Integer Points on A^4+B^4+C^4=13778*D^4

[2025.11.02]A^4+B^4+C^4=13778*D^4の整点

syzygyによる変換を計算するプログラム(Pari/GP)
A^4+B^4+C^4=2m^2D^4の整点を計算するプログラム(Pari/GP)
2次曲線(3a),(3b)がnon-singularであり、共に有理点を持つ有理数uを抽出するプログラム(MAGMA)
楕円曲線の4-descentから有理点を計算するプログラム(MAGMA)

■整点を求める方法は、 "A^4+B^4+C^4=3362*D^4の整点" と同様なので、詳細はそちらを参照すること。ただし、参照する数式のみ記載する。

自然数nを固定したとき、不定方程式
       A^4+B^4+C^4=2*n^2*D^4 ----------(1)
を満たす自明でない整数の組(A,B,C,D) (ただし C!=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])の方法およびTom Womackの文書(参考文献[5])を参考にして、(1)を満たす整数の組(A,B,C,D)を探す。
ここで、整数A,B,C,Dは0以上として良い。

■x=A/C,y=B/C.t=D/Cとすると、
       x^4+y^4+1=2*n^2*t^4 ----------(2)
つまり、(2)を満たす有理数の組(x,y,t)を見つければ良い。

そのためには、nある有理数uに対して、
       ±(u^2-2)*y^2=(-u^2+4*u-2)*x^2-2*(u^2-2*u+2)*x+(-u^2+4*u-2) ----------(3a±)
       ±n*(u^2-2)*t^2=(u^2-2*u+2)*x^2+(-u^2+4*u-2)*x+(u^2-2*u+2) ----------(3b±)
の両方を満たす有理数の組(x,y,t)を見つければ良い。

■任意の有理数uについて、２次曲線(3b+)および(3b-)は、non-singularである。
また、u^2 > 2のとき、(3b+)のみ、u^2 < 2のとき、(3b-)のみが成立する。

■２次曲線(3a)がsingularであるのは、u=0,1,2のときであり。そのときに限る。
u=1のとき、(3a+)はsingularであるが、有理点を持たない。
u-0,2のとき、(3a+)はsingularであり、
       x^2 - x + 1=n*t^2　--------(**)
が有理点をもつかどうかを議論する必要がある。

13778=2*83^2であるので、以下では、n=83とする。

■n=83のとき、２次曲線(**)は、有理点を持たないことが確認できる。

{MAGMAでの計算]

> P2 := ProjectiveSpace(Rationals(), 2);
> N:=83;
> C := Conic(P2,-N*y^2+x^2+x*z+z^2);
> HasRationalPoint(C);
false
>

■有理数u(u!=0,1,2)の高さが小さいものから、順に調べる。
例えば、有理数uの高さが200以下の範囲で、２次曲線(3a+)と２つの２次曲線の和集合(3b±)が共に有理点を持つようなuを選択すると、以下のように138個のuが抽出される。
これらのuについて、(3a+),(3b±)を共に満たす有理数の組(x,y,t)を見つければ良い。

[MAGMAによる計算]

