, 5418167038, 3341373121 ]
Lift of <1, <0, 2>> is [ 1245744907257334112362374121044601, 2386180687038887250859493065400737, 1241605322843810001611154407013866, 1199034612820213430969683790335141 ]
Lift of <1, <0, 3>> is [ 11768193612797843037535612402112753845240274443337542313745910615591817, 27124092798840851670242309414660977133638231450902319579751952946493759, 88654128821066978309956839817571474673240499756204391412509989953247942, 43138146272082536441352878895704167024228813733200822158942004004192317 ]
Lift of <1, <1, -3>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <1, <1, -2>> is [ 5116229387274642538565233425637571354179511, 5981493490547537409678213927870161419673726, 16012611855028525659365480679495496163831911, 7831448293825266049231711312969657728319697 ]
Lift of <1, <1, -1>> is [ 4259434036158583, 6227038270223074, 4430033064864377, 3331706515089377 ]
Lift of <1, <1, 0>> is [ 191, 278, 199, 149 ]
Lift of <1, <1, 1>> is [ 463, 538, 1433, 701 ]
Lift of <1, <1, 2>> is [ 53419645724359129, 641520316904741774, 190923427496780263, 312065608178758649 ]
Lift of <1, <1, 3>> is [ 12139134686958029175557745060773151811617634937, 12213551090194639332012858671072193298820148082, 25410945311503349794693287828558115148151323129, 12649905434394886132967806983661254499715264057 ]
Lift of <1, <2, -3>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <1, <2, -2>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 5028724473680793958525465456720590515991642590439820323981026319061577, 4957852588672482902866271904146679167109963229805184612141323926079002, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <1, <2, -1>> is [ 1835722393106619138500955324013201, 1834484169241706842691587875323671, 3646989810030598874532679932938582, 1824795736290816467165977786779221 ]
Lift of <1, <2, 0>> is [ 25018077593, 223815462953, 67580333026, 108889884353 ]
Lift of <1, <2, 1>> is [ 359, 439, 1214, 593 ]
Lift of <1, <2, 2>> is [ 380847727, 573695561, 391808458, 303601733 ]
Lift of <1, <2, 3>> is [ 21184389997465805360893375679, 29967979746788484219401652247, 22345830106268029802571407366, 16256999363651986688641522541 ]
Lift of <1, <3, -3>> is [ 972903903175447154551234923275314497952665755744057061737581576150907841603870688778515052300339720419437503839126459788922124219146300308641045702674810685456471, 1004070761075828088807130285702132557452798036685804707108085996978667482800288519448712098463587469192558312394281052074837329089465474711081954841640611784384874, 1452133178942316942017460611498633798349671199301221924559191387727046506579932626137609855396378297191384556889734423113392761987039247557387513380360241038968113, 770952097276848510154038790714214392802353200270301199606997486314973209746780766497655577003382535666587263032500292705715788759292290892295543269421432006487749 ]
Lift of <1, <3, -2>> is [ 3740284705655503709280709443725371930814826867446474727657234611909689739603325473445014087990250371887, 12346873828208643226887631629348034460580605096897331415808988455029704340834013066663886896434905708862, 4508172614381761668000687099221260994621755721120742559159978515863879369483751705896559702452761180367, 6033181140211919121899527424999769109933388380976480153742632397668956717387819135318977711635865541249 ]
Lift of <1, <3, -1>> is [ 953129949492712100450229958699009728805700955684762197809, 2747903083677548017031083476303362769113303027292469838978, 9138830669039588260878020798001619084887661590299224495249, 4445996698449893597463494296980735451350727352498158628449 ]
Lift of <1, <3, 0>> is [ 297015007730139032362495433, 598046271556290443753584726, 297234393852523653220392623, 298813986154340838922758373 ]
Lift of <1, <3, 1>> is [ 2748224089, 3228631226, 3315366961, 1997427997 ]
