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Integer Points on A^4+10*B^4=C^4+10*D^4

[2026.05.23]A^4+10*B^4=C^4+10*D^4の整点

■Diophantine Equation
       A^4+n*B^4=C^4+n*D^4 ----------(1)
を満たす自明でない整数の組(A,B,C,D) (ただし 0 < A < C, 0 < D < B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、A/D=y,B/D=x.C/D=tとすると、
       y^4+n*x^4 = t^4+n ----------(2)
       y^4-t^4= y n(1-x^4)
       (y^2+t^2)*(x^2-t^2) = n*(1+x^2)*(1-x^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。

ここで、ある有理数u > 0に対して、
       y^2+t^2 = (n/u)*(1+x^2) ----------(2a)
       y^2-t^2 = u*(1-x^2) ----------(2b)
よって、
       2*u*y^2 = (n+u^2)*x^2+(n-u^2) ----------(3a)
       2*u*t^2 = (n-u^2)*x^2+(n+u^2) ----------(3b)
を満たす有理数の組(x,y,t)が存在すれば、(x,y,t)が(2)を満たすことが分かる。

[pari/gpによる計算]
gp > YY2(n,u,x)
%1 = (1/(2*u)*n - 1/2*u)*x^2 + (1/(2*u)*n + 1/2*u)
gp > TT2(n,u,x)
%2 = (1/(2*u)*n + 1/2*u)*x^2 + (1/(2*u)*n - 1/2*u)
gp > YY2(n,u,x)^2+n*x^4-TT2(n,u,x)^2-n
%3 = 0

■２次曲線(3a),(3b)は、n=u^2のときはsoncularであり、それ以外のときはnon-singularである。
２次曲線(3a)の右辺の判別式は
    -4*(n+u^2)*(n-u^2)
となる。ここで、nが平方数でなければ、任意の有理数uについて、判別式は0にならないので、２次曲線(3a)は常にnon-singularである。
nが平方数であるならば、n=u^2のときに限り、２次曲線(3a)はsingularであり。それ以外のuにつては、non-singularである。

同様に、２次曲線(3b)の右辺の判別式は
    -4*(n-u^2)*(n+u^2)
となる。nが平方数でなければ、任意の有理数uについて、判別式は0にならないので、２次曲線(3b)は常にnon-singularである。
nが平方数であるならば、n=u^2のときに限り、２次曲線(3b)はsingularであり。それ以外のuにつては、non-singularである。

■以下では、n=10とする。

■有理数u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100以下の範囲で、２つの２次曲線(3a)と(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328個のuが抽出される。
[MAGMAによる計算]
> PP(10,1,100);
**u= 1/6 , ht= 19 ; ( -19/2 , 209/4 ), ( -2/19 , -11/2 )
**u= 1/8 , ht= 15 ; ( 35/3 , 74 ), ( 15/4 , 393/16 )
**u= 1/10 , ht= 61 ; ( 61/21 , 152/7 ), ( -159/74 , 2481/148 )
**u= 1/16 , ht= 23 ; ( 3/23 , -415/46 ), ( 11/71 , 1285/142 )
**u= 1/22 , ht= 98 ; ( 43/98 , -2245/196 ), ( 194/29 , 4115/58 )
**u= 1/26 , ht= 278 ; ( -278/177 , -2505/118 ), ( -1269/434 , 30585/868 )
**u= 1/30 , ht= 4 ; ( -1/4 , -101/8 ), ( 4 , -101/2 )
**u= 1/36 , ht= 161 ; ( 161 , 2160 ), ( 3/161 , 6481/483 )
**u= 1/40 , ht= 1 ; ( -1 , 20 ), ( -1 , 20 )
**u= 1/48 , ht= 103 ; ( -505/41 , 7849/41 ), ( -103/23 , 1635/23 )
**u= 1/52 , ht= 479 ; ( -479/431 , 10390/431 ), ( -2029/2911 , -57215/2911 )
**u= 1/58 , ht= 90 ; ( 90/61 , 3703/122 ), ( 22459/5581 , 394100/5581 )
**u= 1/62 , ht= 165 ; ( 83/165 , 1084/55 ), ( -228/445 , -17607/890 )
**u= 1/64 , ht= 77 ; ( 77/57 , 2285/76 ), ( -57/77 , -6855/308 )
**u= 1/66 , ht= 1141 ; ( -1484/769 , -60725/1538 ), ( -107/1141 , -2974/163 )
**u= 1/70 , ht= 167 ; ( 167/118 , 7651/236 ), ( 983/143 , -18584/143 )
**u= 1/84 , ht= 1097 ; ( -1097/121 , 22618/121 ), ( 1979/1703 , -53507/1703 )
**u= 1/90 , ht= 1 ; ( -1 , -30 ), ( -14411/754 , 612247/1508 )
**u= 1/94 , ht= 17669 ; ( 10981/19253 , 480514/19253 ), ( -17669/5643 , -402118/5643 )
**u= 2/5 , ht= 1 ; ( -1131/1229 , 5909/1229 ), ( -1 , -5 )
**u= 2/21 , ht= 125 ; ( 125/83 , 1087/83 ), ( 125/13 , -911/13 )
**u= 2/45 , ht= 1 ; ( 277/27 , 8855/81 ), ( -1 , -15 )
**u= 4 , ht= 3 ; ( 3/2 , 5/4 ), ( -2/3 , -5/6 )
**u= 4/25 , ht= 42 ; ( 42/41 , 3281/410 ), ( 22/53 , 641/106 )
**u= 4/33 , ht= 10 ; ( -2549/1114 , -35715/2228 ), ( -1/10 , 129/20 )
**u= 4/49 , ht= 27 ; ( 26/27 , 1369/126 ), ( -89/58 , 1663/116 )
**u= 4/97 , ht= 1567 ; ( 114819/12290 , 2542859/24580 ), ( -1567/470 , -36031/940 )
**u= 5/2 , ht= 1 ; ( -1 , 2 ), ( -1 , -2 )
**u= 5/8 , ht= 22 ; ( -46/61 , -869/244 ), ( -17/22 , -313/88 )
**u= 5/12 , ht= 11 ; ( 13/19 , -80/19 ), ( -7/11 , -45/11 )
**u= 5/18 , ht= 321 ; ( -286/321 , -10949/1926 ), ( -291/326 , -11119/1956 )
**u= 5/22 , ht= 108 ; ( -13/108 , -341/72 ), ( -108/13 , 1023/26 )
**u= 5/32 , ht= 406 ; ( -406/339 , -7977/904 ), ( -519/679 , -4833/679 )
**u= 5/58 , ht= 837 ; ( 613/837 , -878/93 ), ( -1971/811 , -16236/811 )
**u= 5/72 , ht= 99 ; ( -2/99 , 10085/1188 ), ( -291/106 , -31541/1272 )
**u= 5/92 , ht= 947 ; ( 99027/18527 , 966181/18527 ), ( -157/947 , 9206/947 )
**u= 5/98 , ht= 1 ; ( -474/241 , -73691/3374 ), ( 1 , 14 )
**u= 8/5 , ht= 1 ; ( 1 , 5/2 ), ( 1 , -5/2 )
**u= 8/21 , ht= 41 ; ( -41 , -295/2 ), ( -115/11 , 843/22 )
**u= 8/37 , ht= 713 ; ( 2231/1047 , 7889/698 ), ( 471/713 , 357/62 )
**u= 8/45 , ht= 2867 ; ( -3013/2883 , -132683/17298 ), ( 2037/2867 , -111851/17202 )
**u= 8/69 , ht= 2011 ; ( -1045/2011 , 29777/4022 ), ( -2011/1045 , 2707/190 )
**u= 9/2 , ht= 1 ; ( 0 , 11/6 ), ( -21/11 , 10/3 )
**u= 9/4 , ht= 19 ; ( 13/19 , -112/57 ), ( -7/19 , -71/57 )
**u= 9/10 , ht= 1 ; ( -1 , 10/3 ), ( -1 , -10/3 )
**u= 9/14 , ht= 257 ; ( -177/257 , -2630/771 ), ( -257/177 , 2630/531 )
**u= 9/26 , ht= 134 ; ( 134/69 , 3425/414 ), ( 37/219 , -2518/657 )
**u= 9/28 , ht= 121 ; ( 89/121 , 1780/363 ), ( -2935/1157 , -421/39 )
**u= 9/32 , ht= 83 ; ( -136/75 , 15683/1800 ), ( -83/72 , -11125/1728 )
**u= 9/38 , ht= 10 ; ( -9/10 , -371/60 ), ( -10/9 , 371/54 )
**u= 9/40 , ht= 1 ; ( -39398/39797 , -3167921/477564 ), ( -1 , -20/3 )
**u= 9/56 , ht= 11 ; ( 4421/564 , -297935/6768 ), ( -11/3 , -191/9 )
**u= 9/58 , ht= 65 ; ( 65/7 , -1112/21 ), ( -5/517 , -8794/1551 )
**u= 9/76 , ht= 827 ; ( -827/199 , -16571/597 ), ( -1207/2327 , -51080/6981 )
**u= 9/80 , ht= 229 ; ( 229/159 , 11149/954 ), ( -387/113 , -16135/678 )
**u= 9/88 , ht= 542 ; ( -503/542 , -62045/6504 ), ( -58865/34063 , -1426964/102189 )
**u= 9/98 , ht= 47 ; ( -903/335 , 149192/7035 ), ( 15/47 , -7642/987 )
**u= 9/100 , ht= 313 ; ( -301/313 , -48551/4695 ), ( -349/301 , -10306/903 )
**u= 10 , ht= 1 ; ( 1 , 1 ), ( -1 , 1 )
**u= 10/9 , ht= 179 ; ( -179/141 , -1429/423 ), ( -219/181 , -1829/543 )
**u= 10/13 , ht= 15 ; ( -9/43 , 115/43 ), ( -11/15 , 47/15 )
**u= 10/37 , ht= 115 ; ( -65/1101 , 1587/367 ), ( 33/115 , 513/115 )
**u= 10/49 , ht= 1 ; ( 1401/1591 , 73471/11137 ), ( -1 , 7 )
**u= 10/53 , ht= 155 ; ( 155/57 , 283/19 ), ( -2523/1399 , 14865/1399 )
**u= 10/81 , ht= 13427 ; ( -13427/13101 , -1074445/117909 ), ( -13467/13141 , -1077725/118269 )
**u= 10/93 , ht= 347 ; ( 1375/1933 , -16179/1933 ), ( -25/347 , 2371/347 )
**u= 13/4 , ht= 5 ; ( 27/5 , 4/5 ), ( 1/5 , -1/5 )
**u= 13/6 , ht= 115 ; ( -575/236 , 1541/472 ), ( 29/115 , -6/5 )
**u= 13/12 , ht= 131 ; ( -353/523 , 1385/523 ), ( 131/109 , -370/109 )
**u= 13/14 , ht= 25 ; ( -25/18 , -47/12 ), ( 18/25 , -141/50 )
**u= 13/16 , ht= 673 ; ( -673/275 , -3521/550 ), ( -1123/495 , 6223/990 )
**u= 