> PP(83,1,200);
** u= 5/13 ; tau(u)= 21/8 ; -103*x^2 + 313*y^2 + 466*x*z - 103*z^2
  (-5 : 4 : 1)  C1b (-184902/64699 : 19789/64699 : 1)
** u= -7/13 ; tau(u)= 33/20 ; -751*x^2 + 289*y^2 + 1138*x*z - 751*z^2
  (5/3 : 92/51 : 1)  C2b (662/737 : 1521/12529 : 1)
** u= -7/25 ; tau(u)= 57/32 ; -1999*x^2 + 1201*y^2 + 3298*x*z - 1999*z^2
  (1/7 : 8/7 : 1)  C2b (42122/18199 : -4411/18199 : 1)
** u= -8/29 ; tau(u)= 66/37 ; -2674*x^2 + 1618*y^2 + 4420*x*z - 2674*z^2
  (19/27 : 20/27 : 1)  C2b (-13562/7595 : -491/1519 : 1)
** u= -8/197 ; tau(u)= 402/205 ; -83986*x^2 + 77554*y^2 + 161668*x*z - 83986*z^2
  (327/784 : 71/112 : 1)  C2b (124478/47863 : 12067/47863 : 1)
** u= 13/29 ; tau(u)= 45/16 ; -343*x^2 + 1513*y^2 + 2194*x*z - 343*z^2
  (47/297 : 16/297 : 1)  C1b (-6789/658 : -647/658 : 1)
** u= -15/121 ; tau(u)= 257/136 ; -36767*x^2 + 29057*y^2 + 66274*x*z - 36767*z^2
  (245/223 : 836/1561 : 1)  C2b (54734/50233 : -40767/351631 : 1)
** u= 16/65 ; tau(u)= 114/49 ; -4546*x^2 + 8194*y^2 + 13252*x*z - 4546*z^2
  (3429/1247 : 686/1247 : 1)  C1b (7246/36263 : 3401/36263 : 1)
** u= -16/97 ; tau(u)= 210/113 ; -25282*x^2 + 18562*y^2 + 44356*x*z - 25282*z^2
  (-1115/271 : -1586/271 : 1)  C2b (-875986/233043 : 123199/233043 : 1)
** u= -19/61 ; tau(u)= 141/80 ; -12439*x^2 + 7081*y^2 + 20242*x*z - 12439*z^2
  (-46883/215739 : -338432/215739 : 1)  C2b (2786/1851 : -289/1851 : 1)
** u= -20/29 ; tau(u)= 78/49 ; -4402*x^2 + 1282*y^2 + 6484*x*z - 4402*z^2
  (5/2 : 7/2 : 1)  C2b (-4794/2117 : 1121/2117 : 1)
** u= -20/153 ; tau(u)= 326/173 ; -59458*x^2 + 46418*y^2 + 106676*x*z - 59458*z^2
  (-29/4504 : 5127/4504 : 1)  C2b (3126292379/14280218 : 366911849/14280218 : 1)
** u= 21/8 ; tau(u)= 5/13 ; 103*x^2 - 313*y^2 + 466*x*z + 103*z^2
  (1/5 : -4/5 : 1)  C1a (266/1493 : 149/1493 : 1)
** u= 21/169 ; tau(u)= 317/148 ; -43367*x^2 + 56681*y^2 + 100930*x*z - 43367*z^2
  (7779/3521 : -2648/3521 : 1)  C1b (15779/427555 : -8721/85511 : 1)
** u= 23/153 ; tau(u)= 283/130 ; -33271*x^2 + 46289*y^2 + 80618*x*z - 33271*z^2
  (27697/52501 : -174/52501 : 1)  C1b (1848057/291169 : 179059/291169 : 1)
** u= 24/125 ; tau(u)= 226/101 ; -19826*x^2 + 30674*y^2 + 51652*x*z - 19826*z^2
  (-696/1915 : -15539/13405 : 1)  C1b (21382/36131 : 24111/252917 : 1)
** u= -25/121 ; tau(u)= 267/146 ; -42007*x^2 + 28657*y^2 + 71914*x*z - 42007*z^2
  (-8087/164833 : 208010/164833 : 1)  C2b (-295382/45641 : 39857/45641 : 1)
** u= 28/41 ; tau(u)= 54/13 ; 446*x^2 + 2578*y^2 + 3700*x*z + 446*z^2
  (-668/3443 : 1083/3443 : 1)  C1b (-203774/438351 : 42827/438351 : 1)
** u= -28/61 ; tau(u)= 150/89 ; -15058*x^2 + 6658*y^2 + 23284*x*z - 15058*z^2
  (193/1062 : 1385/1062 : 1)  C2b (79139/12997 : -10367/12997 : 1)
** u= -28/73 ; tau(u)= 174/101 ; -19618*x^2 + 9874*y^2 + 31060*x*z - 19618*z^2
  (548/2247 : 2599/2247 : 1)  C2b (-8472598/764307 : -1235467/764307 : 1)
** u= -31/37 ; tau(u)= 105/68 ; -8287*x^2 + 1777*y^2 + 11986*x*z - 8287*z^2
  (59/17 : 104/17 : 1)  C2b (-247222/219803 : -86601/219803 : 1)
** u= 31/41 ; tau(u)= 51/10 ; 761*x^2 + 2401*y^2 + 3562*x*z + 761*z^2
  (-35/33 : 1534/1617 : 1)  C1b (-170473/99999 : 825749/4899951 : 1)
** u= 32/37 ; tau(u)= 42/5 ; 974*x^2 + 1714*y^2 + 2788*x*z + 974*z^2
  (-9/22 : 1/22 : 1)  C1b (-129337/2854 : 12707/2854 : 1)
** u= 32/45 ; tau(u)= 58/13 ; 686*x^2 + 3026*y^2 + 4388*x*z + 686*z^2
  (-167/1030 : -51/1030 : 1)  C1b (108946/38017 : -11297/38017 : 1)