Lift of <1, <3, 2>> is [ 1497953, 1492558, 2851151, 1433833 ]
Lift of <1, <3, 3>> is [ 28971999793463329, 212289444274950674, 65117799214781713, 103301008591101049 ]
Lift of <2, <-3, -3>> is [ 380847727, 573695561, 391808458, 303601733 ]
Lift of <2, <-3, -2>> is [ 359, 439, 1214, 593 ]
Lift of <2, <-3, -1>> is [ 25018077593, 223815462953, 67580333026, 108889884353 ]
Lift of <2, <-3, 0>> is [ 1835722393106619138500955324013201, 1834484169241706842691587875323671, 3646989810030598874532679932938582, 1824795736290816467165977786779221 ]
Lift of <2, <-3, 1>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 5028724473680793958525465456720590515991642590439820323981026319061577, 4957852588672482902866271904146679167109963229805184612141323926079002, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <2, <-3, 2>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <2, <-3, 3>> is [ 2604371813533642029120177880897647722051828558434955284902239822560336282641843752529860581502866220959121051767579501077312442113081319218314534927632265448240104129479850075053257353, 6160982233387947673812937129173690684648294889828579489895058864373219750124337295626227629312347039700049885860869551934716084169519730912474806357625479982101236489675388401153566121, 20187408102718866119271381799424342231001736248548196340542291947627757088884931666380899338403055980139794314793937206566797692864936196164701342331832386608789309733301513159951745138, 9822681920026626580588678610032546061187161443863531376743622216001282289240218569210545237190376989706691392851765576735937551767767123401540291185813842905529815706186334357021109913 ]
Lift of <2, <-2, -3>> is [ 53419645724359129, 641520316904741774, 190923427496780263, 312065608178758649 ]
Lift of <2, <-2, -2>> is [ 463, 538, 1433, 701 ]
Lift of <2, <-2, -1>> is [ 191, 278, 199, 149 ]
Lift of <2, <-2, 0>> is [ 4259434036158583, 6227038270223074, 4430033064864377, 3331706515089377 ]
Lift of <2, <-2, 1>> is [ 5116229387274642538565233425637571354179511, 5981493490547537409678213927870161419673726, 16012611855028525659365480679495496163831911, 7831448293825266049231711312969657728319697 ]
Lift of <2, <-2, 2>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <2, <-2, 3>> is [ 34020930773066529360747451299085973403020342233104113631044346873266437213433803817229907863126663405827456097965585906263621891786030342129, 34191317102346158645981269995435731421627869496844501442086911977565083952506996803860237656711802187193000170341596288946856962797226296774, 70715084707455271696407680041830258148978097990099430510965457134575934790640960489474770166450309305634838712157837879459008742105905464263, 35224993527274155682589066980541733238289334114500499380412061893291848645637541434466065642353487979486869191409487227231335360021097711149 ]
Lift of <2, <-1, -3>> is [ 1245744907257334112362374121044601, 2386180687038887250859493065400737, 1241605322843810001611154407013866, 1199034612820213430969683790335141 ]
Lift of <2, <-1, -2>> is [ 4598174177, 5529566303, 5418167038, 3341373121 ]
Lift of <2, <-1, -1>> is [ 1, 1, 2, 1 ]
Lift of <2, <-1, 0>> is [ 6617, 61247, 18454, 29797 ]
Lift of <2, <-1, 1>> is [ 49628196412110951312599, 60244793925280254852769, 165799267838517450086714, 81001697183174090229373 ]
Lift of <2, <-1, 2>> is [ 26068327452197351755586417776003396036278658054695315343, 39087678176477024213173181197080641024770665878794825439, 26860551671620357628493544883373042361731871675526624782, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <2, <-1, 3>> is [ 721125922502601138479303582069821317661661495686735600023189156986722193151457750249819291403496383759, 1024448694126496258974105333988441592817148236862052707632628378764757071318953798781775674946380748417, 759184059387327461307263952030989015347125881703200036948010565220279153261631638743232889337140553106, 554693105227654025023841912817201048428239714476218223918997192743853116695875684021868900944287977881 ]
