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Integer Points on A^4+6*B^4=C^4+6*D^4

[2026.05.20]A^4+6*B^4=C^4+6*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=6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362個のuが抽出される。
[MAGMAによる計算]
> PP(6,1,100);
**u= 1/2 , ht= 1 ; ( 0 , 5/2 ), ( 11/5 , -6 )
**u= 1/4 , ht= 13 ; ( 13/3 , -46/3 ), ( -3/13 , 46/13 )
**u= 1/6 , ht= 2 ; ( -359/322 , 4091/644 ), ( -2 , -19/2 )
**u= 1/10 , ht= 67 ; ( -53/67 , 468/67 ), ( -181/34 , -2019/68 )
**u= 1/16 , ht= 11 ; ( -7/53 , -741/106 ), ( -1/11 , 153/22 )
**u= 1/20 , ht= 49 ; ( 49/29 , -441/29 ), ( -41/441 , -70/9 )
**u= 1/24 , ht= 1 ; ( 1 , -12 ), ( 1586/1727 , -79583/6908 )
**u= 1/26 , ht= 69 ; ( -74/17 , 1341/34 ), ( 43/69 , 718/69 )
**u= 1/30 , ht= 227 ; ( 227/92 , -4647/184 ), ( 92/227 , -4647/454 )
**u= 1/46 , ht= 151 ; ( 53/168 , 4139/336 ), ( 36/151 , -3647/302 )
**u= 1/48 , ht= 11 ; ( 5/11 , -145/11 ), ( -101/155 , 444/31 )
**u= 1/52 , ht= 241 ; ( -175/241 , 3720/241 ), ( -387/205 , 1094/41 )
**u= 1/54 , ht= 1 ; ( -1 , -18 ), ( 8426/8747 , -927503/52482 )
**u= 1/60 , ht= 1369 ; ( -491/1369 , 19513/1369 ), ( -1543/407 , 21410/407 )
**u= 1/70 , ht= 197 ; ( 197/22 , 5745/44 ), ( 382/557 , 19575/1114 )
**u= 1/74 , ht= 301 ; ( -17910/331 , 533789/662 ), ( -301/291 , -6238/291 )
**u= 1/84 , ht= 5011 ; ( 9995/1679 , 160887/1679 ), ( 5011/97 , 79563/97 )
**u= 1/94 , ht= 607 ; ( -703/2164 , 76419/4328 ), ( 513/607 , -13346/607 )
**u= 1/96 , ht= 1 ; ( -1 , 24 ), ( -3106/2497 , -49187/1816 )
**u= 1/100 , ht= 289 ; ( -289/193 , 30096/965 ), ( -291/191 , -6029/191 )
**u= 2/3 , ht= 1 ; ( -1 , 3 ), ( -1 , 3 )
**u= 2/27 , ht= 41 ; ( -25/41 , -917/123 ), ( -41/25 , 917/75 )
**u= 2/35 , ht= 39 ; ( -31/39 , 361/39 ), ( -783/127 , -5749/127 )
**u= 2/43 , ht= 389 ; ( -6105/689 , -49333/689 ), ( 389/131 , -3297/131 )
**u= 2/67 , ht= 889 ; ( -1083/1061 , -15199/1061 ), ( -889/585 , -10669/585 )
**u= 2/75 , ht= 4817 ; ( -4439/4817 , -347391/24085 ), ( -15373/14459 , -1119229/72295 )
**u= 3/2 , ht= 1 ; ( -1 , -2 ), ( -1/7 , -8/7 )
**u= 3/8 , ht= 10 ; ( -13/10 , 37/8 ), ( 10 , -115/4 )
**u= 3/32 , ht= 1 ; ( 1 , -8 ), ( -214/143 , 11651/1144 )
**u= 3/38 , ht= 31 ; ( 196/367 , 5131/734 ), ( -31/7 , -28 )
**u= 3/50 , ht= 2498 ; ( -2399/2498 , -244903/24980 ), ( -2402/2501 , -245197/25010 )
**u= 3/56 , ht= 305 ; ( -1585/1991 , 19045/1991 ), ( 199/305 , 545/61 )
**u= 3/68 , ht= 473 ; ( 1943/301 , 377/7 ), ( -473/377 , 172/13 )
**u= 3/92 , ht= 215 ; ( 199/215 , 562/43 ), ( -323/47 , 3131/47 )
**u= 3/98 , ht= 1921 ; ( -9407/9602 , -1862983/134428 ), ( -1882/1921 , -372713/26894 )
**u= 4/31 , ht= 130 ; ( 161/130 , -399/52 ), ( 130/37 , -1305/74 )
**u= 4/55 , ht= 74 ; ( 74/19 , -981/38 ), ( -201/206 , 3697/412 )
**u= 4/71 , ht= 505 ; ( -505/418 , -9567/836 ), ( -931/134 , -13731/268 )
**u= 5/2 , ht= 7 ; ( 7 , 0 ), ( 1/7 , 0 )
**u= 5/4 , ht= 33 ; ( 33 , 44 ), ( -1/33 , -4/3 )
**u= 5/6 , ht= 56 ; ( 23/56 , -239/112 ), ( 272/149 , 1213/298 )
**u= 5/8 , ht= 7 ; ( -7 , 15 ), ( 1/7 , 15/7 )
**u= 5/14 , ht= 183 ; ( -347/101 , -1038/101 ), ( 183 , -536 )
**u= 5/16 , ht= 113 ; ( -113/79 , 426/79 ), ( -7/221 , 1359/442 )
**u= 5/22 , ht= 67 ; ( -2047/956 , 16371/1912 ), ( -67/21 , 256/21 )
**u= 5/28 , ht= 601 ; ( 293/601 , 2745/601 ), ( 2057/1259 , -9897/1259 )
**u= 5/34 , ht= 2693 ; ( 9822/3379 , -93695/6758 ), ( 2693/806 , 25431/1612 )
**u= 5/42 , ht= 439 ; ( -103/439 , 2266/439 ), ( 1213/824 , -143/16 )
**u= 5/44 , ht= 183 ; ( 241/197 , -1599/197 ), ( -183/169 , -1280/169 )
**u= 5/46 , ht= 481 ; ( -481/2 , -5049/4 ), ( 817/311 , 4596/311 )
**u= 5/52 , ht= 101 ; ( 23/101 , 579/101 ), ( 217/659 , 3873/659 )
**u= 5/58 , ht= 1331 ; ( -612/1331 , 17291/2662 ), ( 3929/2508 , -55009/5016 )
**u= 5/62 , ht= 37 ; ( -1439/607 , -9522/607 ), ( -6/37 , -457/74 )
**u= 5/66 , ht= 223 ; ( -206/223 , 3821/446 ), ( -217/311 , -2386/311 )
**u= 5/72 , ht= 509 ; ( 1058/491 , -30657/1964 ), ( -283/509 , -3827/509 )
**u= 5/78 , ht= 2179 ; ( -2401/1538 , -39007/3076 ), ( -278/2179 , -30045/4358 )
**u= 5/86 , ht= 88 ; ( 23813/54171 , -425146/54171 ), ( -88/41 , -1395/82 )
**u= 5/88 , ht= 368 ; ( 9943/14761 , 129336/14761 ), ( 368/291 , 13637/1164 )
**u= 5/96 , ht= 128 ; ( -49/128 , -8323/1024 ), ( 328/91 , -2953/104 )
**u= 5/98 , ht= 23 ; ( 1/23 , 1236/161 ), ( -11109/13082 , -1842383/183148 )
**u= 6 , ht= 13 ; ( -13/11 , 1/11 ), ( -11/13 , -1/13 )
**u= 6/5 , ht= 7 ; ( 29/9 , 43/9 ), ( -7/3 , 13/3 )
**u= 6/25 , ht= 1339 ; ( -1339/1237 , -32213/6185 ), ( -1363/1261 , -32837/6305 )
**u= 6/29 , ht= 99 ; ( 99/41 , 407/41 ), ( -149/33 , -53/3 )
**u= 6/49 , ht= 1 ; ( 1 , 7 ), ( -331/475 , -20051/3325 )
**u= 8/3 , ht= 31 ; ( 7/31 , -97/62 ), ( 13/43 , 11/86 )
**u= 8/27 , ht= 1 ; ( -1 , 9/2 ), ( 1 , 9/2 )
**u= 8/67 , ht= 627 ; ( -1971/2987 , -35893/5974 ), ( -43/627 , -6293/1254 )
**u= 8/75 , ht= 1 ; ( 1 , -15/2 ), ( 353/1079 , -60161/10790 )
**u= 8/83 , ht= 905 ; ( -905/423 , -11141/846 ), ( -7311/823 , 82153/1646 )
**u= 10/7 , ht= 29 ; ( -13/29 , 51/29 ), ( 11/67 , 81/67 )
**u= 10/39 , ht= 889 ; ( -3701/1213 , -13263/1213 ), ( 889/473 , 3455/473 )
**u= 10/47 , ht= 1517 ; ( -739/1517 , 6351/1517 ), ( 7529/3393 , 31087/3393 )
**u= 10/71 , ht= 689 ; ( -4563/1259 , 21815/1259 ), ( -689/447 , 3793/447 )
**u= 10/79 , ht= 301 ; ( -301/173 , 1689/173 ), ( 8391/3967 , 45223/3967 )
**u= 12 , ht= 1 ; ( 0 , 5/2 ), ( -12/5 , 11/2 )
**u= 12/5 , ht= 7 ; ( 7/2 , 7/4 ), ( 1/14 , 1/4 )
**u= 12/49 , ht= 69 ; ( 20/69 , -505/138 ), ( 84/155 , -1723/434 )
**u= 12/53 , ht= 310 ; ( -127/1198 , -8807/2396 ), ( -61/310 , -2291/620 )
**u= 12/73 , ht= 264 ; ( -264/101 , 2411/202 ), ( -7364/3075 , 68291/6150 )
**u= 12/77 , ht= 218 ; ( -167/218 , -2411/436 ), ( -82/611 , -5399/1222 )
**u= 12/97 , ht= 296 ; ( -296/189 , 3457/378 ), ( -1204/2385 , -26293/4770 )
**u= 15/4 , ht= 47 ; ( 41/67 , -101/67 ), ( 31/47 , -14/47 )
**u= 15/28 , ht= 283 ; ( -11/283 , 686/283 ), ( -823/409 , -2206/409 )
**u= 15/52 , ht= 449 ; ( -449/97 , 1472/97 ), ( 1301/53 , -4228/53 )
**u= 15/58 , ht= 1153 ; ( -1153/326 , -8123/652 ), ( 1934/223 , -13333/446 )
**u= 15/76 , ht= 403 ; ( -1459/137 , 5695/137 ), ( 403/61 , -1594/61 )
**u= 15/88 , ht= 317 ; ( -317/101 , -1393/101 ), ( -763/29 , -3211/29 )
**u= 15/94 , ht= 293 ; ( 73/1201 , 5228/1201 ), ( 293/11 , 1274/11 )
**u= 16/7 , ht= 5 ; ( 5/2 , 15/8 ), ( -99/190 , -139/152 )
**u= 16/39 , ht= 226 ; ( -226/203 , -3281/812 ), ( -643/98 , -7129/392 )
**u= 16/55 , ht= 81 ; ( -131/606 , -8015/2424 ), ( 81/26 , 1099/104 )
**u= 16/71 , ht= 449 ; ( 449/94 , -6669/376 ), ( -1973/7754 , 116337/31016 )
**u= 16/95 , ht= 389 ; ( 294/389 , -8237/1556 ), ( -55577/10638 , 957377/42552 )
**u= 20/3 , ht= 2 ; ( -13/94 , 363/188 ), ( -2 , 7/2 )
**u= 20/11 , ht= 9 ; ( -2/9 , -29/18 ), ( -9/2 , 29/4 )
**u= 20/19 , ht= 18 ; ( 18 , 55/2 ), ( -503/1834 , 5889/3668 )
**u= 20/43 , ht= 1762 ; ( -6459/1582 , 33233/3164 ), ( -1549/1762 , 11889/3524 )
**u= 20/51 , ht= 598 ; ( 139/598 , -3435/1196 ), ( 71018/4871 , 398743/9742 )
**u= 20/59 , ht= 283 ; ( 283/106 , -1785/212 ), ( 106/283 , -1785/566 )
**u= 24 , ht= 193 ; ( -193/191 , 1/382 ), ( -191/193 , -1/386 )
**u= 24/5 , ht= 11 ; ( 11/9 , 11/18 ), ( -9/11 , -1/2 )
**u= 24/25 , ht= 1 ; ( -1 , -5/2 ), ( 23/31 , -667/310 )
**u= 24/29 , ht= 183 ; ( 183/67 , -709/134 ), ( 421/15 , -1693/30 )
**u= 24/49 , ht= 1 ; ( -1 , -7/2 ), ( -1 , 7/2 )
**u= 24/77 , ht= 25 ; ( 405/449 , 3755/898 ), ( 3/25 , 31/10 )
**u= 25/2 , ht= 11 ; ( -5/11 , -126/55 ), ( -61/48 , 203/96 )
**u= 25/6 , ht= 1 ; ( -29/22 , -29/44 ), ( -1 , 6/5 )
**u= 25/8 , ht= 8 ; ( -8/5 , 99/100 ), ( -5/8 , 99/160 )
**u= 25/12 , ht= 31 ; ( -17/31 , 50/31 ), ( -25/59 , -271/295 )
**u= 25/16 , ht= 79 ; ( -149/43 , -174/43 ), ( -79/59 , -1443/590 )
**u= 25/24 , ht= 1 ; ( -37/2 , 1139/40 ), ( 1 , -12/5 )
**u= 25/28 , ht= 73 ; ( 73/39 , 146/39 ), ( 39/73 , 2 )
**u= 25/34 , ht= 16 ; ( 223/87 , 2336/435 ), ( -1/16 , 309/160 )
**u= 25/38 , ht= 130 ; ( 73/130 , -3243/1300 ), ( 130/73 , -3243/730 )
**u= 25/54 , ht= 8761 ; ( 8761/8434 , -928067/253020 ), ( 9386/9059 , 996817/271770 )
**u= 25/56 , ht= 61 ; ( 129/31 , -1694/155 ), ( -44/61 , -3879/1220 )
**u= 25/58 , ht= 338 ; ( 338/309 , 2413/618 ), ( 525/554 , 20119/5540 )
**u= 25/62 , ht= 32 ; ( 1/32 , -177/64 ), ( 3155/179 , 43674/895 )
**u= 25/66 , ht= 244 ; ( 244/167 , -1657/334 ), ( -2119/719 , -31784/3595 )
**u= 25/68 , ht= 509 ; ( 315/587 , 9574/2935 ), ( -509/35 , 7368/175 )
**u= 25/76 , ht= 53 ; ( -53/23 , 867/115 ), ( -23/53 , -867/265 )
**u= 25/78 , ht= 379 ; ( 11675/5357 , -195396/26785 ), ( -128/379 , -2431/758 )
**u= 25/82 , ht= 580 ; ( -5735/1153 , -91092/5765 ), ( 457/580 , -23121/5800 )
**u= 25/92 , ht= 295 ; ( -165/617 , -10667/3085 ), ( 19/295 , -4881/1475 )
**u= 25/94 , ht= 556 ; ( -149/556 , 19431/5560 ), ( 7716/2029 , -269329/20290 )
**u= 25/96 , ht= 1 ; ( -1 , -24/5 ), ( 28538/27959 , 1084925/223672 )
**u= 26/11 , ht= 29 ; ( -29/15 , 5/3 ), ( -25/437 , -135/437 )
**u= 26/19 , ht= 1575 ; ( -157899/33431 , 202027/33431 ), ( -211/1575 , -281/225 )
**u= 26/27 , ht= 5 ; ( -155/473 , 65/33 ), ( -1/5 , -5/3 )
**u= 26/35 , ht= 19 ; ( -19/9 , -41/9 ), ( 1671/61 , 3511/61 )
**u= 26/51 , ht= 23 ; ( 23/11 , -61/11 ), ( -2861/1321 , -7751/1321 )
**u= 26/99 , ht= 2503 ; ( 21365/4359 , 219919/13077 ), ( -1413/2503 , 29057/7509 )
**u= 27/2 , ht= 1 ; ( 1 , 2/3 ), ( -67/43 , -416/129 )
**u= 27/8 , ht= 37 ; ( -73/58 , -797/696 ), ( -34/37 , -521/444 )
**u= 27/32 , ht= 5 ; ( 59/38 , -163/48 ), ( 2/5 , -233/120 )
**u= 27/38 , ht= 299 ; ( 109/299 , -88/39 ), ( -529/41 , -3404/123 )
**u= 27/50 , ht= 1 ; ( -287/38 , 3997/228 ), ( -1 , 10/3 )
**u= 27/68 , ht= 107 ; ( 107/73 , 1063/219 ), ( 181/1201 , -9889/3603 )
**u= 27/98 , ht= 157 ; ( -157/14 , 21709/588 ), ( -202/11 , 28213/462 )
**u= 29/4 , ht= 7 ; ( -151/217 , -342/217 ), ( 7/3 , 13/3 )
**u= 29/6 , ht= 175 ; ( -239/200 , -11/16 ), ( -175/41 , 300/41 )
**u= 29/8 , ht= 70 ; ( -35/131 , -210/131 ), ( -57/70 , 7/8 )
**u= 29/14 , ht= 5 ; ( 82277/33998 , 150519/67996 ), ( 2/5 , -9/10 )
**u= 29/16 , ht= 7 ; ( 3453/5747 , -9671/5747 ), ( 7/3 , 23/6 )
**u= 29/20 , ht= 481 ; ( 1/481 , -804/481 ), ( 793/83 , 1329/83 )
**u= 29/24 , ht= 167 ; ( 167/113 , -303/113 ), ( 1172/59 , -8243/236 )
**u= 29/30 , ht= 3239 ; ( -4187/4102 , 20619/8204 ), ( -514/3239 , -135/82 )
**u= 29/40 , ht= 976 ; ( 976/439 , 8451/1756 ), ( -1241/276 , 10747/1104 )
**u= 29/42 , ht= 1055 ; ( -1055/593 , -2470/593 ), ( 3571/860 , 3169/344 )
**u= 29/44 , ht= 557 ; ( -557/15 , 229/3 ), ( 26595/6839 , 60415/6839 )
**u= 29/46 , ht= 611 ; ( 611/266 , -2841/532 ), ( -866/445 , 4329/890 )
**u= 29/56 , ht= 395 ; ( 277/395 , -234/79 ), ( -5115/1811 , 13285/1811 )
**u= 29/60 , ht= 497 ; ( 367/497 , 1548/497 ), ( 191/4001 , 9784/4001 )
**u= 29/66 , ht= 1274 ; ( -1274/1223 , 9223/2446 ), ( -89891/6997 , -239308/6997 )
**u= 29/70 , ht= 2547 ; ( -75309/10189 , -201662/10189 ), ( -2003/2547 , -8690/2547 )
**u= 29/80 , ht= 217 ; ( 26209/1679 , 3249/73 ), ( 23/217 , 621/217 )
**u= 29/86 , ht= 4625 ; ( 38717/7284 , -232931/14568 ), ( -1137/4625 , -14086/4625 )
**u= 29/94 , ht= 4971 ; ( 4971/4523 , -20942/4523 ), ( -300625/147217 , 1048890/147217 )
**u= 29/100 , ht= 1759 ; ( 1359/1759 , 7162/1759 ), ( -7061/5417 , -143379/27085 )
**u= 30 , ht= 3 ; ( -1/3 , -11/3 ), ( 1141/297 , -4283/297 )
**u= 30/29 , ht= 241 ; ( 869/173 , 1379/173 ), ( -241/223 , -563/223 )
**u= 30/49 , ht= 283 ; ( 283/201 , -5323/1407 ), ( -3/559 , -8387/3913 )
**u= 30/53 , ht= 149 ; ( -2381/163 , -5347/163 ), ( 149/43 , 365/43 )
**u= 30/73 , ht= 107 ; ( -39/107 , 311/107 ), ( -107/39 , -311/39 )
**u= 30/77 , ht= 2759 ; ( -2759/167 , 7573/167 ), ( 38099/523 , -107059/523 )
**u= 32/3 , ht= 1 ; ( -1 , -3/4 ), ( 47/7 , -63/4 )
**u= 32/27 , ht= 1 ; ( -1 , -9/4 ), ( 1555/1619 , -42655/19428 )
**u= 32/35 , ht= 647 ; ( -263/647 , 5307/2588 ), ( 933/373 , -7639/1492 )
**u= 32/75 , ht= 33629 ; ( 33629/32723 , 2487401/654460 ), ( 35677/34771 , -2643049/695420 )
**u= 34/7 , ht= 243 ; ( 65/243 , -415/243 ), ( 1017/5 , 355 )
**u= 34/15 , ht= 233 ; ( 601/61 , 279/61 ), ( -37/233 , -117/233 )
**u= 34/31 , ht= 103 ; ( 3475/1553 , 5859/1553 ), ( -21/103 , -157/103 )
**u= 34/39 , ht= 1733 ; ( 751/10579 , -20871/10579 ), ( -1733/217 , -3433/217 )
**u= 34/71 , ht= 583 ; ( -5035/3769 , 15657/3769 ), ( -269/583 , 1587/583 )
**u= 34/79 , ht= 341 ; ( 341/183 , 1013/183 ), ( -8849/3317 , 25239/3317 )
**u= 34/95 , ht= 2477 ; ( 1527/2477 , -8465/2477 ), ( 14087/54747 , 162131/54747 )
**u= 39/4 , ht= 25 ; ( 21/25 , -7/5 ), ( -355/189 , -100/27 )
**u= 39/10 , ht= 1431 ; ( -1431/1009 , 592/1009 ), ( 1426/2151 , 515/4302 )
**u= 39/16 , ht= 9 ; ( -9/7 , -11/7 ), ( 7/9 , 11/9 )
**u= 39/22 , ht= 45 ; ( 465/223 , 550/223 ), ( -43/45 , -16/9 )
**u= 39/28 , ht= 605 ; ( -605/87 , -745/87 ), ( -2601/515 , -887/103 )
**u= 39/34 , ht= 79 ; ( 2629/912 , -431/96 ), ( 57/79 , 152/79 )
**u= 39/70 , ht= 313 ; ( -993/1457 , 4130/1457 ), ( 313/138 , -1615/276 )
**u= 39/94 , ht= 5495 ; ( -5495/3153 , -16912/3153 ), ( 57219/21875 , 23782/3125 )
**u= 40/7 , ht= 159 ; ( 103/159 , 493/318 ), ( -237/211 , 587/422 )
**u= 40/23 , ht= 197 ; ( 197/59 , -411/118 ), ( 49/233 , 459/466 )
**u= 40/31 , ht= 371 ; ( 371/123 , 1051/246 ), ( -1901/1013 , 7059/2026 )
**u= 40/71 , ht= 2207 ; ( 20337/31 , 91343/62 ), ( 361/2207 , -10059/4414 )
**u= 40/79 , ht= 247 ; ( 89/247 , -1299/494 ), ( 247/89 , -1299/178 )
**u= 48/5 , ht= 26 ; ( -1/26 , -235/104 ), ( -106/109 , -257/436 )
**u= 48/25 , ht= 12 ; ( -5/12 , -389/240 ), ( -3/28 , 89/112 )
**u= 48/29 , ht= 65 ; ( -70/41 , -385/164 ), ( 22/65 , -59/52 )
**u= 48/73 , ht= 309 ; ( 368/441 , -4939/1764 ), ( -56/309 , -2591/1236 )
**u= 48/77 , ht= 158 ; ( -4049/802 , -35119/3208 ), ( -158/73 , -1559/292 )
**u= 49/2 , ht= 31 ; ( -15/31 , 670/217 ), ( -31/15 , 134/21 )
**u= 49/6 , ht= 1 ; ( 1 , -6/7 ), ( 1 , 6/7 )
**u= 49/10 , ht= 93 ; ( -77/93 , -874/651 ), ( 471/86 , -11423/1204 )
**u= 49/20 , ht= 69 ; ( -69 , -1/7 ), ( -1/69 , -1/483 )
**u= 49/24 , ht= 1 ; ( -1 , -12/7 ), ( -1786/1757 , 85517/49196 )
**u= 49/30 , ht= 289 ; ( -289/41 , 2096/287 ), ( -31/449 , -3194/3143 )
**u= 49/34 , ht= 21 ; ( -13/21 , 268/147 ), ( -203/307 , 3456/2149 )
**u= 49/36 , ht= 209 ; ( -209/115 , 453/161 ), ( 695/407 , -8980/2849 )
**u= 49/54 , ht= 1 ; ( -1 , -18/7 ), ( -6034/5969 , 648653/250698 )
**u= 49/64 , ht= 31 ; ( 31/15 , -185/42 ), ( -1765/619 , -53775/8666 )
**u= 49/74 , ht= 311 ; ( 1194/491 , -37459/6874 ), ( -281/311 , 6222/2177 )
**u= 49/76 , ht= 35 ; ( -35/3 , -512/21 ), ( 99/1049 , 15359/7343 )
**u= 49/80 , ht= 7 ; ( -3/7 , 241/98 ), ( -7/3 , -241/42 )
**u= 49/90 , ht= 154 ; ( -154/111 , -18569/4662 ), ( -71/314 , 10341/4396 )
**u= 49/96 , ht= 1 ; ( -1 , -24/7 ), ( 29426/28847 , 5596897/1615432 )
**u= 49/100 , ht= 107 ; ( -107/31 , 9483/1085 ), ( 367/229 , 37791/8015 )
**u= 50/3 , ht= 2551 ; ( -7301/7339 , -24457/36695 ), ( 2521/2551 , 5117/12755 )
**u= 50/11 , ht= 27 ; ( -689/785 , 5103/3925 ), ( 27/5 , -229/25 )
**u= 50/19 , ht= 27 ; ( -85/29 , 141/145 ), ( -11/27 , -13/27 )
**u= 50/27 , ht= 719 ; ( 719/233 , 705/233 ), ( 1327/961 , -6791/2883 )
**u= 50/67 , ht= 1001 ; ( -2067/77 , -19753/385 ), ( -1001/607 , -2397/607 )
**u= 50/83 , ht= 483 ; ( -16199/401 , 175257/2005 ), ( -483/181 , 1177/181 )
**u= 50/91 , ht= 2139 ; ( -2139/227 , 4901/227 ), ( -227/2139 , 4901/2139 )
**u= 51/14 , ht= 195 ; ( -134/195 , 115/78 ), ( -375/319 , -520/319 )
**u= 51/20 , ht= 1013 ; ( 1013/453 , -634/453 ), ( -1413/977 , 2191/977 )
**u= 51/26 , ht= 29 ; ( 29/3 , 22/3 ), ( 4/69 , 103/138 )
**u= 51/50 , ht= 5 ; ( 787/813 , -1946/813 ), ( -5/3 , -52/15 )
**u= 51/56 , ht= 191 ; ( 2651/3141 , -7546/3141 ), ( -191/33 , -34/3 )
**u= 51/74 , ht= 21 ; ( -21/16 , 109/32 ), ( 1601/3303 , 7468/3303 )
**u= 52/7 , ht= 131 ; ( -44/131 , 507/262 ), ( 236/257 , 207/514 )
**u= 52/15 , ht= 43 ; ( 752/887 , -2493/1774 ), ( 32/43 , 65/86 )
**u= 52/23 , ht= 256 ; ( 17237/424 , 15339/848 ), ( -227/256 , 747/512 )
**u= 52/47 , ht= 1192 ; ( 23763/1100 , -13987/440 ), ( -1192/487 , 4539/974 )
**u= 52/55 , ht= 43 ; ( 32/43 , 195/86 ), ( 15792/1987 , 60659/3974 )
**u= 52/63 , ht= 8 ; ( -8 , 29/2 ), ( -3/4288 , 46181/25728 )
**u= 52/87 , ht= 3907 ; ( 17444/653 , -75859/1306 ), ( -3907/1400 , 2717/400 )
**u= 53/2 , ht= 254 ; ( -79/1287 , -4696/1287 ), ( -253/254 , 177/508 )
**u= 53/6 , ht= 113 ; ( -55/113 , -220/113 ), ( 10312/4345 , 753/158 )
**u= 53/8 , ht= 105 ; ( 23/105 , -40/21 ), ( 685/781 , 150/781 )
**u= 53/10 , ht= 47 ; ( -32/47 , 3/2 ), ( 47/32 , -141/64 )
**u= 53/12 , ht= 19 ; ( 13/19 , -28/19 ), ( -1213/233 , 2041/233 )
**u= 53/18 , ht= 266 ; ( 266/129 , -581/774 ), ( 433/154 , 193/44 )
**u= 53/20 , ht= 3359 ; ( 6571/1931 , -912/1931 ), ( 1149/3359 , -1033/3359 )
**u= 53/22 , ht= 23 ; ( -1/23 , -36/23 ), ( 23 , -36 )
**u= 53/28 , ht= 111 ; ( 7159/173 , 5727/173 ), ( -97/111 , 178/111 )
**u= 53/30 , ht= 3179 ; ( -3179/931 , 3236/931 ), ( -12653/3547 , -20580/3547 )
**u= 53/40 , ht= 2044 ; ( 908/3973 , 27573/15892 ), ( 319/2044 , -10575/8176 )
**u= 53/42 , ht= 3667 ; ( 7027/1667 , -9726/1667 ), ( -2155/3667 , 6120/3667 )
**u= 53/44 , ht= 7503 ; ( -6995/7503 , 16325/7503 ), ( 8087/4105 , -3060/821 )
**u= 53/48 , ht= 241 ; ( -241/31 , 359/31 ), ( 31/241 , 359/241 )
**u= 53/52 , ht= 1165 ; ( -1165/83 , -1824/83 ), ( 751/5093 , -8067/5093 )
**u= 53/58 , ht= 817 ; ( -43052/8165 , -29631/3266 ), ( 817/24 , -3161/48 )
**u= 53/62 , ht= 54 ; ( -136541/308123 , -656676/308123 ), ( -54/23 , -229/46 )
**u= 53/68 , ht= 30757 ; ( 98393/3771 , 183169/3771 ), ( -30757/20107 , -73542/20107 )
**u= 53/70 , ht= 6771 ; ( -6771/19 , -12818/19 ), ( -58243/44587 , 147816/44587 )
**u= 53/72 , ht= 7 ; ( -2/7 , -61/28 ), ( 1575141/449347 , -10293661/1348041 )
**u= 53/80 , ht= 789 ; ( 551/789 , -4147/1578 ), ( 1549/2639 , 63/26 )
**u= 53/94 , ht= 79 ; ( -14265/56894 , 276845/113788 ), ( 79/15 , 38/3 )
**u= 54 , ht= 973 ; ( -973/971 , 1/2913 ), ( -971/973 , -1/2919 )
**u= 54/25 , ht= 379 ; ( -379/277 , -7253/4155 ), ( -2323/2221 , -57797/33315 )
**u= 54/29 , ht= 205 ; ( 433/27 , -1079/81 ), ( -201/205 , -1087/615 )
**u= 54/49 , ht= 1 ; ( -575/761 , -33949/15981 ), ( 1 , 7/3 )
**u= 58/7 , ht= 57 ; ( -41/57 , 91/57 ), ( -1251/947 , 1913/947 )
**u= 58/47 , ht= 1327 ; ( 4983/2351 , -7867/2351 ), ( -1327/177 , -2329/177 )
**u= 60 , ht= 2 ; ( 1/2 , 19/4 ), ( 3101/1502 , 29759/3004 )
**u= 60/29 , ht= 41 ; ( 6328/2181 , -10669/4362 ), ( 12/41 , 65/82 )
**u= 60/49 , ht= 26 ; ( -26/23 , 107/46 ), ( -47/46 , -1445/644 )
**u= 60/73 , ht= 1078 ; ( 1078/421 , -4235/842 ), ( -3194/3157 , 223/82 )
**u= 60/77 , ht= 444 ; ( -444/73 , -23/2 ), ( -24163/6684 , -102565/13368 )
**u= 60/97 , ht= 59 ; ( 142/239 , 1243/478 ), ( -38/59 , -305/118 )
**u= 64/31 , ht= 874 ; ( 1499/538 , 10323/4304 ), ( -211/874 , 5259/6992 )
**u= 64/39 , ht= 194 ; ( -82/217 , 2901/1736 ), ( -194/157 , -2823/1256 )
**u= 64/47 , ht= 701 ; ( 4450/673 , -44865/5384 ), ( -701/90 , 1913/144 )
**u= 64/55 , ht= 241 ; ( -33551/8394 , -397579/67152 ), ( -186/241 , 3797/1928 )
**u= 68/3 , ht= 208 ; ( -35/208 , 1389/416 ), ( 5273/2912 , 29921/5824 )
**u= 68/11 , ht= 1132 ; ( 725/1132 , -3585/2264 ), ( 2001/1000 , -1369/400 )
**u= 68/27 , ht= 235 ; ( 817/140 , -61/168 ), ( -235/148 , -2195/888 )
**u= 68/43 , ht= 196 ; ( -196/19 , 417/38 ), ( 11/1860 , -3913/3720 )
**u= 68/67 , ht= 2413 ; ( -27453/10804 , 94859/21608 ), ( -2413/200 , -9003/400 )
**u= 73/2 , ht= 443 ; ( -427/443 , 534/443 ), ( -1787/296 , 15093/592 )
**u= 73/4 , ht= 1835 ; ( 1835/1803 , -160/1803 ), ( 9081/1805 , 5429/361 )
**u= 73/8 , ht= 60 ; ( -60/59 , 169/236 ), ( 107/115 , -3/115 )
**u= 73/10 , ht= 