** u= -32/137 ; tau(u)= 306/169 ; -56098*x^2 + 36514*y^2 + 94660*x*z - 56098*z^2
  (4799/10879 : -9048/10879 : 1)  C2b (-4982691/1899019 : 786889/1899019 : 1)
** u= -32/149 ; tau(u)= 330/181 ; -64498*x^2 + 43378*y^2 + 109924*x*z - 64498*z^2
  (-89/61 : 4054/1403 : 1)  C2b (-4094/60821 : 179911/1398883 : 1)
** u= 33/20 ; tau(u)= -7/13 ; 751*x^2 - 289*y^2 + 1138*x*z + 751*z^2
  (-1/583 : 15956/9911 : 1)  C1a (-1017/137 : -2441/2329 : 1)
** u= 42/5 ; tau(u)= 32/37 ; -974*x^2 - 1714*y^2 + 2788*x*z - 974*z^2
  (141/59 : 16/59 : 1)  C1a (2363/539 : -221/539 : 1)
** u= -43/85 ; tau(u)= 213/128 ; -30919*x^2 + 12601*y^2 + 47218*x*z - 30919*z^2
  (-5899/16283 : 33104/16283 : 1)  C2b (-1224711/198721 : -204803/198721 : 1)
** u= 44/49 ; tau(u)= 54/5 ; 1886*x^2 + 2866*y^2 + 4852*x*z + 1886*z^2
  (-1580/2879 : 777/2879 : 1)  C1b (-11423/4607 : -1063/4607 : 1)
** u= 45/16 ; tau(u)= 13/29 ; 343*x^2 - 1513*y^2 + 2194*x*z + 343*z^2
  (-49/2279 : -1008/2279 : 1)  C1a (108946/38017 : -11297/38017 : 1)
** u= 48/73 ; tau(u)= 98/25 ; 1054*x^2 + 8354*y^2 + 11908*x*z + 1054*z^2
  (-1094/9879 : -1715/9879 : 1)  C1b (-17807/87863 : -8163/87863 : 1)
** u= 49/89 ; tau(u)= 129/40 ; -799*x^2 + 13441*y^2 + 19042*x*z - 799*z^2
  (-1383/98507 : 27748/98507 : 1)  C1b (22402/18127 : -2607/18127 : 1)
** u= 49/153 ; tau(u)= 257/104 ; -19231*x^2 + 44417*y^2 + 68450*x*z - 19231*z^2
  (27193/1489843 : 948108/1489843 : 1)  C1b (141494/3139869 : 298987/3139869 : 1)
** u= -49/153 ; tau(u)= 355/202 ; -79207*x^2 + 44417*y^2 + 128426*x*z - 79207*z^2
  (-605/67 : -882/67 : 1)  C2b (-733014/1300237 : 243481/1300237 : 1)
** u= -49/193 ; tau(u)= 435/242 ; -114727*x^2 + 72097*y^2 + 191626*x*z - 114727*z^2
  (2557/76225 : -93478/76225 : 1)  C2b (3345046/92067 : 416399/92067 : 1)
** u= 51/10 ; tau(u)= 31/41 ; -761*x^2 - 2401*y^2 + 3562*x*z - 761*z^2
  (87/89 : 3974/4361 : 1)  C1a (706/283 : -3253/13867 : 1)
** u= 52/113 ; tau(u)= 174/61 ; -4738*x^2 + 22834*y^2 + 32980*x*z - 4738*z^2
  (-4519/10322 : -67765/72254 : 1)  C1b (-724661/14074 : 474573/98518 : 1)
** u= -52/181 ; tau(u)= 414/233 ; -105874*x^2 + 62818*y^2 + 174100*x*z - 105874*z^2
  (-18007/9442 : 239229/66094 : 1)  C2b (-136613/167333 : 246821/1171331 : 1)
** u= 54/5 ; tau(u)= 44/49 ; -1886*x^2 - 2866*y^2 + 4852*x*z - 1886*z^2
  (1580/2879 : -777/2879 : 1)  C1a (-977766/152257 : 105673/152257 : 1)
** u= 54/13 ; tau(u)= 28/41 ; -446*x^2 - 2578*y^2 + 3700*x*z - 446*z^2
  (19/96 : 31/96 : 1)  C1a (-6778/9803 : -1169/9803 : 1)
** u= 55/137 ; tau(u)= 219/82 ; -10423*x^2 + 34513*y^2 + 50986*x*z - 10423*z^2
  (18969/262357 : -116378/262357 : 1)  C1b (-35998962/455711 : 3405067/455711 : 1)
** u= -56/81 ; tau(u)= 218/137 ; -34402*x^2 + 9986*y^2 + 50660*x*z - 34402*z^2
  (4811/983759 : 1819368/983759 : 1)  C2b (2268515/283526 : -21455/16678 : 1)
** u= -56/89 ; tau(u)= 234/145 ; -38914*x^2 + 12706*y^2 + 57892*x*z - 38914*z^2
  (24903/68491 : 92168/68491 : 1)  C2b (3528347/223302 : -560353/223302 : 1)
** u= 56/153 ; tau(u)= 250/97 ; -15682*x^2 + 43682*y^2 + 65636*x*z - 15682*z^2
  (9487/703 : 4740/703 : 1)  C1b (9437/5627 : -929/5627 : 1)
** u= 57/32 ; tau(u)= -7/25 ; 1999*x^2 - 1201*y^2 + 3298*x*z + 1999*z^2
  (-957/5627 : 6280/5627 : 1)  C1a (17102/1587 : -2329/1587 : 1)
** u= -57/185 ; tau(u)= 427/242 ; -113879*x^2 + 65201*y^2 + 185578*x*z - 113879*z^2
  (-110835/1557577 : 2179474/1557577 : 1)  C2b (14488213/926834 : 1825173/926834 : 1)
** u= 58/13 ; tau(u)= 32/45 ; -686*x^2 - 3026*y^2 + 4388*x*z - 686*z^2
  (13/53 : 18/53 : 1)  C1a (-6789/658 : -647/658 : 1)