Lift of <2, <0, -3>> is [ 26068327452197351755586417776003396036278658054695315343, 26860551671620357628493544883373042361731871675526624782, 39087678176477024213173181197080641024770665878794825439, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <2, <0, -2>> is [ 49628196412110951312599, 165799267838517450086714, 60244793925280254852769, 81001697183174090229373 ]
Lift of <2, <0, -1>> is [ 6617, 18454, 61247, 29797 ]
Lift of <2, <0, 0>> is [ 1, 2, 1, 1 ]
Lift of <2, <0, 1>> is [ 4598174177, 5418167038, 5529566303, 3341373121 ]
Lift of <2, <0, 2>> is [ 1245744907257334112362374121044601, 1241605322843810001611154407013866, 2386180687038887250859493065400737, 1199034612820213430969683790335141 ]
Lift of <2, <0, 3>> is [ 11768193612797843037535612402112753845240274443337542313745910615591817, 88654128821066978309956839817571474673240499756204391412509989953247942, 27124092798840851670242309414660977133638231450902319579751952946493759, 43138146272082536441352878895704167024228813733200822158942004004192317 ]
Lift of <2, <1, -3>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <2, <1, -2>> is [ 5116229387274642538565233425637571354179511, 16012611855028525659365480679495496163831911, 5981493490547537409678213927870161419673726, 7831448293825266049231711312969657728319697 ]
Lift of <2, <1, -1>> is [ 4259434036158583, 4430033064864377, 6227038270223074, 3331706515089377 ]
Lift of <2, <1, 0>> is [ 191, 199, 278, 149 ]
Lift of <2, <1, 2>> is [ 53419645724359129, 190923427496780263, 641520316904741774, 312065608178758649 ]
Lift of <2, <1, 3>> is [ 12139134686958029175557745060773151811617634937, 25410945311503349794693287828558115148151323129, 12213551090194639332012858671072193298820148082, 12649905434394886132967806983661254499715264057 ]
Lift of <2, <2, -3>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <2, <2, -2>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 4957852588672482902866271904146679167109963229805184612141323926079002, 5028724473680793958525465456720590515991642590439820323981026319061577, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <2, <2, -1>> is [ 1835722393106619138500955324013201, 3646989810030598874532679932938582, 1834484169241706842691587875323671, 1824795736290816467165977786779221 ]
Lift of <2, <2, 0>> is [ 25018077593, 67580333026, 223815462953, 108889884353 ]
Lift of <2, <2, 1>> is [ 359, 1214, 439, 593 ]
Lift of <2, <2, 2>> is [ 380847727, 391808458, 573695561, 303601733 ]
Lift of <2, <2, 3>> is [ 21184389997465805360893375679, 22345830106268029802571407366, 29967979746788484219401652247, 16256999363651986688641522541 ]
Lift of <2, <3, -3>> is [ 972903903175447154551234923275314497952665755744057061737581576150907841603870688778515052300339720419437503839126459788922124219146300308641045702674810685456471, 1452133178942316942017460611498633798349671199301221924559191387727046506579932626137609855396378297191384556889734423113392761987039247557387513380360241038968113, 1004070761075828088807130285702132557452798036685804707108085996978667482800288519448712098463587469192558312394281052074837329089465474711081954841640611784384874, 770952097276848510154038790714214392802353200270301199606997486314973209746780766497655577003382535666587263032500292705715788759292290892295543269421432006487749 ]
Lift of <2, <3, -2>> is [ 3740284705655503709280709443725371930814826867446474727657234611909689739603325473445014087990250371887, 4508172614381761668000687099221260994621755721120742559159978515863879369483751705896559702452761180367, 12346873828208643226887631629348034460580605096897331415808988455029704340834013066663886896434905708862, 6033181140211919121899527424999769109933388380976480153742632397668956717387819135318977711635865541249 ]