13/18 , ht= 109 ; ( 53/109 , -324/109 ), ( 859/174 , -14165/1044 )
**u= 13/22 , ht= 348 ; ( -59/381 , 380/127 ), ( 348/271 , 2577/542 )
**u= 13/24 , ht= 428 ; ( 385/428 , -7007/1712 ), ( -368/1463 , 235/76 )
**u= 13/28 , ht= 733 ; ( -877/61 , -2854/61 ), ( -733/313 , 2635/313 )
**u= 13/36 , ht= 71 ; ( 355/579 , 7604/1737 ), ( -69/71 , -1105/213 )
**u= 13/42 , ht= 368 ; ( 368/137 , -3145/274 ), ( -62987/10429 , -257764/10429 )
**u= 13/48 , ht= 12847 ; ( 12847/1627 , -110885/3254 ), ( 44527/47629 , 560159/95258 )
**u= 13/58 , ht= 977 ; ( 1396/207 , 4433/138 ), ( -977/647 , -5540/647 )
**u= 13/60 , ht= 173 ; ( -163/173 , -1142/173 ), ( 16907/347 , 81426/347 )
**u= 13/64 , ht= 5 ; ( -418479/81569 , -4222525/163138 ), ( 1/5 , 101/20 )
**u= 13/70 , ht= 296 ; ( -2721/6064 , -69053/12128 ), ( 296/219 , -3823/438 )
**u= 13/72 , ht= 954 ; ( 1226/7391 , -157945/29564 ), ( -467/954 , 67007/11448 )
**u= 13/84 , ht= 1025 ; ( 11549/8189 , -80440/8189 ), ( 1025/367 , 6194/367 )
**u= 13/86 , ht= 373 ; ( -38395/10539 , -25418/1171 ), ( 72/373 , -4365/746 )
**u= 13/100 , ht= 575 ; ( 575/223 , 19112/1115 ), ( -223/575 , -19112/2875 )
**u= 16/9 , ht= 22 ; ( 7/38 , 295/152 ), ( -9/22 , -421/264 )
**u= 16/25 , ht= 7 ; ( 74/33 , -1489/220 ), ( 7/6 , -517/120 )
**u= 16/73 , ht= 110 ; ( 110/37 , 2213/148 ), ( 274/55 , -5351/220 )
**u= 16/81 , ht= 126 ; ( 142/97 , -10375/1164 ), ( 126/113 , -30661/4068 )
**u= 18/5 , ht= 1 ; ( 1 , 5/3 ), ( -2771/2869 , 13781/8607 )
**u= 18/37 , ht= 81 ; ( -63/233 , -2345/699 ), ( 81/7 , 113/3 )
**u= 18/53 , ht= 77 ; ( -77/29 , -943/87 ), ( 2087/9719 , -113821/29157 )
**u= 18/61 , ht= 1911 ; ( -1911/529 , 24395/1587 ), ( -6509/14805 , -28447/6345 )
**u= 18/77 , ht= 139 ; ( -29/139 , -1975/417 ), ( 211/13 , 2941/39 )
**u= 20/9 , ht= 1 ; ( 0 , 11/6 ), ( 20/11 , -7/2 )
**u= 20/41 , ht= 32 ; ( -9/32 , -215/64 ), ( 448/325 , -3557/650 )
**u= 20/81 , ht= 5 ; ( -52/29 , -535/58 ), ( -5/4 , 173/24 )
**u= 20/89 , ht= 53 ; ( -109/20 , 1043/40 ), ( -53/36 , 605/72 )
**u= 20/93 , ht= 146 ; ( 842/67 , -8127/134 ), ( -146/133 , -1905/266 )
**u= 25/4 , ht= 9 ; ( -9/7 , 2/7 ), ( 11/7 , -19/7 )
**u= 25/6 , ht= 2 ; ( 1/2 , -7/4 ), ( 3061/1477 , 5370/1477 )
**u= 25/16 , ht= 9 ; ( 7/9 , -7/3 ), ( 9/7 , -3 )
**u= 25/34 , ht= 9 ; ( 8/9 , 7/2 ), ( 2889/2872 , 21249/5744 )
**u= 25/46 , ht= 171 ; ( -171/148 , 37/8 ), ( 296/923 , 5809/1846 )
**u= 25/56 , ht= 71 ; ( 701/573 , -1008/191 ), ( 48/71 , 1143/284 )
**u= 25/64 , ht= 267 ; ( -3551/413 , 50779/1652 ), ( -59/267 , -1943/534 )
**u= 25/66 , ht= 829 ; ( -5312 , 38321/2 ), ( -829/227 , -3142/227 )
**u= 25/94 , ht= 709 ; ( 333/709 , 3404/709 ), ( -23764/4107 , -5671/222 )
**u= 25/96 , ht= 518 ; ( 787/109 , 3470/109 ), ( 518/191 , -19403/1528 )
**u= 26 , ht= 9 ; ( 7/87 , 105/29 ), ( 9/7 , -3 )
**u= 26/9 , ht= 79 ; ( 727/151 , -473/151 ), ( -79/47 , -143/47 )
**u= 26/17 , ht= 525 ; ( 401/1095 , -763/365 ), ( 525/389 , 1221/389 )
**u= 26/33 , ht= 2117 ; ( -2117/413 , 5275/413 ), ( 5909/35 , -15341/35 )
**u= 26/57 , ht= 1787 ; ( 9755/1091 , 32167/1091 ), ( 685/1787 , 6287/1787 )
**u= 26/73 , ht= 309 ; ( 26653/10155 , -35453/3385 ), ( 309/275 , 141/25 )
**u= 26/89 , ht= 5989 ; ( 