** u= 59/109 ; tau(u)= 159/50 ; -1519*x^2 + 20281*y^2 + 28762*x*z - 1519*z^2
  (-4923/750613 : 217810/750613 : 1)  C1b (24834/653 : -2293/653 : 1)
** u= -59/137 ; tau(u)= 333/196 ; -73351*x^2 + 34057*y^2 + 114370*x*z - 73351*z^2
  (7209/41819 : -53536/41819 : 1)  C2b (-11608803/11124182 : 2929289/11124182 : 1)
** u= -60/61 ; tau(u)= 182/121 ; -25682*x^2 + 3842*y^2 + 36724*x*z - 25682*z^2
  (512045/261178 : -964513/261178 : 1)  C2b (353213/126982 : 67083/126982 : 1)
** u= 61/125 ; tau(u)= 189/64 ; -4471*x^2 + 27529*y^2 + 39442*x*z - 4471*z^2
  (2359/26969 : 5280/26969 : 1)  C1b (1705574/998953 : 174317/998953 : 1)
** u= 64/153 ; tau(u)= 242/89 ; -11746*x^2 + 42722*y^2 + 62660*x*z - 11746*z^2
  (3797/25331 : -6270/25331 : 1)  C1b (-549082615/47179002 : 4048015/3629154 : 1)
** u= -64/157 ; tau(u)= 378/221 ; -93586*x^2 + 45202*y^2 + 146980*x*z - 93586*z^2
  (-122/49 : -235/49 : 1)  C2b (418275758/7679895 : -11572477/1535979 : 1)
** u= 66/37 ; tau(u)= -8/29 ; 2674*x^2 - 1618*y^2 + 4420*x*z + 2674*z^2
  (-172/151 : 125/151 : 1)  C1a (543034/2919 : -69889/2919 : 1)
** u= 67/101 ; tau(u)= 135/34 ; 2177*x^2 + 15913*y^2 + 22714*x*z + 2177*z^2
  (-428387/713321 : 83478/101903 : 1)  C1b (163037/296346 : -32609/296346 : 1)
** u= -67/145 ; tau(u)= 357/212 ; -85399*x^2 + 37561*y^2 + 131938*x*z - 85399*z^2
  (35/81 : -88/81 : 1)  C2b (-10866094/4605527 : -2078013/4605527 : 1)
** u= 68/121 ; tau(u)= 174/53 ; -994*x^2 + 24658*y^2 + 34900*x*z - 994*z^2
  (-18/983 : -253/983 : 1)  C1b (1713502/61137 : -158161/61137 : 1)
** u= 73/89 ; tau(u)= 105/16 ; 4817*x^2 + 10513*y^2 + 16354*x*z + 4817*z^2
  (-15857/12433 : -10984/12433 : 1)  C1b (-32449/25311 : -259/1947 : 1)
** u= -76/61 ; tau(u)= 198/137 ; -31762*x^2 + 1666*y^2 + 44980*x*z - 31762*z^2
  (752/937 : 20403/6559 : 1)  C2b (-661/2826 : -9389/19782 : 1)
** u= -76/153 ; tau(u)= 382/229 ; -99106*x^2 + 41042*y^2 + 151700*x*z - 99106*z^2
  (113/134 : 135/134 : 1)  C2b (-2469781/310421 : -400339/310421 : 1)
** u= 77/85 ; tau(u)= 93/8 ; 5801*x^2 + 8521*y^2 + 14578*x*z + 5801*z^2
  (-11/7 : -4/7 : 1)  C1b (3950973/1779157 : -501161/1779157 : 1)
** u= -77/101 ; tau(u)= 279/178 ; -57439*x^2 + 14473*y^2 + 83770*x*z - 57439*z^2
  (-73329/562759 : 1231694/562759 : 1)  C2b (-148891/118210 : 9169/23642 : 1)
** u= -77/117 ; tau(u)= 311/194 ; -69343*x^2 + 21449*y^2 + 102650*x*z - 69343*z^2
  (441611/851052527 : -1529633370/851052527 : 1)  C2b (10233425/987297 : -1629025/987297 : 1)
** u= 78/49 ; tau(u)= -20/29 ; 4402*x^2 - 1282*y^2 + 6484*x*z + 4402*z^2
  (-2713/2880 : -3773/2880 : 1)  C1a (-23091722/12848211 : -3000361/12848211 : 1)
** u= -79/121 ; tau(u)= 321/200 ; -73759*x^2 + 23041*y^2 + 109282*x*z - 73759*z^2
  (9157/1303 : -14740/1303 : 1)  C2b (-7711/22369 : 4759/22369 : 1)
** u= 84/121 ; tau(u)= 158/37 ; 4318*x^2 + 22226*y^2 + 32020*x*z + 4318*z^2
  (-1068/5543 : -1529/5543 : 1)  C1b (9899/53146 : 5157/53146 : 1)
** u= 88/89 ; tau(u)= 90 ; 7742*x^2 + 8098*y^2 + 15844*x*z + 7742*z^2
  (-823/680 : 73/680 : 1)  C1b (33202/8249 : -4117/8249 : 1)
** u= 88/101 ; tau(u)= 114/13 ; 7406*x^2 + 12658*y^2 + 20740*x*z + 7406*z^2
  (-1377/2188 : 1013/2188 : 1)  C1b (-8695/9946 : 1055/9946 : 1)
** u= -88/193 ; tau(u)= 474/281 ; -150178*x^2 + 66754*y^2 + 232420*x*z - 150178*z^2
  (6401/1788 : -7715/1788 : 1)  C2b (7300130/518429 : 1010355/518429 : 1)