Lift of <2, <3, -1>> is [ 953129949492712100450229958699009728805700955684762197809, 9138830669039588260878020798001619084887661590299224495249, 2747903083677548017031083476303362769113303027292469838978, 4445996698449893597463494296980735451350727352498158628449 ]
Lift of <2, <3, 0>> is [ 297015007730139032362495433, 297234393852523653220392623, 598046271556290443753584726, 298813986154340838922758373 ]
Lift of <2, <3, 1>> is [ 2748224089, 3315366961, 3228631226, 1997427997 ]
Lift of <2, <3, 2>> is [ 1497953, 2851151, 1492558, 1433833 ]
Lift of <2, <3, 3>> is [ 28971999793463329, 65117799214781713, 212289444274950674, 103301008591101049 ]
Lift of <3, <-3, -3>> is [ 380847727, 391808458, 573695561, 303601733 ]
Lift of <3, <-3, -2>> is [ 359, 1214, 439, 593 ]
Lift of <3, <-3, -1>> is [ 25018077593, 67580333026, 223815462953, 108889884353 ]
Lift of <3, <-3, 0>> is [ 1835722393106619138500955324013201, 3646989810030598874532679932938582, 1834484169241706842691587875323671, 1824795736290816467165977786779221 ]
Lift of <3, <-3, 1>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 4957852588672482902866271904146679167109963229805184612141323926079002, 5028724473680793958525465456720590515991642590439820323981026319061577, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <3, <-3, 2>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <3, <-3, 3>> is [ 2604371813533642029120177880897647722051828558434955284902239822560336282641843752529860581502866220959121051767579501077312442113081319218314534927632265448240104129479850075053257353, 20187408102718866119271381799424342231001736248548196340542291947627757088884931666380899338403055980139794314793937206566797692864936196164701342331832386608789309733301513159951745138, 6160982233387947673812937129173690684648294889828579489895058864373219750124337295626227629312347039700049885860869551934716084169519730912474806357625479982101236489675388401153566121, 9822681920026626580588678610032546061187161443863531376743622216001282289240218569210545237190376989706691392851765576735937551767767123401540291185813842905529815706186334357021109913 ]
Lift of <3, <-2, -3>> is [ 53419645724359129, 190923427496780263, 641520316904741774, 312065608178758649 ]
Lift of <3, <-2, -2>> is [ 463, 1433, 538, 701 ]
Lift of <3, <-2, -1>> is [ 191, 199, 278, 149 ]
Lift of <3, <-2, 0>> is [ 4259434036158583, 4430033064864377, 6227038270223074, 3331706515089377 ]
Lift of <3, <-2, 1>> is [ 5116229387274642538565233425637571354179511, 16012611855028525659365480679495496163831911, 5981493490547537409678213927870161419673726, 7831448293825266049231711312969657728319697 ]
Lift of <3, <-2, 2>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <3, <-2, 3>> is [ 34020930773066529360747451299085973403020342233104113631044346873266437213433803817229907863126663405827456097965585906263621891786030342129, 70715084707455271696407680041830258148978097990099430510965457134575934790640960489474770166450309305634838712157837879459008742105905464263, 34191317102346158645981269995435731421627869496844501442086911977565083952506996803860237656711802187193000170341596288946856962797226296774, 35224993527274155682589066980541733238289334114500499380412061893291848645637541434466065642353487979486869191409487227231335360021097711149 ]
Lift of <3, <-1, -3>> is [ 1245744907257334112362374121044601, 1241605322843810001611154407013866, 2386180687038887250859493065400737, 1199034612820213430969683790335141 ]
Lift of <3, <-1, -2>> is [ 4598174177, 5418167038, 5529566303, 3341373121 ]
Lift of <3, <-1, -1>> is [ 1, 2, 1, 1 ]
Lift of <3, <-1, 0>> is [ 6617, 18454, 61247, 29797 ]