57029/10611 , 79675/3537 ), ( 293/5989 , 24701/5989 )
**u= 32/5 , ht= 49 ; ( -49/39 , -21/52 ), ( 93/77 , -81/44 )
**u= 32/45 , ht= 1 ; ( -1 , 15/4 ), ( -10581/9659 , 456919/115908 )
**u= 32/69 , ht= 307 ; ( -245/307 , 5171/1228 ), ( 2801/1145 , -40047/4580 )
**u= 34/9 , ht= 537 ; ( 499/537 , 2659/1611 ), ( -8233/4763 , -14315/4763 )
**u= 34/25 , ht= 27 ; ( 23/27 , -23/9 ), ( -27/23 , 3 )
**u= 34/57 , ht= 24061 ; ( 24061/15055 , -81533/15055 ), ( -89933/6625 , 265637/6625 )
**u= 34/97 , ht= 8629 ; ( 8629/5145 , 12611/1715 ), ( -30895/8099 , -121273/8099 )
**u= 36 , ht= 49 ; ( -22/49 , 1121/294 ), ( 49/22 , 1121/132 )
**u= 36/17 , ht= 54 ; ( -54/31 , 505/186 ), ( -6/79 , -545/474 )
**u= 36/41 , ht= 59 ; ( 18/59 , -911/354 ), ( 2502/739 , 38543/4434 )
**u= 36/49 , ht= 118 ; ( 11117/5678 , -192275/34068 ), ( -118/77 , 15605/3234 )
**u= 36/65 , ht= 114 ; ( -29/114 , -2149/684 ), ( 10419/3334 , -199661/20004 )
**u= 36/73 , ht= 254 ; ( 254/181 , -5935/1086 ), ( -3097/2002 , 10115/1716 )
**u= 37/6 , ht= 67 ; ( -5/67 , -132/67 ), ( 188/167 , 545/334 )
**u= 37/8 , ht= 27 ; ( -14/27 , -7/4 ), ( 513/7 , 135 )
**u= 37/12 , ht= 59 ; ( -53/59 , -106/59 ), ( 41/901 , -5/17 )
**u= 37/16 , ht= 7 ; ( 1319/41 , 1325/41 ), ( 5/7 , -23/14 )
**u= 37/18 , ht= 534 ; ( -1079/73 , -1286/73 ), ( 281/534 , 4925/3204 )
**u= 37/24 , ht= 175 ; ( -175/23 , -279/23 ), ( -23/175 , 279/175 )
**u= 37/28 , ht= 1049 ; ( -3139/3699 , -3190/1233 ), ( 477/1049 , 2109/1049 )
**u= 37/34 , ht= 2221 ; ( -2221/635 , 4696/635 ), ( -2939/2501 , -8350/2501 )
**u= 37/40 , ht= 832 ; ( -832/293 , 7925/1172 ), ( 97/972 , 8695/3888 )
**u= 37/42 , ht= 5087 ; ( 404/5087 , -25229/10174 ), ( 128381/48413 , 336266/48413 )
**u= 37/44 , ht= 2757 ; ( 15665/639 , 12286/213 ), ( 2757/2075 , -8496/2075 )
**u= 37/46 , ht= 63366 ; ( 108673/2022 , 174725/1348 ), ( 63366/4831 , -326865/9662 )
**u= 37/54 , ht= 445 ; ( 445/68 , -2377/136 ), ( -68/445 , 2377/890 )
**u= 37/58 , ht= 517 ; ( 47201/5209 , 130280/5209 ), ( -517/179 , -1556/179 )
**u= 37/60 , ht= 949 ; ( -2759/311 , -7758/311 ), ( 329/949 , 2817/949 )
**u= 37/72 , ht= 382 ; ( 382/113 , -4915/452 ), ( 2317/454 , -29817/1816 )
**u= 37/76 , ht= 1975 ; ( -1975/343 , 6352/343 ), ( -75535/22049 , -254678/22049 )
**u= 37/80 , ht= 9017 ; ( 9017/2487 , -10157/829 ), ( 4191/9569 , -34098/9569 )
**u= 37/84 , ht= 263 ; ( 263/235 , -1187/235 ), ( 95/577 , -1952/577 )
**u= 37/88 , ht= 183 ; ( -183/136 , 3137/544 ), ( 15289/2727 , 53998/2727 )
**u= 37/90 , ht= 5576 ; ( -5576/2619 , 128207/15714 ), ( 2487/12628 , -931/264 )
**u= 40 , ht= 321 ; ( -321/319 , 1/638 ), ( -319/321 , -1/642 )
**u= 40/9 , ht= 1 ; ( -1 , 3/2 ), ( -519/481 , -4829/2886 )
**u= 40/49 , ht= 1 ; ( -1 , 7/2 ), ( 1 , 7/2 )
**u= 40/77 , ht= 29 ; ( -29/15 , 67/10 ), ( -447/125 , 2913/250 )
**u= 40/81 , ht= 13127 ; ( -13127/12801 , -1049845/230418 ), ( -13767/13441 , -1102325/241938 )
**u= 41/2 , ht= 233 ; ( -300/293 , -37/586 ), ( 233/47 , -740/47 )
**u= 41/4 , ht= 39 ; ( 1207/3237 , 185/83 ), ( -39/23 , 78/23 )
**u= 41/6 , ht= 119 ; ( 1111/938 , 1159/1876 ), ( 118/119 , -281/238 )
**u= 41/12 , ht= 43 ; ( 1781/695 , 869/695 ), ( -23/43 , -35/43 )
**u= 41/14 , ht= 116 ; ( -419/83 , 254/83 ), ( 116/101 , 425/202 )
**u= 41/20 , ht= 271 ; ( 21/271 , 505/271 ), ( -139/389 , -530/389 )
**u= 41/32 , ht= 75 ; ( -19/75 , 109/50 ), ( -771/400 , -14361/3200 )
**u= 41/36 , ht= 429 ; ( -429/89 , 2585/267 ), ( 3877/9801 , -521/243 )
**u= 41/40 , ht= 9 ; ( -1/9 , 7/3 ), ( -380787/44182 , -3555603/176728 )
**u= 41/42 , ht= 1390 ; ( 7367/11855 , -32248/11855 ), ( -907/1390 , 53/20 )
**u= 41/50 , ht= 157 ; ( -60/157 , 4253/1570 ), ( 275/179 , -4106/895 )
**u= 41/54 , ht= 434 ; ( -10091/406 , -150965/2436 ), ( 434/257 , -7873/1542 )
**u= 41/58 , ht= 849 ; ( 559/849 , 910/283 ), ( -4803/8515 , 1974/655 )
**u= 41/60 , ht= 4399 ; ( 9499/2999 , -26425/2999 ), ( -4399/2639 , -2004/377 )
**u= 41/78 , ht= 9064 ; ( -23305/8129 , -75296/8129 ), ( 9064/601 , 56795/1202 )
**u= 41/84 , ht= 2339 ; ( 757/5419 , -17712/5419 ), ( -2339/1007 , 8217/1007 )
**u= 41/86 , ht= 329 ; ( -329/114 , 745/76 ), ( 23766/10247 , -168933/20494 )
**u= 45/2 , ht= 1 ; ( -1 , -2/3 ), ( -1 , 2/3 )
**u= 45/8 , ht= 154 ; ( -154/139 , -1931/1668 ), ( -251/266 , -3739/3192 )
**u= 45/22 , ht= 63 ; ( 13767/2063 , 50576/6189 ), ( -47/63 , 346/189 )
**u= 45/28 , ht= 157 ; ( 3/157 , -932/471 ), ( -237/13 , -1408/39 )
**u= 45/32 , ht= 1 ; ( 1 , 8/3 ), ( -1 , 8/3 )
**u= 45/52 , ht= 189 ; ( -561/401 , 4912/1203 ), ( -189/97 , 1565/291 )
**u= 45/58 , ht= 72 ; ( 72/43 , -1259/258 ), ( -603/1427 , 11548/4281 )
**u= 45/62 , ht= 142 ; ( -41/641 , 5188/1923 ), ( -142/13 , 2303/78 )
**u= 45/68 , ht= 41 ; ( -7/41 , -350/123 ), ( -41/7 , 50/3 )
**u= 45/98 , ht= 9401 ; ( -9206/9401 , -1823989/394842 ), ( -9611/9806 , -1902559/411852 )
**u= 49/4 , ht= 3 ; ( 23/39 , 194/91 ), ( -3 , 51/7 )
**u= 49/10 , ht= 1 ; ( -1 , -10/7 ), ( -1 , 10/7 )
**u= 49/16 , ht= 297 ; ( -3949/317 , -9647/2219 ), ( -41/297 , -76/189 )
**u= 49/18 , ht= 115 ; ( 396/155 , 16343/6510 ), ( -83/115 , -1178/805 )
**u= 49/22 , ht= 79 ; ( -79/31 , -710/217 ), ( -31/79 , 710/553 )
**u= 49/24 , ht= 49 ; ( -133/122 , -7765/3416 ), ( 17/49 , 466/343 )
**u= 49/30 , ht= 197 ; ( 197/157 , 2992/1099 ), ( 296/101 , -8431/1414 )
**u= 49/36 , ht= 151 ; ( -97/381 , -17063/8001 ), ( -9/151 , -5500/3171 )
**u= 49/40 , ht= 11734 ; ( -11734/11331 , -306773/105756 ), ( 31203/30667 , 619527/214669 )
**u= 49/46 , ht= 219 ; ( 219/104 , 545/112 ), ( -1144/673 , -41353/9422 )
**u= 49/48 , ht= 7 ; ( -5/7 , 271/98 ), ( -667/245 , -11437/1715 )
**u= 49/52 , ht= 205 ; ( 92485/21349 , 1468148/149443 ), ( 23/205 , -3179/1435 )
**u= 49/76 , ht= 673 ; ( 673/581 , -17279/4067 ), ( 1761/3521 , -75780/24647 )
**u= 49/78 , ht= 4832 ; ( 5515/4072 , 269189/57008 ), ( 4832/1205 , -200091/16870 )
**u= 49/86 , ht= 801 ; ( -2497/906 , 36261/4228 ), ( -801/113 , 17034/791 )
**u= 49/88 , ht= 419 ; ( 419/166 , -37385/4648 ), ( 1489/1049 , 38405/7343 )
**u= 49/90 , ht= 609 ; ( -26/609 , 78721/25578 ), ( -3951/3934 , 709697/165228 )
**u= 49/96 , ht= 74 ; ( -1439/658 , 274959/36848 ), ( 74/11 , -13275/616 )
**u= 49/100 , ht= 419 ; ( -419/73 , -47011/2555 ), ( 73/419 , -47011/14665 )
**u= 50/9 , ht= 143 ; ( -19/143 , 21/11 ), ( 169/173 , -221/173 )
**u= 50/13 , ht= 17 ; ( 2453/1179 , -95/131 ), ( 9/17 , -9/17 )
**u= 50/41 , ht= 149 ; ( 149/97 , -349/97 ), ( 223/141 , 551/141 )
**u= 50/49 , ht= 1419 ; ( 823/1419 , -8689/3311 ), ( -6243/2399 , -107565/16793 )
**u= 50/81 , ht= 29 ; ( 7/29 , -259/87 ), ( -209/147 , -941/189 )
**u= 50/89 , ht= 287 ; ( 287/109 , -905/109 ), ( -4943/4669 , -20303/4669 )
**u= 50/93 , ht= 919 ; ( 16561/7 , -49769/7 ), ( 919/113 , 2863/113 )
**u= 52/21 , ht= 11 ; ( 50/29 , 137/58 ), ( 10/11 , -41/22 )
**u= 52/29 , ht= 35 ; ( -35/6 , 33/4 ), ( 2595/14 , -9963/28 )
**u= 52/37 , ht= 134 ; ( 97/134 , 643/268 ), ( 27127/3854 , -112739/7708 )
**u= 52/69 , ht= 470 ; ( -3713/818 , 19075/1636 ), ( 263/470 , -2733/940 )
**u= 53/6 , ht= 128 ; ( 1061/1355 , 2194/1355 ), ( 128/77 , 485/154 )
**u= 53/8 , ht= 71 ; ( -741/613 , -352/613 ), ( 63/71 , -57/71 )
**u= 53/12 , ht= 403 ; ( 403/259 , 6/7 ), ( -407/73 , -740/73 )
**u= 53/18 , ht= 91 ; ( 91/27 , 194/81 ), ( -3249/686 , -34765/4116 )
**u= 53/26 , ht= 6389 ; ( -6389/1515 , 2718/505 ), ( -4443/6620 , -22923/13240 )
**u= 53/30 , ht= 7 ; ( -7/2 , -21/4 ), ( -7162/2807 , 4099/802 )
**u= 53/32 , ht= 463 ; ( -463/49 , 692/49 ), ( -731/1178 , -18059/9424 )
**u= 53/36 , ht= 755 ; ( 755/19 , -1232/19 ), ( 24437/13317 , 162650/39951 )
**u= 53/38 , ht= 6807 ; ( -6807/5087 , 15640/5087 ), ( 204332/36391 , 854671/72782 )
**u= 53/42 , ht= 677 ; ( -572/677 , -3575/1354 ), ( 347/1144 , 31/16 )
**u= 53/46 , ht= 1645 ; ( 1645/219 , 1076/73 ), ( 3786/7949 , -35115/15898 )
**u= 53/48 , ht= 1249 ; ( 637/1249 , 6177/2498 ), ( 5189/2129 , 24885/4258 )
**u= 53/60 , ht= 6181 ; ( 1379/7549 , 18912/7549 ), ( 6181/2561 , 83/13 )
**u= 53/62 , ht= 1843 ; ( -1043/6499 , 16462/6499 ), ( -1843/541 , 4786/541 )
**u= 53/68 , ht= 47 ; ( -30289/1151 , 74410/1151 ), ( 47/21 , 19/3 )
**u= 53/78 , ht= 1829 ; ( 1829/386 , 9925/772 ), ( -7223/7129 , -27538/7129 )
**u= 53/82 , ht= 5259 ; ( 4622/5259 , -13017/3506 ), ( 20487/11426 , 131913/22852 )
**u= 53/84 , ht= 695 ; ( 75107/18995 , 214234/18995 ), ( 695/233 , 2096/233 )
**u= 53/92 , ht= 78881 ; ( 78881/9467 , 230251/9467 ), ( 99161/7943 , 297832/7943 )
**u= 53/98 , ht= 243 ; ( -77/243 , -610/189 ), ( -77709/22643 , 249126/22643 )
**u= 58/5 , ht= 93 ; ( -73/93 , -53/31 ), ( 1419/991 , -2697/991 )
**u= 58/13 , ht= 209 ; ( 209/131 , 95/131 ), ( 6117/2489 , -573/131 )
**u= 58/21 , ht= 611 ; ( -611/121 , -455/121 ), ( 1613/767 , 225/59 )
**u= 58/37 , ht= 5299 ; ( 21909/3631 , -34745/3631 ), ( 5299/2561 , -11285/2561 )
**u= 58/45 , ht= 1209 ; ( -42651/3239 , -231059/9717 ), ( -1181/1209 , -9967/3627 )
**u= 58/61 , ht= 4225 ; ( 28625/9227 , 66389/9227 ), ( 443/4225 , -9301/4225 )
**u= 64 , ht= 27 ; ( -14/27 , 349/72 ), ( 27/14 , 1047/112 )
**u= 64/17 , ht= 209 ; ( -209/90 , 113/240 ), ( 279/670 , 267/5360 )
**u= 64/57 , ht= 8890 ; ( -8890/4049 , 157943/32392 ), ( -9577/1082 , 172415/8656 )
**u= 64/73 , ht= 149 ; ( -149/70 , 3067/560 ), ( -1257/4250 , -81897/34000 )
**u= 65/6 , ht= 22 ; ( 551/506 , -51/1012 ), ( 22/19 , -65/38 )
**u= 65/14 , ht= 113 ; ( -97/113 , 178/113 ), ( -209/281 , 224/281 )
**u= 65/16 , ht= 9 ; ( 7/9 , 5/3 ), ( 3369/2519 , -5652/2519 )
**u= 65/24 , ht= 49 ; ( -49/41 , 81/41 ), ( -121/383 , 345/383 )
**u= 65/34 , ht= 