** u= 89/121 ; tau(u)= 153/32 ; 5873*x^2 + 21361*y^2 + 31330*x*z + 5873*z^2
  (-4161/6553 : 4840/6553 : 1)  C1b (12641170/774791 : 1204015/774791 : 1)
** u= 90 ; tau(u)= 88/89 ; -7742*x^2 - 8098*y^2 + 15844*x*z - 7742*z^2
  (1783/2152 : -201/2152 : 1)  C1a (-79/102 : -1/6 : 1)
** u= 91/173 ; tau(u)= 255/82 ; -5167*x^2 + 51577*y^2 + 73306*x*z - 5167*z^2
  (56029/798909 : -25202/798909 : 1)  C1b (-1170517/723299 : -131271/723299 : 1)
** u= 92/193 ; tau(u)= 294/101 ; -11938*x^2 + 66034*y^2 + 94900*x*z - 11938*z^2
  (4664/42941 : 7033/42941 : 1)  C1b (-858225/67001 : -80875/67001 : 1)
** u= 93/8 ; tau(u)= 77/85 ; -5801*x^2 - 8521*y^2 + 14578*x*z - 5801*z^2
  (44093/83833 : -14692/83833 : 1)  C1a (-1193329/96582 : -125371/96582 : 1)
** u= -93/149 ; tau(u)= 391/242 ; -108479*x^2 + 35753*y^2 + 161530*x*z - 108479*z^2
  (-2020857/957443 : 4890182/957443 : 1)  C2b (362182/276323 : -44607/276323 : 1)
** u= -95/149 ; tau(u)= 393/244 ; -110047*x^2 + 35377*y^2 + 163474*x*z - 110047*z^2
  (11575/13669 : -16336/13669 : 1)  C2b (2298119/2729703 : 345439/2729703 : 1)
** u= 96/113 ; tau(u)= 130/17 ; 8638*x^2 + 16322*y^2 + 26116*x*z + 8638*z^2
  (-3161/1527 : 1096/1527 : 1)  C1b (25671262/25127 : 2521161/25127 : 1)
** u= 98/25 ; tau(u)= 48/73 ; -1054*x^2 - 8354*y^2 + 11908*x*z - 1054*z^2
  (8/3 : -5/3 : 1)  C1a (-821578/29201 : 76419/29201 : 1)
** u= 100/113 ; tau(u)= 126/13 ; 9662*x^2 + 15538*y^2 + 25876*x*z + 9662*z^2
  (-1663/2862 : -1055/2862 : 1)  C1b (-9319/8494 : 1001/8494 : 1)
** u= 101/117 ; tau(u)= 133/16 ; 9689*x^2 + 17177*y^2 + 27890*x*z + 9689*z^2
  (-10351/25511 : -1104/25511 : 1)  C1b (304231849/2597121 : 30178741/2597121 : 1)
** u= 101/181 ; tau(u)= 261/80 ; -2599*x^2 + 55321*y^2 + 78322*x*z - 2599*z^2
  (-16429/21613 : 162264/151291 : 1)  C1b (-4394374/2803807 : 3420101/19626649 : 1)
** u= -104/101 ; tau(u)= 306/205 ; -73234*x^2 + 9586*y^2 + 104452*x*z - 73234*z^2
  (608/3991 : -9903/3991 : 1)  C2b (-68634/97631 : 39391/97631 : 1)
** u= 105/16 ; tau(u)= 73/89 ; -4817*x^2 - 10513*y^2 + 16354*x*z - 4817*z^2
  (8949/26485 : -3256/26485 : 1)  C1a (364179/129206 : 33757/129206 : 1)
** u= 105/68 ; tau(u)= -31/37 ; 8287*x^2 - 1777*y^2 + 11986*x*z + 8287*z^2
  (-887/617 : -1324/617 : 1)  C1a (-85691/40883 : -13017/40883 : 1)
** u= 105/137 ; tau(u)= 169/32 ; 8977*x^2 + 26513*y^2 + 39586*x*z + 8977*z^2
  (-123991/510351 : 32968/510351 : 1)  C1b (-17311754/285209 : -1635909/285209 : 1)
** u= -107/153 ; tau(u)= 413/260 ; -123751*x^2 + 35369*y^2 + 182018*x*z - 123751*z^2
  (-889/6197 : 12864/6197 : 1)  C2b (88204079/28314554 : -12679319/28314554 : 1)
** u= 114/13 ; tau(u)= 88/101 ; -7406*x^2 - 12658*y^2 + 20740*x*z - 7406*z^2
  (1479/656 : -241/656 : 1)  C1a (1181387/125363 : 113521/125363 : 1)
** u= 114/49 ; tau(u)= 16/65 ; 4546*x^2 - 8194*y^2 + 13252*x*z + 4546*z^2
  (-487/150 : -161/150 : 1)  C1a (-3727/11313 : 1049/11313 : 1)
** u= -116/173 ; tau(u)= 462/289 ; -153586*x^2 + 46402*y^2 + 226900*x*z - 153586*z^2
  (36268/70407 : 90967/70407 : 1)  C2b (-472642271/7469086 : 81993051/7469086 : 1)
** u= -120/101 ; tau(u)= 322/221 ; -83282*x^2 + 6002*y^2 + 118084*x*z - 83282*z^2
  (821/4465 : 14624/4465 : 1)  C2b (65941/18578 : 18747/18578 : 1)
** u= -124/117 ; tau(u)= 358/241 ; -100786*x^2 + 12002*y^2 + 143540*x*z - 100786*z^2
  (172/169 : 375/169 : 1)  C2b (-46642938/4380313 : -13371403/4380313 : 1)
** u= 125/193 ; tau(u)= 261/68 ; 6377*x^2 + 58873*y^2 + 83746*x*z + 6377*z^2
  (-8077/14913 : 11840/14913 : 1)  C1b (2686678/1234797 : 281513/1234797 : 1)