Lift of <3, <-1, 1>> is [ 49628196412110951312599, 165799267838517450086714, 60244793925280254852769, 81001697183174090229373 ]
Lift of <3, <-1, 2>> is [ 26068327452197351755586417776003396036278658054695315343, 26860551671620357628493544883373042361731871675526624782, 39087678176477024213173181197080641024770665878794825439, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <3, <-1, 3>> is [ 721125922502601138479303582069821317661661495686735600023189156986722193151457750249819291403496383759, 759184059387327461307263952030989015347125881703200036948010565220279153261631638743232889337140553106, 1024448694126496258974105333988441592817148236862052707632628378764757071318953798781775674946380748417, 554693105227654025023841912817201048428239714476218223918997192743853116695875684021868900944287977881 ]
Lift of <3, <0, -3>> is [ 26068327452197351755586417776003396036278658054695315343, 39087678176477024213173181197080641024770665878794825439, 26860551671620357628493544883373042361731871675526624782, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <3, <0, -2>> is [ 49628196412110951312599, 60244793925280254852769, 165799267838517450086714, 81001697183174090229373 ]
Lift of <3, <0, -1>> is [ 6617, 61247, 18454, 29797 ]
Lift of <3, <0, 0>> is [ 1, 1, 2, 1 ]
Lift of <3, <0, 1>> is [ 4598174177, 5529566303, 5418167038, 3341373121 ]
Lift of <3, <0, 2>> is [ 1245744907257334112362374121044601, 2386180687038887250859493065400737, 1241605322843810001611154407013866, 1199034612820213430969683790335141 ]
Lift of <3, <0, 3>> is [ 11768193612797843037535612402112753845240274443337542313745910615591817, 27124092798840851670242309414660977133638231450902319579751952946493759, 88654128821066978309956839817571474673240499756204391412509989953247942, 43138146272082536441352878895704167024228813733200822158942004004192317 ]
Lift of <3, <1, -3>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <3, <1, -2>> is [ 5116229387274642538565233425637571354179511, 5981493490547537409678213927870161419673726, 16012611855028525659365480679495496163831911, 7831448293825266049231711312969657728319697 ]
Lift of <3, <1, -1>> is [ 4259434036158583, 6227038270223074, 4430033064864377, 3331706515089377 ]
Lift of <3, <1, 0>> is [ 191, 278, 199, 149 ]
Lift of <3, <1, 1>> is [ 463, 538, 1433, 701 ]
Lift of <3, <1, 2>> is [ 53419645724359129, 641520316904741774, 190923427496780263, 312065608178758649 ]
Lift of <3, <1, 3>> is [ 12139134686958029175557745060773151811617634937, 12213551090194639332012858671072193298820148082, 25410945311503349794693287828558115148151323129, 12649905434394886132967806983661254499715264057 ]
Lift of <3, <2, -3>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <3, <2, -2>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 5028724473680793958525465456720590515991642590439820323981026319061577, 4957852588672482902866271904146679167109963229805184612141323926079002, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <3, <2, -1>> is [ 1835722393106619138500955324013201, 1834484169241706842691587875323671, 3646989810030598874532679932938582, 1824795736290816467165977786779221 ]
Lift of <3, <2, 0>> is [ 25018077593, 223815462953, 67580333026, 108889884353 ]
Lift of <3, <2, 1>> is [ 359, 439, 1214, 593 ]
Lift of <3, <2, 2>> is [ 380847727, 573695561, 391808458, 303601733 ]
Lift of <3, <2, 3>> is [ 21184389997465805360893375679, 29967979746788484219401652247, 22345830106268029802571407366, 16256999363651986688641522541 ]
