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Integer Points on A^4+2*B^4=C^4+2*D^4

[2026.05.16]A^4+2*B^4=C^4+2*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=2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305個のuが抽出される。
[MAGMAによる計算]
> PP(2,1,100);
**u= 1/2 , ht= 1 ; ( 0 , 3/2 ), ( 1 , -2 )
**u= 1/6 , ht= 2 ; ( 1/2 , -11/4 ), ( 187/109 , -532/109 )
**u= 1/8 , ht= 21 ; ( -16/21 , -299/84 ), ( -49/64 , -911/256 )
**u= 1/18 , ht= 323 ; ( -288/323 , -11017/1938 ), ( -289/324 , -11051/1944 )
**u= 1/24 , ht= 11 ; ( 11/2 , -219/8 ), ( -2/11 , 219/44 )
**u= 1/30 , ht= 317 ; ( 239/344 , 4589/688 ), ( 92/317 , -3615/634 )
**u= 1/32 , ht= 341 ; ( -320/341 , -21163/2728 ), ( -961/1024 , -63551/8192 )
**u= 1/34 , ht= 81 ; ( 81/47 , 546/47 ), ( 611/120 , 2421/80 )
**u= 1/36 , ht= 63 ; ( 31/63 , -1264/189 ), ( 33/65 , 1312/195 )
**u= 1/38 , ht= 17 ; ( 17/2 , 211/4 ), ( -379/75 , 794/25 )
**u= 1/42 , ht= 37 ; ( -5/37 , 242/37 ), ( -1511/505 , 10326/505 )
**u= 1/44 , ht= 25 ; ( -25/3 , -167/3 ), ( 147/29 , -994/29 )
**u= 1/50 , ht= 833 ; ( -800/833 , -81667/8330 ), ( -2401/2500 , -245099/25000 )
**u= 1/52 , ht= 181 ; ( -525/151 , 3939/151 ), ( -181/105 , 503/35 )
**u= 1/62 , ht= 43 ; ( 3307/517 , 26354/517 ), ( 12/43 , 703/86 )
**u= 1/66 , ht= 4397 ; ( 4715/6827 , -67406/6827 ), ( -4397/1939 , -39042/1939 )
**u= 1/72 , ht= 888 ; ( -888/737 , 117503/8844 ), ( -5041/5184 , -736271/62208 )
**u= 1/74 , ht= 298 ; ( -897/601 , -9288/601 ), ( 109/298 , -5459/596 )
**u= 1/86 , ht= 99 ; ( 397/741 , 7796/741 ), ( 99/13 , 926/13 )
**u= 1/98 , ht= 140 ; ( 140 , 19403/14 ), ( -9409/9604 , -1863371/134456 )
**u= 2 , ht= 1 ; ( -5/3 , 1/3 ), ( 1 , 1 )
**u= 2/9 , ht= 195 ; ( -195/157 , -1589/471 ), ( -203/165 , -1669/495 )
**u= 2/25 , ht= 1 ; ( -1 , -5 ), ( 45/29 , -947/145 )
**u= 2/33 , ht= 37 ; ( -29/37 , -191/37 ), ( -37/29 , 191/29 )
**u= 2/49 , ht= 1 ; ( 1 , 7 ), ( -1 , -7 )
**u= 2/81 , ht= 13443 ; ( -13443/13117 , -1075757/118053 ), ( -13451/13125 , -1076413/118125 )
**u= 4 , ht= 1 ; ( 0 , 3/2 ), ( -4/3 , 3/2 )
**u= 4/9 , ht= 5 ; ( 5/4 , -19/8 ), ( -4/5 , 19/10 )
**u= 4/17 , ht= 25 ; ( 15/32 , 147/64 ), ( -8/25 , -107/50 )
**u= 4/57 , ht= 20 ; ( -25/1856 , -14031/3712 ), ( 1/20 , -151/40 )
**u= 4/81 , ht= 76 ; ( 13/76 , -2083/456 ), ( 17/80 , 2207/480 )
**u= 8 , ht= 65 ; ( -65/63 , 1/126 ), ( -63/65 , -1/130 )
**u= 8/9 , ht= 15 ; ( -15/7 , -13/6 ), ( 7/15 , -91/90 )
**u= 8/25 , ht= 1 ; ( 1 , 5/2 ), ( -541/435 , 4113/1450 )
**u= 8/49 , ht= 1645 ; ( -1645/1579 , -78983/22106 ), ( -5063/4865 , -243349/68110 )
**u= 8/73 , ht= 137 ; ( 515/1829 , -11509/3658 ), ( -9/137 , -827/274 )
**u= 8/81 , ht= 1 ; ( -1 , -9/2 ), ( -13511/13185 , -1081333/237330 )
**u= 8/97 , ht= 129 ; ( 2907/2131 , 