** u= 126/13 ; tau(u)= 100/113 ; -9662*x^2 - 15538*y^2 + 25876*x*z - 9662*z^2
  (716/1529 : -225/1529 : 1)  C1a (-150402/82171 : -19679/82171 : 1)
** u= -128/101 ; tau(u)= 330/229 ; -88498*x^2 + 4018*y^2 + 125284*x*z - 88498*z^2
  (2661/2885 : 69968/20195 : 1)  C2b (24258/7781 : -59321/54467 : 1)
** u= 129/40 ; tau(u)= 49/89 ; 799*x^2 - 13441*y^2 + 19042*x*z + 799*z^2
  (-349/27159 : -5516/27159 : 1)  C1a (-143067/24979 : 13321/24979 : 1)
** u= 130/17 ; tau(u)= 96/113 ; -8638*x^2 - 16322*y^2 + 26116*x*z - 8638*z^2
  (59271/31369 : 24392/31369 : 1)  C1a (-79591/83822 : -13089/83822 : 1)
** u= 133/16 ; tau(u)= 101/117 ; -9689*x^2 - 17177*y^2 + 27890*x*z - 9689*z^2
  (227/499 : -120/499 : 1)  C1a (17471079/671870 : -341189/134374 : 1)
** u= 133/149 ; tau(u)= 165/16 ; 17177*x^2 + 26713*y^2 + 44914*x*z + 17177*z^2
  (-7623/3959 : 1816/3959 : 1)  C1b (290201/32086 : -30587/32086 : 1)
** u= -133/157 ; tau(u)= 447/290 ; -150511*x^2 + 31609*y^2 + 217498*x*z - 150511*z^2
  (-506251/59374565 : -130362878/59374565 : 1)  C2b (35477/5234 : -6531/5234 : 1)
** u= 135/34 ; tau(u)= 67/101 ; -2177*x^2 - 15913*y^2 + 22714*x*z - 2177*z^2
  (175/57 : 98/57 : 1)  C1a (-396651/28451 : 37127/28451 : 1)
** u= -136/121 ; tau(u)= 378/257 ; -113602*x^2 + 10786*y^2 + 161380*x*z - 113602*z^2
  (-373/15104 : -49885/15104 : 1)  C2b (-1029658/230883 : 361289/230883 : 1)
** u= -140/117 ; tau(u)= 374/257 ; -112498*x^2 + 7778*y^2 + 159476*x*z - 112498*z^2
  (4045/39874 : 141159/39874 : 1)  C2b (-490663/158646 : -215429/158646 : 1)
** u= -140/197 ; tau(u)= 534/337 ; -207538*x^2 + 58018*y^2 + 304756*x*z - 207538*z^2
  (692/269 : 997/269 : 1)  C2b (178069/157547 : -24077/157547 : 1)
** u= 141/80 ; tau(u)= -19/61 ; 12439*x^2 - 7081*y^2 + 20242*x*z + 12439*z^2
  (-65107/357717 : 407008/357717 : 1)  C1a (-325462/106621 : 35853/106621 : 1)
** u= 145/153 ; tau(u)= 161/8 ; 20897*x^2 + 25793*y^2 + 46946*x*z + 20897*z^2
  (-47137/46037 : -20796/46037 : 1)  C1b (-46713/106231 : 9961/106231 : 1)
** u= -148/109 ; tau(u)= 366/257 ; -110194*x^2 + 1858*y^2 + 155860*x*z - 110194*z^2
  (-5784/2489 : -59659/2489 : 1)  C2b (73890/40487 : 38075/40487 : 1)
** u= 149/181 ; tau(u)= 213/32 ; 20153*x^2 + 43321*y^2 + 67570*x*z + 20153*z^2
  (-63077/63853 : -50344/63853 : 1)  C1b (29251/284554 : 28557/284554 : 1)
** u= 150/89 ; tau(u)= -28/61 ; 15058*x^2 - 6658*y^2 + 23284*x*z + 15058*z^2
  (-31/84 : -95/84 : 1)  C1a (-340738/883879 : -103431/883879 : 1)
** u= -151/121 ; tau(u)= 393/272 ; -125167*x^2 + 6481*y^2 + 177250*x*z - 125167*z^2
  (40129/72447 : -230120/72447 : 1)  C2b (-3109142/250423 : 1335717/250423 : 1)
** u= 153/32 ; tau(u)= 89/121 ; -5873*x^2 - 21361*y^2 + 31330*x*z - 5873*z^2
  (20507/37859 : 25080/37859 : 1)  C1a (-549082615/47179002 : 4048015/3629154 : 1)
** u= 158/37 ; tau(u)= 84/121 ; -4318*x^2 - 22226*y^2 + 32020*x*z - 4318*z^2
  (2757/18968 : -1991/18968 : 1)  C1a (7249/1930 : -135/386 : 1)
** u= 159/50 ; tau(u)= 59/109 ; 1519*x^2 - 20281*y^2 + 28762*x*z + 1519*z^2
  (-8669/168801 : 8030/168801 : 1)  C1a (-2581/7933 : 759/7933 : 1)
** u= -160/153 ; tau(u)= 466/313 ; -170338*x^2 + 21218*y^2 + 242756*x*z - 170338*z^2
  (965/1109 : 232728/114227 : 1)  C2b (9/2 : -209/206 : 1)
** u= 161/8 ; tau(u)= 145/153 ; -20897*x^2 - 25793*y^2 + 46946*x*z - 20897*z^2
  (79009/128645 : 5844/128645 : 1)  C1a (291886/215027 : -28783/215027 : 1)