154 ; ( -154/149 , 231/298 ), ( 149/154 , -3/4 )
**u= 73/18 , ht= 337 ; ( -120/337 , -3263/2022 ), ( 613/53 , 1018/53 )
**u= 73/22 , ht= 450 ; ( -397/450 , 253/180 ), ( 738/727 , -1997/1454 )
**u= 73/28 , ht= 131 ; ( 131/33 , -7/33 ), ( -1175/2087 , 1650/2087 )
**u= 73/30 , ht= 203 ; ( -5129/436 , 1889/872 ), ( 67/203 , -108/203 )
**u= 73/40 , ht= 153 ; ( 103/153 , -260/153 ), ( -153/103 , 260/103 )
**u= 73/42 , ht= 6422 ; ( -2594/9313 , -30387/18626 ), ( 3923/6422 , 17353/12844 )
**u= 73/48 , ht= 1477 ; ( 497/1739 , -5853/3478 ), ( 1147/1477 , 2498/1477 )
**u= 73/62 , ht= 126812 ; ( -163703/26668 , 467913/53336 ), ( 126812/14313 , 450967/28626 )
**u= 73/68 , ht= 7935 ; ( 32027/6833 , -49713/6833 ), ( -3421/7935 , 13459/7935 )
**u= 73/70 , ht= 196318 ; ( 492551/172966 , -1640817/345932 ), ( 196318/63493 , 749565/126986 )
**u= 73/72 , ht= 5424 ; ( 5424/1835 , -21971/4404 ), ( 17215/8409 , -103945/25227 )
**u= 73/82 , ht= 872 ; ( 872/257 , 3147/514 ), ( -48967/34304 , 224403/68608 )
**u= 73/90 , ht= 5162 ; ( -5162/3487 , 23465/6974 ), ( -6603/85322 , -932455/511932 )
**u= 73/92 , ht= 1431 ; ( -27305/6781 , -52107/6781 ), ( -823/1431 , 3124/1431 )
**u= 73/100 , ht= 32185 ; ( 32185/149 , -311409/745 ), ( -202689/86639 , -460366/86639 )
**u= 74/19 , ht= 499 ; ( 1323/2311 , -3529/2311 ), ( -499/71 , 819/71 )
**u= 74/35 , ht= 5951 ; ( -5951/2589 , 5423/2589 ), ( -2927/10373 , 705/943 )
**u= 74/51 , ht= 517 ; ( -1585/149 , 1853/149 ), ( -517/401 , 981/401 )
**u= 74/75 , ht= 109 ; ( -109/25 , -901/125 ), ( -35857/5315 , -339689/26575 )
**u= 74/91 , ht= 137 ; ( -32007/12037 , -62899/12037 ), ( -137/45 , -289/45 )
**u= 74/99 , ht= 321 ; ( 6521/427 , -12473/427 ), ( 321/35 , 2027/105 )
**u= 75/2 , ht= 1 ; ( 131/142 , 487/284 ), ( -1 , -2/5 )
**u= 75/8 , ht= 889 ; ( -889/874 , -12221/17480 ), ( -986/1001 , -14029/20020 )
**u= 75/14 , ht= 145 ; ( -314/295 , -2701/2950 ), ( -121/145 , -268/725 )
**u= 75/32 , ht= 1 ; ( -1 , -8/5 ), ( -1 , 8/5 )
**u= 75/44 , ht= 215 ; ( -137/215 , -1856/1075 ), ( -109/463 , 475/463 )
**u= 75/56 , ht= 73 ; ( 10/73 , -2503/1460 ), ( 149/275 , 2141/1375 )
**u= 75/86 , ht= 67 ; ( -67/19 , -122/19 ), ( 2105/1073 , 22712/5365 )
**u= 75/92 , ht= 163 ; ( 6151/3943 , 13688/3943 ), ( 163/113 , 1939/565 )
**u= 75/98 , ht= 1 ; ( -1 , -14/5 ), ( 574/349 , -95171/24430 )
**u= 78/53 , ht= 8709 ; ( 8709/4363 , 12313/4363 ), ( -14337/2623 , -24067/2623 )
**u= 78/73 , ht= 161 ; ( -161/131 , 341/131 ), ( -743/133 , 1373/133 )
**u= 78/77 , ht= 3879 ; ( 9029/867 , 14239/867 ), ( 3879/337 , 7243/337 )
**u= 80/11 , ht= 862 ; ( -689/862 , -4863/3448 ), ( -2658/1049 , 20023/4196 )
**u= 80/27 , ht= 527 ; ( 527/514 , -8725/6168 ), ( 81361/40402 , 501895/161608 )
**u= 80/43 , ht= 702 ; ( 702/229 , 2741/916 ), ( 1261/3282 , -13501/13128 )
**u= 80/51 , ht= 2282 ; ( -9457/3446 , -46117/13784 ), ( 1721/2282 , -14893/9128 )
**u= 80/67 , ht= 254 ; ( 254/53 , -1455/212 ), ( -386/753 , 4979/3012 )
**u= 80/83 , ht= 1022 ; ( -8251/298 , 53577/1192 ), ( -389/1022 , 7257/4088 )
**u= 80/91 , ht= 334 ; ( -1398/19 , -9643/76 ), ( -334/303 , -3353/1212 )
**u= 80/99 , ht= 1643 ; ( -929/2162 , 18803/8648 ), ( -1206/1643 , -46349/19716 )
**u= 82/7 , ht= 387 ; ( 121/387 , -913/387 ), ( -387/121 , 83/11 )
**u= 82/23 , ht= 