25089/4262 ), ( 129/65 , -1007/130 )
**u= 9/2 , ht= 40 ; ( -40/37 , -71/222 ), ( -41/44 , -91/264 )
**u= 9/4 , ht= 9 ; ( 9/7 , 2/3 ), ( -7/9 , -14/27 )
**u= 9/8 , ht= 23 ; ( -8/23 , -337/276 ), ( -89/104 , -1471/1248 )
**u= 9/16 , ht= 33 ; ( 33/79 , -361/237 ), ( -33/31 , -182/93 )
**u= 9/20 , ht= 73 ; ( 13/73 , 347/219 ), ( 119/89 , 674/267 )
**u= 9/22 , ht= 111 ; ( -485/54 , -4387/324 ), ( 111 , -542/3 )
**u= 9/32 , ht= 1 ; ( -1 , 8/3 ), ( -1 , -8/3 )
**u= 9/46 , ht= 165 ; ( -165/157 , -1544/471 ), ( -1347/539 , -9908/1617 )
**u= 9/50 , ht= 2459 ; ( -2360/2459 , -241081/73770 ), ( -2441/2540 , -249019/76200 )
**u= 9/64 , ht= 111 ; ( -65/129 , 1159/387 ), ( -47/111 , 961/333 )
**u= 9/76 , ht= 315 ; ( -315/103 , 2881/309 ), ( -111/383 , 3466/1149 )
**u= 9/82 , ht= 168 ; ( -168/37 , 3107/222 ), ( -37/168 , -3107/1008 )
**u= 9/98 , ht= 451 ; ( 520/293 , 82631/12306 ), ( 451/304 , -10777/1824 )
**u= 10/9 , ht= 19 ; ( 143/21 , 263/63 ), ( 19/17 , -25/17 )
**u= 10/17 , ht= 31 ; ( -31/3 , -37/3 ), ( 109/143 , -229/143 )
**u= 10/41 , ht= 159 ; ( 77/159 , -361/159 ), ( -899/167 , 1877/167 )
**u= 10/97 , ht= 3241 ; ( 29769/41797 , -22851/5971 ), ( -3241/1107 , 3563/369 )
**u= 16/33 , ht= 31 ; ( 20/31 , 7/4 ), ( 124/91 , 899/364 )
**u= 16/41 , ht= 25 ; ( -25/16 , 187/64 ), ( -209/168 , -577/224 )
**u= 16/49 , ht= 32 ; ( 15/32 , 1761/896 ), ( -31/48 , -923/448 )
**u= 16/57 , ht= 103 ; ( -68/103 , 939/412 ), ( 2363/368 , -18391/1472 )
**u= 16/81 , ht= 80 ; ( 80 , -713/4 ), ( -1/80 , 713/320 )
**u= 16/89 , ht= 1112 ; ( 1739/48 , -16279/192 ), ( -1112/401 , 11221/1604 )
**u= 17/4 , ht= 23 ; ( 83/93 , -86/93 ), ( -23/7 , 34/7 )
**u= 17/6 , ht= 133 ; ( -133/107 , 38/107 ), ( 523/475 , 26/25 )
**u= 17/10 , ht= 26 ; ( -26/19 , 37/38 ), ( 27/62 , -13/124 )
**u= 17/12 , ht= 37 ; ( -5/37 , 44/37 ), ( 37/5 , 44/5 )
**u= 17/16 , ht= 215 ; ( -215/27 , 283/54 ), ( -13/1293 , 276/431 )
**u= 17/18 , ht= 5 ; ( -5/3 , 16/9 ), ( 3/5 , 16/15 )
**u= 17/20 , ht= 33 ; ( 33/47 , -66/47 ), ( -13/33 , 1 )
**u= 17/28 , ht= 219 ; ( -1009/8133 , 11419/8133 ), ( -219/145 , -349/145 )
**u= 17/30 , ht= 23 ; ( 907/112 , 2231/224 ), ( -23/7 , 34/7 )
**u= 17/38 , ht= 123 ; ( 123/86 , -441/172 ), ( 2849/1065 , -1572/355 )
**u= 17/52 , ht= 653 ; ( -113637/4949 , 193566/4949 ), ( 653/159 , 401/53 )
**u= 17/60 , ht= 91 ; ( 4139/869 , -7798/869 ), ( 41/91 , -185/91 )
**u= 17/66 , ht= 139 ; ( 29/139 , 284/139 ), ( 191/3656 , -14187/7312 )
**u= 17/70 , ht= 382 ; ( -587/342 , -2737/684 ), ( 382/87 , -537/58 )
**u= 17/72 , ht= 329 ; ( -95/329 , 713/329 ), ( 329/95 , 713/95 )
**u= 17/74 , ht= 59250 ; ( -59250/1567 , -244041/3134 ), ( 256097/117825 , -197726/39275 )
**u= 17/76 , ht= 407 ; ( 217/407 , -982/407 ), ( -9031/5371 , 22349/5371 )
**u= 17/84 , ht= 227 ; ( -739/899 , 2592/899 ), ( -227/143 , 599/143 )
**u= 17/88 , ht= 249 ; ( 855/1306 , 14259/5224 ), ( 249/7 , 572/7 )
**u= 17/94 , ht= 608 ; ( 10177/3832 , 50827/7664 ), ( 608/373 , -3367/746 )
**u= 18 , ht= 325 ; ( -325/323 , 1/969 ), ( -323/325 , -1/975 )
**u= 18/25 , ht= 1027 ; ( -1027/925 , -24101/13875 ), ( -1675/1573 , -40949/23595 )
**u= 18/49 , ht= 1 ; ( -1 , 7/3 ), ( -1 , -7/3 )
**u= 18/73 , ht= 523 ; ( -85/523 , -3247/1569 ), ( -4553/119 , 1643/21 )
**u= 20 , ht= 83 ; ( -46/83 , 439/166 ), ( -261/98 , 1535/196 )
**u= 20/33 , ht= 61 ; ( -61/38 , 177/76 ), ( -347/566 , -1633/1132 )
**u= 20/41 , ht= 39 ; ( -39/2 , -105/4 ), ( 229/222 , 305/148 )
**u= 20/49 , ht= 34 ; ( 51/2 , -153/4 ), ( -33/34 , -61/28 )
**u= 20/57 , ht= 34 ; ( -847/46 , 2775/92 ), ( -23/34 , -137/68 )
**u= 20/81 , ht= 238 ; ( -131/238 , 3307/1428 ), ( -242/321 , -14501/5778 )
**u= 20/97 , ht= 83 ; ( -769/3402 , -15509/6804 ), ( -14/83 , -367/166 )
**u= 25/2 , ht= 104 ; ( -104/103 , -41/206 ), ( -313/316 , -127/632 )
**u= 25/4 , ht= 65 ; ( 87/91 , -69/91 ), ( 65/33 , -172/55 )
**u= 25/8 , ht= 88 ; ( -88/83 , -1157/1660 ), ( -361/376 , -5279/7520 )
**u= 25/14 , ht= 4 ; ( -93/205 , -1206/1025 ), ( 4/3 , -3/2 )
**u= 25/18 , ht= 24 ; ( 24/11 , -409/330 ), ( -11/48 , 91/288 )
**u= 25/26 , ht= 434 ; ( -434/125 , -3593/1250 ), ( -553/110 , 6869/1100 )
**u= 25/32 , ht= 49 ; ( -9/49 , -639/490 ), ( -217/144 , -277/128 )
**u= 25/36 , ht= 93 ; ( 1/93 , -373/279 ), ( -93 , 373/3 )
**u= 25/38 , ht= 161 ; ( -211/74 , -2513/740 ), ( 66/161 , 1973/1610 )
**u= 25/42 , ht= 79 ; ( 682/157 , 8317/1570 ), ( -47/79 , -114/79 )
**u= 25/56 , ht= 104 ; ( 55/104 , 3619/2080 ), ( -380/187 , 1187/340 )
**u= 25/66 , ht= 1517 ; ( 5344/1945 , -89827/19450 ), ( -1517/595 , 13582/2975 )
**u= 25/68 , ht= 91 ; ( -91/79 , 989/395 ), ( 113/91 , -241/91 )
**u= 25/72 , ht= 1 ; ( -1 , 12/5 ), ( -5353/5496 , -156115/65952 )
**u= 25/84 , ht= 155 ; ( -659/55 , 5927/275 ), ( 43/155 , 1446/775 )
**u= 25/94 , ht= 32 ; ( -15/32 , 693/320 ), ( -236189/40923 , 157496/13641 )
**u= 25/98 , ht= 513 ; ( -3032/3097 , -600883/216790 ), ( 513/284 , 16393/3976 )
**u= 26 , ht= 23 ; ( 133/249 , -761/249 ), ( 23/11 , -73/11 )
**u= 26/9 , ht= 19 ; ( -19/17 , -11/17 ), ( -211/193 , -197/193 )
**u= 26/33 , ht= 1681 ; ( -3125/703 , -3061/703 ), ( 1681/1021 , -2369/1021 )
**u= 26/73 , ht= 1873 ; ( -317/1873 , 3277/1873 ), ( -737/3205 , 5351/3205 )
**u= 26/81 , ht= 205 ; ( 205/119 , -59/17 ), ( -679/1077 , -19999/9693 )
**u= 32 , ht= 1 ; ( -1 , -1/4 ), ( -43/3 , 229/4 )
**u= 32/9 , ht= 1 ; ( -825/863 , -8611/10356 ), ( -1 , -3/4 )
**u= 32/17 , ht= 93 ; ( 93/59 , -159/236 ), ( -59/93 , 53/124 )
**u= 32/25 , ht= 1 ; ( -1 , -5/4 ), ( -2375/2273 , -59149/45460 )
**u= 32/33 , ht= 95 ; ( -95/7 , -283/28 ), ( -7/95 , -283/380 )
**u= 32/49 , ht= 1 ; ( -1325/1259 , -62983/35252 ), ( -1 , 7/4 )
**u= 32/81 , ht= 12423 ; ( -12423/12097 , -992117/435492 ), ( -14471/14145 , -1160053/509220 )
**u= 34/33 , ht= 179 ; ( -179/19 , -123/19 ), ( -241/199 , 323/199 )
**u= 34/41 , ht= 623 ; ( 1483/171 , 1337/171 ), ( 623/279 , -277/93 )
**u= 34/49 , ht= 141 ; ( 405/709 , 7269/4963 ), ( 37/141 , 363/329 )
**u= 34/57 , ht= 79 ; ( -41/79 , -121/79 ), ( -251/205 , -427/205 )
**u= 34/89 , ht= 763 ; ( 763/229 , 1249/229 ), ( 604873/1631 , -1013713/1631 )
**u= 36/17 , ht= 28 ; ( -23/28 , 179/168 ), ( 844/385 , 6011/2310 )
**u= 36/49 , ht= 81 ; ( -840/193 , -36749/8106 ), ( -81/32 , 667/192 )
**u= 36/65 , ht= 67 ; ( 67/28 , 79/24 ), ( -5992/1013 , 52423/6078 )
**u= 36/97 , ht= 85 ; ( -85/12 , 817/72 ), ( -3504/1055 , 37061/6330 )
**u= 40/9 , ht= 19 ; ( -17/19 , -35/38 ), ( 51/7 , 475/42 )
**u= 40/17 , ht= 23 ; ( 11/23 , 55/46 ), ( -109/143 , 11/26 )
**u= 40/33 , ht= 119 ; ( -53/119 , -17/14 ), ( -119/163 , 323/326 )
**u= 40/41 , ht= 59 ; ( -23/59 , -149/118 ), ( -583/1139 , 2201/2278 )
**u= 41/6 , ht= 179 ; ( -73/1106 , 4167/2212 ), ( 179/53 , -324/53 )
**u= 41/12 , ht= 61 ; ( 73/83 , -79/83 ), ( -61/25 , -81/25 )
**u= 41/24 , ht= 46 ; ( -37/46 , -9/8 ), ( -1081/1447 , 1058/1447 )
**u= 41/26 , ht= 159 ; ( -159/127 , 138/127 ), ( -383/735 , -118/245 )
**u= 41/28 , ht= 171 ; ( 513/107 , -57/107 ), ( 101/171 , 2/3 )
**u= 41/34 , ht= 103 ; ( 4400/949 , 4763/1898 ), ( 103/16 , 247/32 )
**u= 41/36 , ht= 233 ; ( 621/13 , 1036/39 ), ( -233/35 , 281/35 )
**u= 41/40 , ht= 514 ; ( -514/89 , -1465/356 ), ( 777/802 , 4375/3208 )
**u= 41/42 , ht= 655 ; ( -1651/1541 , -2248/1541 ), ( -73/655 , -488/655 )
**u= 41/46 , ht= 293 ; ( 1041/512 , 2139/1024 ), ( -259/293 , -404/293 )
**u= 41/58 , ht= 22 ; ( 1963/1267 , 376/181 ), ( 7/22 , 49/44 )
**u= 41/60 , ht= 97 ; ( -97/53 , 125/53 ), ( 5441/11249 , 13978/11249 )
**u= 41/66 , ht= 32 ; ( -32 , 73/2 ), ( 151/47 , -216/47 )
**u= 41/70 , ht= 23 ; ( 353/143 , -466/143 ), ( -2/23 , -55/46 )
**u= 41/78 , ht= 301 ; ( -301/284 , -1137/568 ), ( 43432/11023 , -130899/22046 )
**u= 41/80 , ht= 1439 ; ( 267/1583 , 4755/3166 ), ( 1439/159 , -716/53 )
**u= 41/90 , ht= 427 ; ( 1206/71 , -10171/426 ), ( -427/93 , -2032/279 )
**u= 41/94 , ht= 1105 ; ( 2899/2147 , -5386/2147 ), ( -23/1105 , 1592/1105 )
**u= 41/96 , ht= 73 ; ( 1415/1009 , -5239/2018 ), ( -1/73 , 213/146 )
**u= 49/2 , ht= 400 ; ( -400/399 , -797/5586 ), ( -1201/1204 , -2411/16856 )
**u= 49/8 , ht= 253 ; ( 245/253 , 1261/1771 ), ( -439/420 , -3099/3920 )
**u= 49/10 , ht= 103 ; ( -102/103 , 969/1442 ), ( 107/102 , -23/28 )
**u= 49/12 , ht= 31 ; ( 1/31 , 328/217 ), ( 