** u= -161/153 ; tau(u)= 467/314 ; -171271*x^2 + 20897*y^2 + 244010*x*z - 171271*z^2
  (-28021/1811231 : -5242746/1811231 : 1)  C2b (-173758/1297197 : 377819/1297197 : 1)
** u= 161/169 ; tau(u)= 177/8 ; 25793*x^2 + 31201*y^2 + 57250*x*z + 25793*z^2
  (-47169/42661 : -18668/42661 : 1)  C1b (1467451/1208386 : -240009/1208386 : 1)
** u= -164/117 ; tau(u)= 398/281 ; -131026*x^2 + 482*y^2 + 185300*x*z - 131026*z^2
  (-193/98 : 639/14 : 1)  C2b (781/17646 : -1531/1038 : 1)
** u= 165/16 ; tau(u)= 133/149 ; -17177*x^2 - 26713*y^2 + 44914*x*z - 17177*z^2
  (11987/10795 : -7088/10795 : 1)  C1a (10230034/1693 : 1029497/1693 : 1)
** u= -168/157 ; tau(u)= 482/325 ; -183026*x^2 + 21074*y^2 + 260548*x*z - 183026*z^2
  (-123016/5320071 : 15938425/5320071 : 1)  C2b (1922629/58034 : 513699/58034 : 1)
** u= 168/185 ; tau(u)= 202/17 ; 27646*x^2 + 40226*y^2 + 69028*x*z + 27646*z^2
  (-1042/895 : 551/895 : 1)  C1b (-565127/182954 : 52833/182954 : 1)
** u= 169/32 ; tau(u)= 105/137 ; -8977*x^2 - 26513*y^2 + 39586*x*z - 8977*z^2
  (174929/435449 : -197912/435449 : 1)  C1a (8947/69574 : -6459/69574 : 1)
** u= 172/173 ; tau(u)= 174 ; 29582*x^2 + 30274*y^2 + 59860*x*z + 29582*z^2
  (-287/264 : 35/264 : 1)  C1b (-610793/203145 : -11789/40629 : 1)
** u= -172/193 ; tau(u)= 558/365 ; -236866*x^2 + 44914*y^2 + 340948*x*z - 236866*z^2
  (60368/116509 : -193431/116509 : 1)  C2b (-133159/235942 : -73163/235942 : 1)
** u= 173/181 ; tau(u)= 189/8 ; 29801*x^2 + 35593*y^2 + 65650*x*z + 29801*z^2
  (-26497/20221 : -7620/20221 : 1)  C1b (-32825814/2719075 : 667967/543815 : 1)
** u= 174 ; tau(u)= 172/173 ; -29582*x^2 - 30274*y^2 + 59860*x*z - 29582*z^2
  (764/861 : 11/123 : 1)  C1a (86859109/44087885 : -1648443/8817577 : 1)
** u= 174/53 ; tau(u)= 68/121 ; 994*x^2 - 24658*y^2 + 34900*x*z + 994*z^2
  (-409/15336 : -11/216 : 1)  C1a (-46714/1274199 : -337/3651 : 1)
** u= 174/61 ; tau(u)= 52/113 ; 4738*x^2 - 22834*y^2 + 32980*x*z + 4738*z^2
  (-1896/13439 : 8339/94073 : 1)  C1a (5894/3431 : 4723/24017 : 1)
** u= 174/101 ; tau(u)= -28/73 ; 19618*x^2 - 9874*y^2 + 31060*x*z + 19618*z^2
  (-1546/1179 : 1333/1179 : 1)  C1a (429388290/18897941 : 60785135/18897941 : 1)
** u= -176/197 ; tau(u)= 570/373 ; -247282*x^2 + 46642*y^2 + 355876*x*z - 247282*z^2
  (21039/1523209 : 3472558/1523209 : 1)  C2b (-1249409/248606 : -307297/248606 : 1)
** u= 177/8 ; tau(u)= 161/169 ; -25793*x^2 - 31201*y^2 + 57250*x*z - 25793*z^2
  (5209/7661 : -1508/7661 : 1)  C1a (156278/225583 : -21939/225583 : 1)
** u= 182/121 ; tau(u)= -60/61 ; 25682*x^2 - 3842*y^2 + 36724*x*z + 25682*z^2
  (2120/8643 : 26543/8643 : 1)  C1a (-23302/9173 : 4347/9173 : 1)
** u= 184/185 ; tau(u)= 186 ; 33854*x^2 + 34594*y^2 + 68452*x*z + 33854*z^2
  (-2037/1787 : 916/12509 : 1)  C1b (657/12319 : -9673/86233 : 1)
** u= 186 ; tau(u)= 184/185 ; -33854*x^2 - 34594*y^2 + 68452*x*z - 33854*z^2
  (482/501 : 487/3507 : 1)  C1a (6814/649 : -4981/4543 : 1)
** u= -187/169 ; tau(u)= 525/356 ; -218503*x^2 + 22153*y^2 + 310594*x*z - 218503*z^2
  (36403/54837 : -121420/54837 : 1)  C2b (743161/190646 : -181327/190646 : 1)
** u= 189/8 ; tau(u)= 173/181 ; -29801*x^2 - 35593*y^2 + 65650*x*z - 29801*z^2
  (15399/22699 : 3844/22699 : 1)  C1a (116111/93249 : -901/7173 : 1)
** u= 189/64 ; tau(u)= 61/125 ; 4471*x^2 - 27529*y^2 + 39442*x*z + 4471*z^2
  (1233/10177 : -5920/10177 : 1)  C1a (663774/243427 : -68047/243427 : 1)
** u= 198/137 ; tau(u)= -76/61 ; 31762*x^2 - 1666*y^2 + 44980*x*z + 31762*z^2
  (-269/126 : 6131/882 : 1)  C1a (-3914/5799 : -11603/40593 : 1)
138
>