25 ; ( -21/25 , 7/5 ), ( 1811/63 , 419/9 )
**u= 82/39 , ht= 625 ; ( -625/131 , -435/131 ), ( -3109/145 , 979/29 )
**u= 82/63 , ht= 257 ; ( 229/257 , -531/257 ), ( -1147/337 , -2019/337 )
**u= 82/95 , ht= 4927 ; ( -4927/3053 , 10503/3053 ), ( 17657/5573 , 36231/5573 )
**u= 87/22 , ht= 494 ; ( -494/331 , -97/662 ), ( 2797/3875 , 1762/3875 )
**u= 87/28 , ht= 1033 ; ( -1243/619 , -238/619 ), ( -1033/361 , 1616/361 )
**u= 87/40 , ht= 1081 ; ( 1741/46 , 3773/184 ), ( -49/1081 , -589/1081 )
**u= 87/76 , ht= 2675 ; ( -2675/61 , 3830/61 ), ( 7559/605 , 2707/121 )
**u= 87/82 , ht= 1120 ; ( -1171/1376 , -6167/2752 ), ( 1120/13 , -4105/26 )
**u= 87/88 , ht= 89 ; ( 125/37 , 211/37 ), ( 89/34 , 703/136 )
**u= 87/100 , ht= 3835 ; ( 4019/1681 , -7723/1681 ), ( -3835/1679 , -40499/8395 )
**u= 96 , ht= 3073 ; ( -3073/3071 , 1/12284 ), ( -3071/3073 , -1/12292 )
**u= 96/25 , ht= 1 ; ( -1721/1823 , -47347/36460 ), ( -1 , -5/4 )
**u= 96/29 , ht= 127 ; ( 127/69 , -43/276 ), ( -143/147 , -761/588 )
**u= 96/49 , ht= 1 ; ( -1 , -7/4 ), ( -1 , 7/4 )
**u= 97/2 , ht= 89 ; ( -79/89 , -204/89 ), ( 1697/1369 , -4968/1369 )
**u= 97/8 , ht= 95 ; ( 44/95 , 9/4 ), ( 5111/2153 , -11742/2153 )
**u= 97/10 , ht= 647 ; ( 647/607 , 12/607 ), ( 2179/659 , -4746/659 )
**u= 97/12 , ht= 161 ; ( -47/161 , 326/161 ), ( 161/47 , 326/47 )
**u= 97/18 , ht= 140 ; ( 140/121 , 151/242 ), ( 853/731 , 1106/731 )
**u= 97/22 , ht= 337 ; ( 337/249 , -76/249 ), ( 2107/2437 , -1938/2437 )
**u= 97/28 , ht= 41 ; ( 1627/969 , 383/969 ), ( 29/41 , 27/41 )
**u= 97/40 , ht= 163 ; ( -1633177/30873 , -260740/30873 ), ( -3/163 , -26/163 )
**u= 97/42 , ht= 22 ; ( -286529/11345 , -22050/2269 ), ( -1/22 , -17/44 )
**u= 97/44 , ht= 7395 ; ( -17835/7561 , -14935/7561 ), ( 173/7395 , -26/51 )
**u= 97/50 , ht= 829 ; ( -829/293 , -3912/1465 ), ( -1057/71 , 8388/355 )
**u= 97/56 , ht= 8605 ; ( 8605/131 , -8010/131 ), ( -9019/17125 , -4314/3425 )
**u= 97/58 , ht= 1898 ; ( -1898/1695 , 1327/678 ), ( 2255/442 , 7365/884 )
**u= 97/70 , ht= 14381 ; ( 14381/2166 , 35657/4332 ), ( 1889/16351 , 102/83 )
**u= 97/72 , ht= 4392 ; ( 19408/697 , 96867/2788 ), ( -473/4392 , -66391/52704 )
**u= 97/78 , ht= 4361 ; ( 2009/7988 , 28343/15976 ), ( 3113/4361 , 1138/623 )
**u= 97/88 , ht= 1857 ; ( -1729/1857 , 4216/1857 ), ( -1713/2863 , 5234/2863 )
**u= 97/92 , ht= 8739 ; ( -34261/26533 , 71382/26533 ), ( 7085/8739 , 18610/8739 )
**u= 97/94 , ht= 29728 ; ( -92695/12184 , -290205/24368 ), ( 29728/23335 , 26313/9334 )
**u= 97/98 , ht= 3233 ; ( -7331/1523 , -84138/10661 ), ( -3233/2353 , -7134/2353 )
**u= 98/3 , ht= 275 ; ( 259/275 , 547/385 ), ( 305/193 , -6725/1351 )
**u= 98/27 , ht= 71 ; ( -301/293 , 2593/2051 ), ( 71/41 , 2267/861 )
**u= 98/43 , ht= 1275 ; ( -1375 , -4047/7 ), ( 157/1275 , 4129/8925 )
**u= 98/67 , ht= 115 ; ( -115/19 , -951/133 ), ( 4357/2883 , 55907/20181 )
**u= 98/75 , ht= 173 ; ( -91/173 , 11171/6055 ), ( 77/659 , 29921/23065 )
**u= 100/63 , ht= 235 ; ( -235/102 , 8923/3060 ), ( 243/250 , -14293/7500 )
**u= 100/87 , ht= 430 ; ( -430/353 , -8793/3530 ), ( 1783/130 , -31873/1300 )
362
>

■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。

[2026.05.21追記]u=15/4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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