221/125 , 2024/875 )
**u= 49/18 , ht= 877 ; ( -840/877 , 33289/36834 ), ( -1561/1524 , 57949/64008 )
**u= 49/26 , ht= 17 ; ( 285/181 , 852/1267 ), ( 9/17 , 4/119 )
**u= 49/30 , ht= 169 ; ( 169/64 , 53/896 ), ( 64/169 , -53/2366 )
**u= 49/32 , ht= 1 ; ( -80/59 , -3637/3304 ), ( -1 , 8/7 )
**u= 49/36 , ht= 103 ; ( 103/21 , -727/441 ), ( -129/95 , -3256/1995 )
**u= 49/46 , ht= 135 ; ( -135/49 , 732/343 ), ( 7/375 , 558/875 )
**u= 49/48 , ht= 331 ; ( 331/181 , -2216/1267 ), ( 3467/1397 , 30372/9779 )
**u= 49/50 , ht= 140 ; ( -400/433 , -42467/30310 ), ( 47/140 , -8203/9800 )
**u= 49/54 , ht= 263 ; ( 169/263 , 2486/1841 ), ( 509/83 , 13406/1743 )
**u= 49/60 , ht= 49 ; ( -49 , -310/7 ), ( 1/49 , -310/343 )
**u= 49/66 , ht= 260 ; ( 260/73 , 3837/1022 ), ( -703/65 , 6466/455 )
**u= 49/72 , ht= 1 ; ( 0 , 113/84 ), ( 3983/113 , -332/7 )
**u= 49/80 , ht= 13 ; ( -561/131 , 9399/1834 ), ( 13/3 , -43/7 )
**u= 49/90 , ht= 3 ; ( -61/99 , 3418/2079 ), ( -3/2 , 211/84 )
**u= 50 , ht= 1 ; ( 1 , 1/5 ), ( -1 , 1/5 )
**u= 50/9 , ht= 1 ; ( 1 , 3/5 ), ( 1 , 3/5 )
**u= 50/17 , ht= 189 ; ( -81/217 , -9/7 ), ( 155/189 , -31/105 )
**u= 50/49 , ht= 1 ; ( -1 , 7/5 ), ( -7499/7301 , -365149/255535 )
**u= 50/81 , ht= 597 ; ( 597/179 , 6547/1611 ), ( -15947/15621 , -256217/140589 )
**u= 52/17 , ht= 18 ; ( 1/18 , 49/36 ), ( -231/262 , 257/524 )
**u= 52/33 , ht= 166 ; ( -166/67 , -93/134 ), ( 458/535 , 1009/1070 )
**u= 52/41 , ht= 35 ; ( -35/6 , -31/12 ), ( 298/645 , 291/430 )
**u= 52/89 , ht= 642 ; ( -337/642 , -1987/1284 ), ( -7859/1486 , 22529/2972 )
**u= 58/41 , ht= 755 ; ( 755/17 , 13/17 ), ( 557/3263 , 659/3263 )
**u= 58/57 , ht= 4325 ; ( 4325/1489 , -3489/1489 ), ( -157259/50335 , 195159/50335 )
**u= 58/65 , ht= 1493 ; ( -15857/653 , 13049/653 ), ( -847/1493 , -1621/1493 )
**u= 64/9 , ht= 41 ; ( -91/108 , -2923/2592 ), ( 41/40 , 219/320 )
**u= 64/41 , ht= 989 ; ( 989/948 , 8543/7584 ), ( -496/1103 , 3389/8824 )
**u= 64/57 , ht= 91 ; ( -80/91 , -951/728 ), ( -91/80 , -951/640 )
**u= 64/81 , ht= 47 ; ( 47/16 , -1163/384 ), ( -17/80 , -1867/1920 )
**u= 65/4 , ht= 711 ; ( -487/711 , -1492/711 ), ( 1063/141 , -1005/47 )
**u= 65/6 , ht= 263 ; ( -41/263 , 610/263 ), ( 427/419 , -264/419 )
**u= 65/12 , ht= 239 ; ( 157/239 , 321/239 ), ( -32917/1841 , 55911/1841 )
**u= 65/16 , ht= 1309 ; ( 1387/1579 , -1498/1579 ), ( -1309/227 , 1952/227 )
**u= 65/22 , ht= 383 ; ( -197/419 , -524/419 ), ( 383/9 , -172/3 )
**u= 65/28 , ht= 4001 ; ( 2081/13967 , 17530/13967 ), ( 3343/4001 , 2470/4001 )
**u= 65/34 , ht= 129 ; ( 112/129 , 277/258 ), ( -32009/54572 , -30091/109144 )
**u= 65/46 , ht= 5453 ; ( -18097/671 , -1010/671 ), ( -251/5453 , 352/5453 )
**u= 65/48 , ht= 583 ; ( -451/1073 , 2563/2146 ), ( 29/583 , 27/106 )
**u= 65/54 , ht= 151 ; ( 883/86 , -2609/516 ), ( -151/103 , 562/309 )
**u= 65/58 , ht= 46 ; ( 46/17 , 67/34 ), ( -2359/1257 , 978/419 )
**u= 65/64 , ht= 229 ; ( 6319/5023 , -15059/10046 ), ( 207/229 , -1193/916 )
**u= 65/72 , ht= 11 ; ( 11/3 , 29/9 ), ( 1431/8572 , 86047/102864 )
**u= 65/86 , ht= 13538 ; ( -546303/116174 , -1104609/232348 ), ( 13538/5511 , 12301/3674 )
**u= 65/88 , ht= 919 ; ( -17767/6169 , -19400/6169 ), ( -919/137 , 1214/137 )
**u= 68 , ht= 88 ; ( 67/208 , -2297/416 ), ( -88/63 , -239/42 )
**u= 68/33 , ht= 65 ; ( -65/56 , 99/112 ), ( 4208/1205 , -10207/2410 )
**u= 68/49 , ht= 535 ; ( 535/36 , 1363/504 ), ( -523/1072 , -9047/15008 )
**u= 68/57 , ht= 2336 ; ( 223/5296 , 12689/10592 ), ( 113/2336 , -2313/4672 )
**u= 68/73 , ht= 885 ; ( 885/64 , 1389/128 ), ( -592/1671 , 997/1114 )
**u= 68/97 , ht= 1384 ; ( -3095/3404 , -11117/6808 ), ( 317/1384 , -2993/2768 )
**u= 72 , ht= 5185 ; ( -5185/5183 , 1/31098 ), ( -5183/5185 , -1/31110 )
**u= 72/17 , ht= 321 ; ( 321/287 , -1/246 ), ( -399/433 , -889/2598 )
**u= 72/25 , ht= 505 ; ( 655/569 , 9941/17070 ), ( -505/457 , 14311/13710 )
**u= 72/41 , ht= 189 ; ( -189/139 , 781/834 ), ( -1179/461 , 8371/2766 )
**u= 72/49 , ht= 1 ; ( -1 , 7/6 ), ( -10183/9985 , -499349/419370 )
**u= 72/89 , ht= 105 ; ( -22711/12921 , 2179/1062 ), ( 73/105 , 803/630 )
**u= 73/14 , ht= 148 ; ( 199/1431 , -2374/1431 ), ( 148/127 , -299/254 )
**u= 73/16 , ht= 751 ; ( -129/751 , -1173/751 ), ( -2039/2055 , 433/685 )
**u= 73/20 , ht= 241 ; ( 381/359 , 213/359 ), ( -209/241 , 40/241 )
**u= 73/24 , ht= 742 ; ( -1025/1306 , 5517/5224 ), ( -742/187 , -3953/748 )
**u= 73/30 , ht= 493 ; ( 5579/5144 , -8483/10288 ), ( 413/493 , 286/493 )
**u= 73/38 , ht= 927 ; ( 37915/21707 , -8086/21707 ), ( 574/927 , 221/618 )
**u= 73/44 , ht= 1137 ; ( -623/1137 , 1328/1137 ), ( 1237/2447 , 913/2447 )
**u= 73/48 , ht= 11 ; ( -347/121 , -183/242 ), ( 11/5 , -13/5 )
**u= 73/52 , ht= 27 ; ( 27 , 3 ), ( 1747/28605 , 1194/9535 )
**u= 73/54 , ht= 1363 ; ( 1363/82 , 2147/492 ), ( -397/3005 , 894/3005 )
**u= 73/62 , ht= 2225 ; ( 2225/799 , 1486/799 ), ( -18567/140215 , -74962/140215 )
**u= 73/68 , ht= 65 ; ( 65/3 , 41/3 ), ( 2711/1657 , -3446/1657 )
**u= 73/74 , ht= 981 ; ( -981/499 , -936/499 ), ( -20927/15777 , 3122/1753 )
**u= 73/84 , ht= 2755 ; ( -2755/221 , 2348/221 ), ( -15289/1399 , -19286/1399 )
**u= 73/86 , ht= 217 ; ( -13957/8747 , 16414/8747 ), ( -217/127 , 296/127 )
**u= 73/88 , ht= 405 ; ( 17361/278 , -61767/1112 ), ( 277/405 , -56/45 )
**u= 73/90 , ht= 39 ; ( 3974/4149 , 38545/24894 ), ( 39/31 , 172/93 )
**u= 73/96 , ht= 965 ; ( 965/797 , 2791/1594 ), ( 1171/811 , -3429/1622 )
**u= 73/98 , ht= 3757 ; ( 3757/261 , 3716/261 ), ( 3989/10832 , 166301/151648 )
**u= 74/9 , ht= 421 ; ( 417/421 , -713/1263 ), ( -421/417 , -713/1251 )
**u= 74/49 , ht= 203 ; ( 203/71 , -403/497 ), ( -2173/1281 , -5967/2989 )
**u= 74/57 , ht= 1991 ; ( 17819/719 , 6261/719 ), ( 661/1991 , 1049/1991 )
**u= 74/73 , ht= 1943 ; ( -455/2027 , 2497/2027 ), ( -1943/1853 , 2699/1853 )
**u= 80/33 , ht= 766 ; ( -1163/1114 , 3867/4456 ), ( -697/766 , 2263/3064 )
**u= 80/57 , ht= 26 ; ( 26/7 , -5/4 ), ( 7/26 , 35/104 )
**u= 80/73 , ht= 2146 ; ( 9106/857 , -22379/3428 ), ( -847/2146 , 6605/8584 )
**u= 80/81 , ht= 39 ; ( 39/2 , 1015/72 ), ( -118/199 , -815/796 )
**u= 80/97 , ht= 1573 ; ( -1573/1434 , 9227/5736 ), ( -4726/4047 , 9371/5396 )
**u= 81/2 , ht= 3280 ; ( -3280/3277 , -6551/58986 ), ( -3281/3284 , -6571/59112 )
**u= 81/8 , ht= 3232 ; ( -3232/3217 , -45023/115812 ), ( -3329/3344 , -46831/120384 )
**u= 81/10 , ht= 43 ; ( -238/267 , -4933/4806 ), ( 43/18 , -1445/324 )
**u= 81/28 , ht= 79 ; ( -159/191 , -1744/1719 ), ( 79/51 , -821/459 )
**u= 81/32 , ht= 1 ; ( -1 , 8/9 ), ( -1 , -8/9 )
**u= 81/44 , ht= 193 ; ( 817/619 , 5003/5571 ), ( -193/115 , 2003/1035 )
**u= 81/50 , ht= 80 ; ( -80/71 , -6949/6390 ), ( -5681/5780 , -566539/520200 )
**u= 81/52 , ht= 87 ; ( -87/43 , -359/387 ), ( -423/253 , 4459/2277 )
**u= 81/62 , ht= 55 ; ( -55/7 , -26/9 ), ( 1309/8005 , 27916/72045 )
**u= 81/64 , ht= 31 ; ( 951/73 , 3484/657 ), ( 3/31 , 461/1116 )
**u= 81/68 , ht= 2503 ; ( -2503/523 , 12473/4707 ), ( -3671/1225 , 39952/11025 )
**u= 81/74 , ht= 457 ; ( 346/457 , -10633/8226 ), ( -391/793 , 6062/7137 )
**u= 81/86 , ht= 67 ; ( 2843/5251 , -61724/47259 ), ( -67/19 , -758/171 )
**u= 81/98 , ht= 1 ; ( -6128/6323 , -1226857/796698 ), ( -1 , -14/9 )
**u= 81/100 , ht= 33 ; ( -33/17 , 334/153 ), ( 17/33 , 334/297 )
**u= 82/17 , ht= 59 ; ( 43/59 , -71/59 ), ( -59/43 , 71/43 )
**u= 82/33 , ht= 317 ; ( 317/229 , -43/229 ), ( 3077/3011 , 2821/3011 )
**u= 82/57 , ht= 559 ; ( -2405/331 , -123/331 ), ( 217/559 , -243/559 )
**u= 82/65 , ht= 2649 ; ( 2649/881 , -1497/881 ), ( 4997/6843 , 2189/2281 )
**u= 89/6 , ht= 527 ; ( -505/527 , 452/527 ), ( 527/505 , 452/505 )
**u= 89/10 , ht= 561 ; ( -239/561 , -1090/561 ), ( 2241/2126 , 3637/4252 )
**u= 89/24 , ht= 214 ; ( 67/214 , 1201/856 ), ( 2542/1949 , 11103/7796 )
**u= 89/28 , ht= 2897 ; ( -3221/2981 , -1922/2981 ), ( -2897/1933 , -3349/1933 )
**u= 89/30 , ht= 6157 ; ( -6157/5528 , 6979/11056 ), ( 16067/15757 , -13614/15757 )
**u= 89/36 , ht= 99 ; ( 99/73 , -1/3 ), ( 6643/9325 , -292/9325 )
**u= 89/38 , ht= 1551 ; ( -1551/1114 , -879/2228 ), ( 222038/86997 , 180311/57998 )
**u= 89/40 , ht= 1751 ; ( -1138/1757 , 7963/7028 ), ( -1609/1751 , 1418/1751 )
**u= 89/42 , ht= 767 ; ( -535/767 , 856/767 ), ( 2519/4066 , 1/76 )
**u= 89/52 , ht= 2913 ; ( -2913/2833 , 3042/2833 ), ( -9407/4155 , 3693/1385 )
**u= 89/58 , ht= 1243 ; ( 1243/547 , -496/547 ), ( 4653/2710 , -10931/5420 )
**u= 89/66 , ht= 4480 ; ( -4480/41 , 2327/82 ), ( 10973/1964 , 26133/3928 )
**u= 89/72 , ht= 2165 ; ( -2165/363 , 3122/1089 ), ( 363/2165 , 3122/6495 )
**u= 89/76 , ht= 26321 ; ( -26321/5003 , -14899/5003 ), ( -33169/17827 , 40853/17827 )
**u= 90 , ht= 41 ; ( 3/41 , -823/123 ), ( 1051/567 , -17813/1701 )
**u= 90/17 , ht= 2281 ; ( -1187/2281 , 10081/6843 ), ( -2803/1039 , -13291/3117 )
**u= 90/73 , ht= 107 ; ( -5077/471 , 6929/1413 ), ( -39/107 , -199/321 )
**u= 90/89 , ht= 217 ; ( 139/217 , 121/93 ), ( -49/307 , 665/921 )
**u= 97/4 , ht= 497 ; ( -285/2099 , 7254/2099 ), ( 497/27 , 577/9 )
**u= 97/6 , ht= 1462 ; ( 1271/2399 , 5824/2399 ), ( -1462/1415 , 2323/2830 )
**u= 97/8 , ht= 595 ; ( 595/587 , -17/587 ), ( 1007/544 , 497/128 )
**u= 97/12 , ht= 143 ; ( 6575/6379 , 347/6379 ), ( 139/143 , 19/143 )
**u= 97/14 , ht= 47 ; ( -33/47 , -66/47 ), ( 47/33 , -2 )
**u= 97/18 , ht= 329 ; ( 1053/1181 , 3334/3543 ), ( 329/328 , -409/656 )
**u= 97/22 , ht= 5623 ; ( -4431/5623 , -6168/5623 ), ( -9413/5811 , -4064/1937 )
**u= 97/38 , ht= 15874 ; ( 5557/15874 , 39649/31748 ), ( -43619/11443 , -55298/11443 )
**u= 97/44 , ht= 1351 ; ( -251/1351 , 239/193 ), ( 21287/3397 , 26411/3397 )
**u= 97/48 , ht= 451 ; ( -4199/2653 , -2453/5306 ), ( -451/125 , -546/125 )
**u= 97/50 , ht= 111 ; ( -276/349 , 765/698 ), ( -111/17 , -674/85 )
**u= 97/54 , ht= 221 ; ( -221/109 , -74/327 ), ( 2366/3335 , -12509/20010 )
**u= 97/56 , ht= 1788 ; ( 6595/4096 , 13657/16384 ), ( 1788/143 , -8587/572 )
**u= 97/68 , ht= 31 ; ( -20055/3643 , -3723/3643 ), ( 3/31 , 1/31 )
**u= 97/74 , ht= 1297 ; ( -4091/1829 , 2558/1829 ), ( 113/1297 , -446/1297 )
**u= 97/78 , ht= 10991 ; ( 77177/5911 , -33702/5911 ), ( -10991/2503 , -13168/2503 )
**u= 97/90 , ht= 1559 ; ( 22826/4071 , 90391/24426 ), ( 639/1559 , -3728/4677 )
**u= 97/92 , ht= 11269 ; ( -11269/277 , -7322/277 ), ( -93/31049 , -20153/31049 )
**u= 97/96 , ht= 122 ; ( -409/1529 , 1891/1529 ), ( -122/65 , -1247/520 )
**u= 98 , ht= 9605 ; ( -9605/9603 , 1/67221 ), ( -9603/9605 , -1/67235 )
**u= 98/9 , ht= 1 ; ( -1 , -3/7 ), ( -1 , -3/7 )
**u= 98/25 , ht= 155 ; ( 147/155 , 4479/5425 ), ( -1315/1197 , -4581/4655 )
**u= 98/81 , ht= 3843 ; ( -3843/3517 , -288557/221571 ), ( -23051/22725 , -1863613/1431675 )
**u= 100/9 , ht= 41 ; ( 17/144 , -2039/864 ), ( 41/40 , -273/400 )
**u= 100/33 , ht= 241 ; ( -188/241 , 511/482 ), ( 10336/3571 , 134911/35710 )
**u= 100/97 , ht= 417 ; ( -417/68 , -2931/680 ), ( 553/708 , -2753/2360 )
305
>

■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。

[参考文献]

Last Update: 2026.06.16
H.Nakao

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