Lift of <3, <3, -3>> is [ 972903903175447154551234923275314497952665755744057061737581576150907841603870688778515052300339720419437503839126459788922124219146300308641045702674810685456471, 1004070761075828088807130285702132557452798036685804707108085996978667482800288519448712098463587469192558312394281052074837329089465474711081954841640611784384874, 1452133178942316942017460611498633798349671199301221924559191387727046506579932626137609855396378297191384556889734423113392761987039247557387513380360241038968113, 770952097276848510154038790714214392802353200270301199606997486314973209746780766497655577003382535666587263032500292705715788759292290892295543269421432006487749 ]
Lift of <3, <3, -2>> is [ 3740284705655503709280709443725371930814826867446474727657234611909689739603325473445014087990250371887, 12346873828208643226887631629348034460580605096897331415808988455029704340834013066663886896434905708862, 4508172614381761668000687099221260994621755721120742559159978515863879369483751705896559702452761180367, 6033181140211919121899527424999769109933388380976480153742632397668956717387819135318977711635865541249 ]
Lift of <3, <3, -1>> is [ 953129949492712100450229958699009728805700955684762197809, 2747903083677548017031083476303362769113303027292469838978, 9138830669039588260878020798001619084887661590299224495249, 4445996698449893597463494296980735451350727352498158628449 ]
Lift of <3, <3, 0>> is [ 297015007730139032362495433, 598046271556290443753584726, 297234393852523653220392623, 298813986154340838922758373 ]
Lift of <3, <3, 1>> is [ 2748224089, 3228631226, 3315366961, 1997427997 ]
Lift of <3, <3, 2>> is [ 1497953, 1492558, 2851151, 1433833 ]
Lift of <3, <3, 3>> is [ 28971999793463329, 212289444274950674, 65117799214781713, 103301008591101049 ]
Lift of <4, <-3, -3>> is [ 380847727, 573695561, 391808458, 303601733 ]
Lift of <4, <-3, -2>> is [ 359, 439, 1214, 593 ]
Lift of <4, <-3, -1>> is [ 25018077593, 223815462953, 67580333026, 108889884353 ]
Lift of <4, <-3, 0>> is [ 1835722393106619138500955324013201, 1834484169241706842691587875323671, 3646989810030598874532679932938582, 1824795736290816467165977786779221 ]
Lift of <4, <-3, 1>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 5028724473680793958525465456720590515991642590439820323981026319061577, 4957852588672482902866271904146679167109963229805184612141323926079002, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <4, <-3, 2>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <4, <-3, 3>> is [ 2604371813533642029120177880897647722051828558434955284902239822560336282641843752529860581502866220959121051767579501077312442113081319218314534927632265448240104129479850075053257353, 6160982233387947673812937129173690684648294889828579489895058864373219750124337295626227629312347039700049885860869551934716084169519730912474806357625479982101236489675388401153566121, 20187408102718866119271381799424342231001736248548196340542291947627757088884931666380899338403055980139794314793937206566797692864936196164701342331832386608789309733301513159951745138, 9822681920026626580588678610032546061187161443863531376743622216001282289240218569210545237190376989706691392851765576735937551767767123401540291185813842905529815706186334357021109913 ]
Lift of <4, <-2, -3>> is [ 53419645724359129, 641520316904741774, 190923427496780263, 312065608178758649 ]
Lift of <4, <-2, -2>> is [ 463, 538, 1433, 701 ]
Lift of <4, <-2, -1>> is [ 191, 278, 199, 149 ]
Lift of <4, <-2, 0>> is [ 4259434036158583, 6227038270223074, 4430033064864377, 3331706515089377 ]
Lift of <4, <-2, 1>> is [ 5116229387274642538565233425637571354179511, 5981493490547537409678213927870161419673726, 16012611855028525659365480679495496163831911, 7831448293825266049231711312969657728319697 ]