573 ; ( 1516/1119 , 1919/746 ), ( 573/148 , -2199/296 )
**u= 65/56 , ht= 83 ; ( 181/216 , 263/96 ), ( 27/83 , -171/83 )
**u= 65/66 , ht= 59 ; ( 4/59 , -279/118 ), ( 59/4 , -279/8 )
**u= 65/84 , ht= 211 ; ( 1531/3547 , -10020/3547 ), ( -41/211 , 531/211 )
**u= 68/5 , ht= 28 ; ( 113/432 , -83/32 ), ( 27/28 , 27/56 )
**u= 68/21 , ht= 152 ; ( 1681/484 , 1455/968 ), ( 47/152 , -145/304 )
**u= 68/29 , ht= 84 ; ( -2268/23 , 4445/46 ), ( 55/84 , 37/24 )
**u= 68/53 , ht= 1321 ; ( 8736/7199 , 43985/14398 ), ( 1321/632 , 6073/1264 )
**u= 68/69 , ht= 220 ; ( -427/128 , -1925/256 ), ( 220/217 , 199/62 )
**u= 68/77 , ht= 1223 ; ( 1223/684 , 2177/456 ), ( -1971/3056 , 17025/6112 )
**u= 68/93 , ht= 2956 ; ( -1924/21109 , -113735/42218 ), ( -2956/383 , -15987/766 )
**u= 72/5 , ht= 1 ; ( -1 , -5/6 ), ( 583/353 , -7849/2118 )
**u= 72/13 , ht= 513 ; ( -599/441 , 1265/2646 ), ( -513/449 , -4609/2694 )
**u= 72/37 , ht= 769 ; ( 1149/187 , -8963/1122 ), ( 177/769 , -6163/4614 )
**u= 74/25 , ht= 19 ; ( 19/3 , 17/5 ), ( 57/1193 , 555/1193 )
**u= 74/33 , ht= 175 ; ( 175/23 , 189/23 ), ( 461/749 , -165/107 )
**u= 74/65 , ht= 1463 ; ( 1463/657 , -357/73 ), ( 3789/539 , 1215/77 )
**u= 74/97 , ht= 31493 ; ( -31493/1035 , -8699/115 ), ( -39483/24925 , -121023/24925 )
**u= 80 , ht= 9 ; ( 8/9 , -35/12 ), ( -96/65 , -1791/260 )
**u= 80/37 , ht= 91 ; ( -266/81 , 49/12 ), ( -54/91 , 81/52 )
**u= 80/41 , ht= 156 ; ( 1279/8 , -6445/32 ), ( 115/156 , -1169/624 )
**u= 80/53 , ht= 219 ; ( 46/219 , 597/292 ), ( -219/46 , 1791/184 )
**u= 80/77 , ht= 43 ; ( -43/34 , 475/136 ), ( 106/909 , 7597/3636 )
**u= 80/93 , ht= 1217 ; ( -16777/6938 , 170433/27752 ), ( -178/1217 , -11433/4868 )
**u= 81/8 , ht= 5 ; ( -3/5 , -89/45 ), ( 4161/2399 , -75250/21591 )
**u= 81/10 , ht= 1 ; ( -13802/13903 , 283601/250254 ), ( -1 , 10/9 )
**u= 81/14 , ht= 11 ; ( -3/11 , -188/99 ), ( 212/9 , 7393/162 )
**u= 81/22 , ht= 59 ; ( -79/31 , 70/279 ), ( 26/59 , -395/1062 )
**u= 81/28 , ht= 149 ; ( -149/89 , -1595/801 ), ( -1951/191 , 31300/1719 )
**u= 81/34 , ht= 92 ; ( 92/41 , -2069/738 ), ( 107/56 , 3623/1008 )
**u= 81/38 , ht= 653 ; ( -4183/738 , -88645/13284 ), ( 558/653 , -22825/11754 )
**u= 81/40 , ht= 1 ; ( -7433/6737 , -138991/60633 ), ( -1 , -20/9 )
**u= 81/44 , ht= 483 ; ( -161/513 , -9016/4617 ), ( -109/483 , -38/27 )
**u= 81/52 , ht= 193 ; ( 193/115 , 3407/1035 ), ( 167/505 , -7696/4545 )
**u= 81/56 , ht= 89 ; ( -89/72 , 7493/2592 ), ( 261/37 , -4834/333 )
**u= 81/64 , ht= 103 ; ( 373/59 , -12439/1062 ), ( -103/83 , -9625/2988 )
**u= 81/76 , ht= 823 ; ( 4249/4477 , -120677/40293 ), ( -823/203 , 17335/1827 )
**u= 81/80 , ht= 897 ; ( -897/563 , 41407/10134 ), ( 5713/4353 , 145589/39177 )
**u= 81/82 , ht= 569 ; ( 569/241 , -12080/2169 ), ( 78901/17155 , -1705972/154395 )
**u= 82/45 , ht= 9 ; ( 9 , -37/3 ), ( -8327/61423 , 84665/61423 )
**u= 82/53 , ht= 8281 ; ( -26507/2763 , 13975/921 ), ( -729/8281 , 1005/637 )
**u= 82/61 , ht= 50833 ; ( -92255/27671 , -171169/27671 ), ( -32737/50833 , -112165/50833 )
**u= 85/22 , ht= 35 ; ( -1369/1260 , 1321/840 ), ( 35/13 , -62/13 )
**u= 85/28 , ht= 297 ; ( 385335/29273 , -147974/29273 ), ( 185/297 , -346/297 )
**u= 85/42 , ht= 148 ; ( 400/253 , 1351/506 ), ( -71/148 , -445/296 )
**u= 85/48 , ht= 25 ; ( -4187/2825 , 15947/5650 ), ( 19/25 , -101/50 )
**u= 85/58 , ht= 1735 ; ( 3329/3225 , -2844/1075 ), ( -1735/208 , 7097/416 )
**u= 85/72 , ht= 165 ; ( 165/37 , 976/111 ), ( 853/430 , -8183/1720 )
**u= 85/78 , ht= 23 ; ( -23/10 , 103/20 ), ( -3151/3713 , -10330/3713 )
**u= 89/6 , ht= 5297 ; ( -1729/5297 , 14014/5297 ), ( -23174/12103 , 1229/266 )
**u= 89/8 , ht= 375 ; ( 23/375 , -306/125 ), ( -437/440 , 1591/1760 )
**u= 89/10 , ht= 669 ; ( 229/669 , -476/223 ), ( -669/229 , -1428/229 )
**u= 89/14 , ht= 436 ; ( 317/436 , 1433/872 ), ( -1307/1451 , 1318/1451 )
**u= 89/24 , ht= 103 ; ( 31/103 , 183/103 ), ( 9118/10477 , -58065/41908 )
**u= 89/28 , ht= 1639 ; ( 3379/369 , -55/41 ), ( -279/1639 , 450/1639 )
**u= 89/32 , ht= 49 ; ( -18900/3847 , -111029/30776 ), ( -25/49 , -109/98 )
**u= 89/36 , ht= 499 ; ( -1249/191 , 1160/191 ), ( 499/211 , 920/211 )
**u= 89/42 , ht= 2990 ; ( 3967/313 , 4560/313 ), ( 1891/2990 , -9767/5980 )
**u= 89/44 , ht= 199 ; ( -87/199 , -386/199 ), ( -79181/2307 , 147806/2307 )
**u= 89/46 , ht= 1717 ; ( -1717/261 , 746/87 ), ( 312192/47761 , -1182957/95522 )
**u= 89/50 , ht= 953 ; ( 1103/145 , 7766/725 ), ( -953/95 , 9188/475 )
**u= 89/54 , ht= 646 ; ( 881/2798 , -11299/5596 ), ( 646/553 , -9071/3318 )
**u= 89/64 , ht= 4129 ; ( -4129/33 , -2344/11 ), ( -6009/407 , 24933/814 )
**u= 89/68 , ht= 3035 ; ( 6729/745 , -12076/745 ), ( 3035/1029 , 6676/1029 )
**u= 89/72 , ht= 499 ; ( 235/3491 , 7551/3491 ), ( -211/499 , -1030/499 )
**u= 89/84 , ht= 11801 ; ( 11801/1847 , 24522/1847 ), ( 55969/19009 , -134000/19009 )
**u= 89/92 , ht= 793 ; ( 23331/12641 , 58765/12641 ), ( 173/793 , -1765/793 )
**u= 89/96 , ht= 20402 ; ( 16289/20402 , -489627/163216 ), ( 23183/4679 , -114117/9358 )
**u= 89/98 , ht= 12755 ; ( -12755/11187 , 13198/3729 ), ( -16975/6393 , 307024/44751 )
**u= 90 , ht= 1621 ; ( -1621/1619 , 1/4857 ), ( -1619/1621 , -1/4863 )
**u= 90/49 , ht= 1 ; ( 169/71 , -793/213 ), ( 1 , 7/3 )
**u= 90/53 , ht= 1377 ; ( -1377/955 , -8179/2865 ), ( -2389/1215 , -14923/3645 )
**u= 90/77 , ht= 111 ; ( 333/25 , -1927/75 ), ( 111/43 , -775/129 )
**u= 98/5 , ht= 1 ; ( 1 , 5/7 ), ( -733/597 , -9901/4179 )
**u= 98/29 , ht= 465 ; ( -481/505 , -6101/3535 ), ( 233/465 , -2491/3255 )
**u= 98/45 , ht= 271 ; ( 271/231 , 10897/4851 ), ( 2313/1183 , -93439/24843 )
**u= 98/69 , ht= 421 ; ( -85/421 , 6143/2947 ), ( 4423/3457 , -75505/24199 )
**u= 100/13 , ht= 56 ; ( 43/56 , -181/112 ), ( -179/172 , 445/344 )
**u= 100/37 , ht= 28 ; ( 28/9 , 17/6 ), ( -36/73 , -165/146 )
**u= 100/81 , ht= 106 ; ( -194/87 , -7301/1566 ), ( 63/106 , 4301/1908 )
**u= 100/93 , ht= 76 ; ( -17/76 , 353/152 ), ( -76/17 , 353/34 )
328
>

■これらのuについて、(2),(3a),(3b)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を順に調べれば良い。

ここからは、A^4+B^4=C^4+D^4と同様なので、最終的に得られた整点のみ記述する。
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, 0 < C < D, 0 < A < Cを満たすように、
A,B,C,Dの符号を変更したり、A,B,C,Dを交換して、Dの小さい順に並び替えると、以下のようになる。

[2026.05.25追記]u=2/5,5/2,5/8,9/2,10/9,4/25,10/37,2/45,9/58のときの整数解を追加した。

[参考文献]

Last Update: 2026.06.16
H.Nakao

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