ここからは、 "A^4+B^4+C^4=3362*D^4の整点" と同様なので、最終的に得られた(1)の整点のみを記述する。
ここで、対応する整点が見つかった各有理数uについて、0 <= A <= B <=C 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。

u=-8/29のとき

222601^4+2408274^4+2863999^4=13778*292549^4
33774164352954757181603324246847925129901613009143833375760453^4+38585427006986034364571206015869031658082115694811115164617122^4+46666762183537493518960131104507003277403678502338208198211253^4=13778*4948299025116306621234264359633998540151926368640469567839803^4
1647696054852734331539658863167013373712004728933048175539840897850633629191102700278133196163412532959453725275224109899648516913228851579954119369835527325905673256141834^4+2440335681426518905318169117936509552180290509854043790132056414377663784070295603821222139192715287767359497360401285183644308592016604495739456052752259948522543486778559^4+3017978408691407090334137134584089183354419346555433302852540129646991985263579830350371512350894384399794064879955245609775602525558824424463570016816048083602578843375041^4=13778*309114290430890840191309105911115646532491076534340619164241037145482992089089732716557637920195210840556319510082002227031178403145397476558243506580583199410246805081609^4
...

u=96/113のとき

119495384902^4+710769404341^4+1863324838933^4=13778*172889470611^4
10858539351671^4+18812592313607^4+43672407082658^4=13778*4068988742601^4
899845672612156501940358798814179175950237961^4+2848880022885609948243685649965165968885824642^4+7290073773006129542485909078008929822010591911^4=13778*676804286491662129399515488528046774580078393^4
631687913623090903089667595831516784423047195476678^4+804951234902616487477212258124739811143472775708273^4+1497481886471447136650176791050619619045018374649561^4=13778*142036082302511592325069503522034652734943191396869^4
2971337102647465424269573409708455481200899933261669359728768414874498419129249^4+4011199880783500963728708761603033071641575266616442977129661802609886773279346^4+8062733767433340968458452582531736015678372203642659814371975481500150833535567^4=13778*758598401363612632870759533310638407976843212900858728190950967627188033395057^4
873786834988008868696443175663211964259325858586525240931193489221266456502947193898^4+1060251999075905785372738302168971788742701144576846921286946060025427258289657464977^4+1648268022632209611192655312063266625599226865666229850181180387231230372202488348969^4=13778*160869996358558772208960121115545311703448325466783526581638187496320626966808499901^4
10979889977766935135998839677842650807330424708721917905534431330218084038922964691047146255548660436731662762^4+23508014031133890210387772586643270786054308663705336895601160513214009567615666021443377005879456785991744301^4+57501462330226830569777675723990800075709414853451977617014446661815884709526834664326277935799796214930780429^4=13778*5345816976019897114409380834901803972621947594164467640976217199806559574936094073912813226027746344457608347^4
12762978579416288948274047213686540612997077401703678289451355774814761349841028157650347333657023250742894568883002^4+19175127697903284298076633715737574428421107634051190570370162526371895207870830664205476558828620990370607562677561^4+41822697629138236879060751579889390282947759417663120149153521418412600184123411841997535426354955519332755846815249^4=13778*3910283479791508203608477079309482797217831638150421678415464528487157371588166912035724119460105716568095594869637^4
2695115167384878828303663796900058486602503956102894967783118815259275332062603521871141141315187737405432338183339819705719^4+3277511096465092385810862801759487319005076619237301576777153384435283948666371402128020126565570075326631417122156284302314^4+4670344358249726444524300809484758872660870005076317838837215785542577959037452860367213019256241096391284583726907336227263^4=13778*464955769924017994358119574740254579722891889992674597404775296440430496399124080420252408014189525077221863018404685832773^4
20433281997143442191584049013355057316919740878176726771779692380633668348505702232916748150853116933579516139638387001099681^4+25928741445418524253105516849566255255722052178012940042845474243433772912963701503906756338803169070307959280862452988311303^4+30528943902617961780990340250697877775287935434920319657136842862064685469226991000647696949485862687521636748250225939808442^4=13778*3227461749111690709343271426402367727740600381899203951992767632876728710050214473774126355749037881744106784821110349543789^4
169406191608333654643982459483321319581782030620476715388016250136784315193192954552784578137078013988631131005218857534074948144156770706104210198405554225817021837019739681^4+14192690607254497183623183266587279411399304691389160460910975754005331394770933967840252889603465782266630060518175897568736597145477863656774164188286332374225989973252454447^4+37543167959902925567032213884742769528805583022464576843907565248618959011289398328828648770986229673710157476734866111670516127405172943362576883319436732923082052985154550546^4=13778*3482807811034607672642738382540045994053704136995258408477512742530117263675974852805157218150706646018605670627960140533690753486744325423121511025221126477247529222744366913^4
...

[2025.11.05追記]u=96/113のときの整点を追加した。

[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.
[8]StarkExchange MATHEMATICS, "a^4+b^4+c^4=2*d^2 such that a,b,c,d are all nonzero Integers & a+b+c!=0", 2024/04/26.

Last Update: 2025.11.10

H.Nakao