Lift of <4, <-2, 2>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <4, <-2, 3>> is [ 34020930773066529360747451299085973403020342233104113631044346873266437213433803817229907863126663405827456097965585906263621891786030342129, 34191317102346158645981269995435731421627869496844501442086911977565083952506996803860237656711802187193000170341596288946856962797226296774, 70715084707455271696407680041830258148978097990099430510965457134575934790640960489474770166450309305634838712157837879459008742105905464263, 35224993527274155682589066980541733238289334114500499380412061893291848645637541434466065642353487979486869191409487227231335360021097711149 ]
Lift of <4, <-1, -3>> is [ 1245744907257334112362374121044601, 2386180687038887250859493065400737, 1241605322843810001611154407013866, 1199034612820213430969683790335141 ]
Lift of <4, <-1, -2>> is [ 4598174177, 5529566303, 5418167038, 3341373121 ]
Lift of <4, <-1, -1>> is [ 1, 1, 2, 1 ]
Lift of <4, <-1, 0>> is [ 6617, 61247, 18454, 29797 ]
Lift of <4, <-1, 1>> is [ 49628196412110951312599, 60244793925280254852769, 165799267838517450086714, 81001697183174090229373 ]
Lift of <4, <-1, 2>> is [ 26068327452197351755586417776003396036278658054695315343, 39087678176477024213173181197080641024770665878794825439, 26860551671620357628493544883373042361731871675526624782, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <4, <-1, 3>> is [ 721125922502601138479303582069821317661661495686735600023189156986722193151457750249819291403496383759, 1024448694126496258974105333988441592817148236862052707632628378764757071318953798781775674946380748417, 759184059387327461307263952030989015347125881703200036948010565220279153261631638743232889337140553106, 554693105227654025023841912817201048428239714476218223918997192743853116695875684021868900944287977881 ]
Lift of <4, <0, -3>> is [ 26068327452197351755586417776003396036278658054695315343, 26860551671620357628493544883373042361731871675526624782, 39087678176477024213173181197080641024770665878794825439, 20718443749580280029668047139828586381922904131732511657 ]
Lift of <4, <0, -2>> is [ 49628196412110951312599, 165799267838517450086714, 60244793925280254852769, 81001697183174090229373 ]
Lift of <4, <0, -1>> is [ 6617, 18454, 61247, 29797 ]
Lift of <4, <0, 0>> is [ 1, 2, 1, 1 ]
Lift of <4, <0, 1>> is [ 4598174177, 5418167038, 5529566303, 3341373121 ]
Lift of <4, <0, 2>> is [ 1245744907257334112362374121044601, 1241605322843810001611154407013866, 2386180687038887250859493065400737, 1199034612820213430969683790335141 ]
Lift of <4, <0, 3>> is [ 11768193612797843037535612402112753845240274443337542313745910615591817, 88654128821066978309956839817571474673240499756204391412509989953247942, 27124092798840851670242309414660977133638231450902319579751952946493759, 43138146272082536441352878895704167024228813733200822158942004004192317 ]
Lift of <4, <1, -3>> is [ 311325838227436170333415508000369638599291690919644405388095536642968808307920308097, 1066510770274663105416193937112570053702153646091482940336082022879046262329684733849, 3577595488008232702618705641964599342069939281003615222521351751481953354549135888842, 1740337144757435551266343505361681238491657092675181716837199389238893540994685665317 ]
Lift of <4, <1, -2>> is [ 5116229387274642538565233425637571354179511, 16012611855028525659365480679495496163831911, 5981493490547537409678213927870161419673726, 7831448293825266049231711312969657728319697 ]
Lift of <4, <1, -1>> is [ 4259434036158583, 4430033064864377, 6227038270223074, 3331706515089377 ]
Lift of <4, <1, 0>> is [ 191, 199, 278, 149 ]
Lift of <4, <1, 1>> is [ 463, 1433, 538, 701 ]
Lift of <4, <1, 2>> is [ 53419645724359129, 190923427496780263, 641520316904741774, 312065608178758649 ]
Lift of <4, <1, 3>> is [ 12139134686958029175557745060773151811617634937, 25410945311503349794693287828558115148151323129, 12213551090194639332012858671072193298820148082, 12649905434394886132967806983661254499715264057 ]
Lift of <4, <2, -3>> is [ 564868513361807163567972782095746615593724648665952699074665044601404253863365827279447776324203589576392617076478571561, 563181036081399023712272812886986297094694455141367000265575993871109398163502150122054642911584316240524234345534931406, 1088919141189854774445518475219687053009109812623753025913350410671427476897996899935447837743394521405492613013242735927, 546739368847907289662159221548377023401029027965676740395813314132851052084995454599654590767378283313979390023017881081 ]
Lift of <4, <2, -2>> is [ 4194849334924041895484625917876792791474735661026148381890604611957313, 4957852588672482902866271904146679167109963229805184612141323926079002, 5028724473680793958525465456720590515991642590439820323981026319061577, 3047876519083863133503426870657720276590285566463024338715050166598749 ]
Lift of <4, <2, -1>> is [ 1835722393106619138500955324013201, 3646989810030598874532679932938582, 1834484169241706842691587875323671, 1824795736290816467165977786779221 ]
Lift of <4, <2, 0>> is [ 25018077593, 67580333026, 223815462953, 108889884353 ]
Lift of <4, <2, 1>> is [ 359, 1214, 439, 593 ]
Lift of <4, <2, 2>> is [ 380847727, 391808458, 573695561, 303601733 ]
Lift of <4, <2, 3>> is [ 21184389997465805360893375679, 22345830106268029802571407366, 29967979746788484219401652247, 16256999363651986688641522541 ]
Lift of <4, <3, -3>> is [ 972903903175447154551234923275314497952665755744057061737581576150907841603870688778515052300339720419437503839126459788922124219146300308641045702674810685456471, 1452133178942316942017460611498633798349671199301221924559191387727046506579932626137609855396378297191384556889734423113392761987039247557387513380360241038968113, 1004070761075828088807130285702132557452798036685804707108085996978667482800288519448712098463587469192558312394281052074837329089465474711081954841640611784384874, 770952097276848510154038790714214392802353200270301199606997486314973209746780766497655577003382535666587263032500292705715788759292290892295543269421432006487749 ]
Lift of <4, <3, -2>> is [ 3740284705655503709280709443725371930814826867446474727657234611909689739603325473445014087990250371887, 4508172614381761668000687099221260994621755721120742559159978515863879369483751705896559702452761180367, 12346873828208643226887631629348034460580605096897331415808988455029704340834013066663886896434905708862, 6033181140211919121899527424999769109933388380976480153742632397668956717387819135318977711635865541249 ]
Lift of <4, <3, -1>> is [ 953129949492712100450229958699009728805700955684762197809, 9138830669039588260878020798001619084887661590299224495249, 2747903083677548017031083476303362769113303027292469838978, 4445996698449893597463494296980735451350727352498158628449 ]
Lift of <4, <3, 0>> is [ 297015007730139032362495433, 297234393852523653220392623, 598046271556290443753584726, 298813986154340838922758373 ]
Lift of <4, <3, 1>> is [ 2748224089, 3315366961, 3228631226, 1997427997 ]
Lift of <4, <3, 2>> is [ 1497953, 2851151, 1492558, 1433833 ]
Lift of <4, <3, 3>> is [ 28971999793463329, 65117799214781713, 212289444274950674, 103301008591101049 ]
[2024.12.10追記]
u=-800/97のとき、方程式系(5a),(5b)が有理数解(x,y,t)を持つことを確認できた。
楕円曲線
y^2=6060056701806149841*x^4+77967010154082816000*x^3-32880985420958691354*x^2+66868751094884194560*x-254345673531010056887
の有理点から、A^4+B^4+C^4=D^4の整数解を見つけることができた。
[参考文献]
- [1]Noam Elkies, "On A^4+B^4+C^4=D^4", Math Comp. 51(184), p824-835, 1988.
- [2]StarkExchange MATHEMATICS, "Distribution of Primitive Pytagorean Triples (PPT) and of solutions of A^4+B^4+C^4=D^4", 2016/07/08.
- [3]StarkExchange MATHEMATICS, "More elliptic curves fpr x^4+y^4+z^4=1?", 2017/07/28.
- [4]Tom Womack, "The quartic surfaces x^4+y^4+z^4=N", 2013/05/17.
- [5]Tom Womack, "elk18.mag", 2013/06/07.
- [6]Tom Womack, "elk18.pts", 2013/06/07.
- [7]Tom Womack, "Integer points on x^4+y^4+z^4=Nt^4", 2013/06/07.
| Last Update: 2024.12.10 |
| H.Nakao |