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Integer Points on A^4+8*B^4=C^4+8*D^4

[2026.05.22]A^4+8*B^4=C^4+8*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=8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507個のuが抽出される。
[MAGMAによる計算]
> PP(8,1,100);
**u= 1 , ht= 5 ; ( 3/5 , -12/5 ), ( -11/3 , -8 )
**u= 1/2 , ht= 1 ; ( -1 , 4 ), ( -1 , -4 )
**u= 1/3 , ht= 11 ; ( -11 , 38 ), ( 25/13 , 98/13 )
**u= 1/5 , ht= 23 ; ( 23/13 , -118/13 ), ( 13/23 , -118/23 )
**u= 1/8 , ht= 1 ; ( 1 , -8 ), ( 33/8 , -769/32 )
**u= 1/12 , ht= 59 ; ( 149/119 , 1321/119 ), ( -59 , 409 )
**u= 1/14 , ht= 59 ; ( 59/52 , -1177/104 ), ( 52/59 , -1177/118 )
**u= 1/15 , ht= 67 ; ( 77/53 , -724/53 ), ( -67/47 , -634/47 )
**u= 1/17 , ht= 129 ; ( -129/41 , 1116/41 ), ( -589/249 , 1758/83 )
**u= 1/18 , ht= 1 ; ( -1224/1295 , -90721/7770 ), ( -1 , -12 )
**u= 1/21 , ht= 179 ; ( 137/179 , 2066/179 ), ( 2705/1277 , -27418/1277 )
**u= 1/26 , ht= 98 ; ( 98/59 , 2333/118 ), ( -105/649 , -6704/649 )
**u= 1/29 , ht= 337 ; ( 119/403 , 146/13 ), ( 279/337 , -4712/337 )
**u= 1/32 , ht= 1 ; ( -1 , 16 ), ( -1 , -16 )
**u= 1/33 , ht= 2815 ; ( 5681/5 , -65266/5 ), ( 2815/2023 , -39828/2023 )
**u= 1/36 , ht= 141 ; ( 17/141 , -5113/423 ), ( 15/143 , -5176/429 )
**u= 1/37 , ht= 389 ; ( 603/877 , 12948/877 ), ( 389/129 , -1662/43 )
**u= 1/38 , ht= 122 ; ( 103/122 , 3937/244 ), ( 122/103 , 3937/206 )
**u= 1/39 , ht= 187 ; ( -187/161 , 134/7 ), ( -161/187 , -3082/187 )
**u= 1/46 , ht= 55 ; ( -12987/1181 , 176886/1181 ), ( 55/9 , 84 )
**u= 1/50 , ht= 1 ; ( -1 , 20 ), ( -113/88 , -4051/176 )
**u= 1/51 , ht= 1181 ; ( -757/1181 , 20036/1181 ), ( 3755/1817 , 59582/1817 )
**u= 1/56 , ht= 144 ; ( -137/144 , 11899/576 ), ( -1751/727 , -28376/727 )
**u= 1/59 , ht= 4891 ; ( 24043/6579 , -382928/6579 ), ( 4891/407 , 75398/407 )
**u= 1/63 , ht= 139 ; ( -3637/465 , -174614/1395 ), ( -15/139 , -6658/417 )
**u= 1/65 , ht= 211 ; ( -101/211 , 3772/211 ), ( 177/307 , -5714/307 )
**u= 1/66 , ht= 133 ; ( 248/1469 , -48413/2938 ), ( 109/133 , 2794/133 )
**u= 1/71 , ht= 323 ; ( 323/245 , 976/35 ), ( 931/347 , 16744/347 )
**u= 1/72 , ht= 1 ; ( -1 , 24 ), ( 4079/776 , -281861/3104 )
**u= 1/80 , ht= 991 ; ( -1067/27 , -19093/27 ), ( 991/721 , -21923/721 )
**u= 1/81 , ht= 317 ; ( -67/319 , 17602/957 ), ( -69/317 , -52556/2853 )
**u= 1/82 , ht= 348 ; ( -1041/1009 , -26256/1009 ), ( -307/348 , -5603/232 )
**u= 1/84 , ht= 625 ; ( 289/625 , -12622/625 ), ( -4362961/338687 , 80215702/338687 )
**u= 1/86 , ht= 8789 ; ( 8789/6338 , 401951/12676 ), ( -24615/1529 , -41584/139 )
**u= 1/89 , ht= 1477 ; ( -11487/1685 , -219054/1685 ), ( -765/1477 , 31384/1477 )
**u= 1/93 , ht= 8903 ; ( 119501/8149 , 2310188/8149 ), ( 8903/4997 , -196914/4997 )
**u= 1/98 , ht= 8887 ; ( -8887/8535 , 1707704/59745 ), ( -12415/12804 , 1647837/59752 )
**u= 2 , ht= 1 ; ( 1 , -2 ), ( 0 , 1 )
**u= 2/9 , ht= 1 ; ( -1 , 6 ), ( -1 , -6 )
**u= 2/13 , ht= 17 ; ( -17/9 , -98/9 ), ( -263/49 , 1366/49 )
**u= 2/21 , ht= 97 ; ( -97/50 , 707/50 ), ( -50/97 , -707/97 )
**u= 2/25 , ht= 185 ; ( -185/137 , 8138/685 ), ( 260/207 , 3917/345 )
**u= 2/45 , ht= 377 ; ( 377/98 , 3695/98 ), ( 1599/1049 , 54430/3147 )
**u= 2/49 , ht= 1599 ; ( -1471/1599 , -150562/11193 ), ( -2500/2401 , 240199/16807 )
**u= 2/69 , ht= 62 ; ( -7/62 , -733/62 ), ( 86/109 , 1631/109 )
**u= 2/81 , ht= 1 ; ( 1 , 18 ), ( -1 , 18 )
**u= 2/85 , ht= 407 ; ( -563/1222 , -17543/1222 ), ( 118/407 , 5525/407 )
**u= 2/93 , ht= 811 ; ( 811/253 , 11586/253 ), ( -1925/307 , -26586/307 )
**u= 4/15 , ht= 13 ; ( 13/8 , -59/8 ), ( 277/82 , -1123/82 )
**u= 4/39 , ht= 205 ; ( 478/595 , 681/85 ), ( 205/94 , 1409/94 )
**u= 4/63 , ht= 247 ; ( 247/68 , 2033/68 ), ( 68/247 , 107/13 )
**u= 4/71 , ht= 82 ; ( -82/11 , 697/11 ), ( -11/82 , -17/2 )
**u= 8 , ht= 1 ; ( -1 , -1 ), ( -5/3 , -3 )
**u= 8/9 , ht= 1 ; ( -1 , -3 ), ( -143/177 , 1433/531 )
**u= 8/25 , ht= 1 ; ( -1 , -5 ), ( -1367/1265 , -32941/6325 )
**u= 8/49 , ht= 1 ; ( -151/145 , -7253/1015 ), ( 1 , -7 )
**u= 8/57 , ht= 253 ; ( 253/101 , -1453/101 ), ( -289/167 , 1783/167 )
**u= 8/81 , ht= 1 ; ( -1 , 9 ), ( -1 , -9 )
**u= 9 , ht= 5 ; ( 3/5 , 28/15 ), ( 21/13 , -116/39 )
**u= 9/2 , ht= 1 ; ( -1 , 4/3 ), ( -7/8 , -49/48 )
**u= 9/7 , ht= 27 ; ( -27/13 , -148/39 ), ( 443/93 , 2612/279 )
**u= 9/8 , ht= 1 ; ( -1 , 8/3 ), ( -265/296 , -8911/3552 )
**u= 9/10 , ht= 109 ; ( -109/51 , 736/153 ), ( -97/198 , -2701/1188 )
**u= 9/11 , ht= 899 ; ( 899/503 , -6682/1509 ), ( 949/487 , -7246/1461 )
**u= 9/16 , ht= 63 ; ( 511/387 , 10201/2322 ), ( 1/63 , 494/189 )
**u= 9/25 , ht= 99 ; ( 19/117 , 5972/1755 ), ( -1/99 , 982/297 )
**u= 9/29 , ht= 41 ; ( -173/289 , 214/51 ), ( -17/41 , 476/123 )
**u= 9/32 , ht= 1 ; ( -3928/4055 , -511057/97320 ), ( -1 , -16/3 )
**u= 9/34 , ht= 9 ; ( 444/155 , 10931/930 ), ( -9/8 , 281/48 )
**u= 9/38 , ht= 3655 ; ( -7007/3082 , 188303/18492 ), ( 3655/121 , -45244/363 )
**u= 9/49 , ht= 5 ; ( 3/5 , 572/105 ), ( 111/173 , 20126/3633 )
**u= 9/50 , ht= 1 ; ( -1 , 20/3 ), ( -1 , -20/3 )
**u= 9/52 , ht= 317 ; ( -1189/735 , -20143/2205 ), ( 57/317 , 4637/951 )
**u= 9/53 , ht= 1355 ; ( 401/1355 , 20606/4065 ), ( 3119/151 , 45548/453 )
**u= 9/55 , ht= 309 ; ( 1433/993 , 25844/2979 ), ( -241/309 , -5810/927 )
**u= 9/58 , ht= 163 ; ( 18/163 , -5003/978 ), ( -375/2 , -11441/12 )
**u= 9/62 , ht= 427 ; ( 1061/131 , 16814/393 ), ( 148/427 , -14219/2562 )
**u= 9/67 , ht= 225 ; ( -459/211 , -8264/633 ), ( -225/43 , -3754/129 )
**u= 9/74 , ht= 169 ; ( -271/25 , -4678/75 ), ( -169/127 , 3638/381 )
**u= 9/77 , ht= 103 ; ( -8401/2071 , 151736/6213 ), ( -103/37 , -1922/111 )
**u= 9/85 , ht= 27 ; ( 1120913/280263 , -21291700/840789 ), ( 7/27 , 514/81 )
**u= 9/94 , ht= 1279 ; ( -743/2265 , 46244/6795 ), ( -225/1279 , -25168/3837 )
**u= 9/98 , ht= 1672 ; ( -37984/38375 , -14966641/1611750 ), ( -1655/1672 , -652097/70224 )
**u= 9/100 , ht= 57 ; ( 1399/357 , -20617/765 ), ( -7/57 , -1148/171 )
**u= 10/3 , ht= 2 ; ( 119/53 , -38/53 ), ( -1/2 , -1/2 )
**u= 10/23 , ht= 152 ; ( 152/101 , -551/101 ), ( -433/19 , 70 )
**u= 10/27 , ht= 157 ; ( 179/473 , -5018/1419 ), ( -157/86 , 591/86 )
**u= 10/51 , ht= 5687 ; ( -5687/1861 , 26974/1861 ), ( -204223/20194 , -929081/20194 )
**u= 10/71 , ht= 313 ; ( 2376/317 , -12759/317 ), ( 313/171 , 634/57 )
**u= 10/99 , ht= 12546 ; ( 12546/1337 , 238043/4011 ), ( -27227/966 , 514657/2898 )
**u= 16/9 , ht= 1 ; ( 8 , -19/2 ), ( 0 , 7/6 )
**u= 16/17 , ht= 28 ; ( -23/28 , -151/56 ), ( 35/52 , 253/104 )
**u= 16/25 , ht= 24 ; ( 24 , 117/2 ), ( 1/24 , -39/16 )
**u= 16/65 , ht= 89 ; ( 89/76 , -943/152 ), ( -136/189 , -625/126 )
**u= 16/97 , ht= 8968 ; ( 1887/8968 , -90399/17936 ), ( -8968/1887 , -30133/1258 )
**u= 17 , ht= 319 ; ( -169/319 , -808/319 ), ( 5887/3363 , -4822/1121 )
**u= 17/3 , ht= 25 ; ( -19/25 , 38/25 ), ( -157/101 , -256/101 )
**u= 17/6 , ht= 43 ; ( -43/2 , 3/4 ), ( -10/227 , -11/454 )
**u= 17/7 , ht= 97 ; ( -457/145 , -388/145 ), ( -71/97 , 136/97 )
**u= 17/8 , ht= 5 ; ( 3/5 , 9/5 ), ( 3989/2268 , 9529/3024 )
**u= 17/9 , ht= 5 ; ( 3/5 , -28/15 ), ( -5/3 , 28/9 )
**u= 17/10 , ht= 22 ; ( 22/3 , 55/6 ), ( -3/22 , 5/4 )
**u= 17/12 , ht= 13 ; ( -23/41 , -84/41 ), ( 1/13 , 19/13 )
**u= 17/13 , ht= 1427 ; ( -9313/1399 , -14692/1399 ), ( 1427/361 , -2806/361 )
**u= 17/15 , ht= 41 ; ( -41/11 , 74/11 ), ( 11/41 , 74/41 )
**u= 17/18 , ht= 79 ; ( -164/7 , -91/2 ), ( 7/79 , -154/79 )
**u= 17/19 , ht= 421 ; ( 1527/22945 , -50976/22945 ), ( 35/421 , -848/421 )
**u= 17/22 , ht= 354 ; ( -97/354 , -1723/708 ), ( -19961/5537 , -48614/5537 )
**u= 17/24 , ht= 517 ; ( 517/142 , 4957/568 ), ( -622/227 , -6443/908 )
**u= 17/30 , ht= 383 ; ( -383/172 , 2201/344 ), ( 2011/371 , 5534/371 )
**u= 17/31 , ht= 735 ; ( -641/6055 , 2392/865 ), ( -735/443 , -2338/443 )
**u= 17/33 , ht= 2383 ; ( 401/3173 , -9054/3173 ), ( -2383/1049 , 7336/1049 )
**u= 17/36 , ht= 21 ; ( 21 , -181/3 ), ( -1/21 , 181/63 )
**u= 17/37 , ht= 247 ; ( -423/349 , -1614/349 ), ( 247/49 , 752/49 )
**u= 17/40 , ht= 203 ; ( -2257/103 , -6853/103 ), ( -203/13 , 631/13 )
**u= 17/42 , ht= 895 ; ( -43861/17123 , 146898/17123 ), ( 895/874 , -7867/1748 )
**u= 17/44 , ht= 717 ; ( -717/97 , -2307/97 ), ( -829/579 , 1088/193 )
**u= 17/45 , ht= 779 ; ( -6203/4327 , 24534/4327 ), ( -779/131 , 2592/131 )
**u= 17/52 , ht= 1663 ; ( -1663/1535 , 7912/1535 ), ( 2855/4441 , 18416/4441 )
**u= 17/55 , ht= 2411 ; ( -471/2411 , 8886/2411 ), ( 5581/35391 , 42718/11797 )
**u= 17/58 , ht= 1063 ; ( 1063/831 , -4978/831 ), ( 3551/9902 , 77399/19804 )
**u= 17/63 , ht= 179 ; ( -12351/2195 , 144274/6585 ), ( 53/179 , 4 )
**u= 17/64 , ht= 857 ; ( -1651/25 , 25517/100 ), ( 857/567 , 2663/378 )
**u= 17/67 , ht= 10325 ; ( -18941/1253 , -10724/179 ), ( 10325/8167 , 52318/8167 )
**u= 17/69 , ht= 58585 ; ( -61783/6803 , -249518/6803 ), ( 58585/14149 , 243662/14149 )
**u= 17/70 , ht= 313 ; ( -13931/58869 , 246320/58869 ), ( -142/313 , -2783/626 )
**u= 17/71 , ht= 69663 ; ( -69663/6023 , -284784/6023 ), ( 157807/9417 , -216148/3139 )
**u= 17/72 , ht= 379 ; ( -932/39 , -45913/468 ), ( 37/379 , 1562/379 )
**u= 17/75 , ht= 61 ; ( 61/23 , 1366/115 ), ( -7039/12533 , -301418/62665 )
**u= 17/78 , ht= 248 ; ( 36007/43040 , 481053/86080 ), ( -248/181 , 2633/362 )
**u= 17/79 , ht= 2371 ; ( -3289/8719 , -40264/8719 ), ( 2055/2371 , 13522/2371 )
**u= 17/82 , ht= 879 ; ( 428/879 , 8603/1758 ), ( 46949/66252 , 237571/44168 )
**u= 17/84 , ht= 87043 ; ( 87043/15475 , -392094/15475 ), ( 110995/27821 , 509868/27821 )
**u= 17/89 , ht= 6879 ; ( -20121/7469 , 98046/7469 ), ( 775/6879 , 10536/2293 )
**u= 17/93 , ht= 1289 ; ( -3871/2887 , 22576/2887 ), ( -607/1289 , -6656/1289 )
**u= 18 , ht= 1 ; ( -1 , 2/3 ), ( -1 , -2/3 )
**u= 18/5 , ht= 9 ; ( -21/19 , 82/57 ), ( -9 , 46/3 )
**u= 18/13 , ht= 25 ; ( -1/25 , -142/75 ), ( 46/7 , -263/21 )
**u= 18/25 , ht= 55 ; ( 55/16 , 1969/240 ), ( 103/55 , -382/75 )
**u= 18/37 , ht= 341 ; ( -587/634 , 7441/1902 ), ( 82/341 , 2977/1023 )
**u= 18/49 , ht= 236 ; ( -2459/2360 , 236099/49560 ), ( -236/119 , -18407/2499 )
**u= 18/77 , ht= 870 ; ( -883/69 , 10954/207 ), ( 870/89 , 10889/267 )
**u= 18/89 , ht= 2339 ; ( 381/2339 , -31694/7017 ), ( -4484/4149 , 81521/12447 )
**u= 20/7 , ht= 4 ; ( -9/2 , 3/2 ), ( 3/4 , 5/4 )
**u= 20/31 , ht= 134 ; ( -151/438 , -1177/438 ), ( -134/83 , -397/83 )
**u= 20/39 , ht= 28 ; ( -1648/1199 , -5663/1199 ), ( -11/28 , -83/28 )
**u= 20/63 , ht= 34 ; ( -3917/1186 , 14451/1186 ), ( -34/27 , -463/81 )
**u= 25/2 , ht= 1 ; ( -1 , -4/5 ), ( 1 , 4/5 )
**u= 25/6 , ht= 37 ; ( -13/59 , -102/59 ), ( -35/37 , -234/185 )
**u= 25/7 , ht= 261 ; ( 261/131 , -336/655 ), ( 131/261 , 112/435 )
**u= 25/8 , ht= 1 ; ( -1 , 8/5 ), ( -601/568 , 3869/2272 )
**u= 25/11 , ht= 15 ; ( 15/7 , 12/5 ), ( 35/53 , -364/265 )
**u= 25/18 , ht= 1 ; ( -912/983 , -68881/29490 ), ( -1 , 12/5 )
**u= 25/21 , ht= 619 ; ( 1051/505 , 10078/2525 ), ( 619/367 , -1374/367 )
**u= 25/23 , ht= 179 ; ( 137/459 , -974/459 ), ( -179/97 , -406/97 )
**u= 25/26 , ht= 29 ; ( -574/557 , -3257/1114 ), ( -3/29 , 56/29 )
**u= 25/32 , ht= 1 ; ( -1 , 16/5 ), ( -1 , -16/5 )
**u= 25/33 , ht= 811 ; ( 481/811 , -11018/4055 ), ( 44939/6527 , -107874/6527 )
**u= 25/34 , ht= 169 ; ( 189/835 , -10284/4175 ), ( -169/33 , -138/11 )
**u= 25/36 , ht= 145 ; ( -31/195 , 7309/2925 ), ( -81/145 , -5884/2175 )
**u= 25/41 , ht= 1425 ; ( -461/1425 , -19538/7125 ), ( -2273/3139 , 9854/3139 )
**u= 25/42 , ht= 113 ; ( -59/179 , 2486/895 ), ( -113/34 , 623/68 )
**u= 25/51 , ht= 269 ; ( -3415/869 , -49662/4345 ), ( 269/235 , 5112/1175 )
**u= 25/53 , ht= 1889 ; ( -761/1889 , -29948/9445 ), ( 7307/1901 , -22252/1901 )
**u= 25/69 , ht= 3977 ; ( 114683/24569 , 386770/24569 ), ( -3977/3061 , -16710/3061 )
**u= 25/72 , ht= 1 ; ( -1 , 24/5 ), ( -1 , -24/5 )
**u= 25/78 , ht= 821 ; ( -1979/803 , 7510/803 ), ( 704/821 , -38169/8210 )
**u= 25/81 , ht= 301 ; ( 125/351 , 60638/15795 ), ( -75/301 , 49988/13545 )
**u= 25/83 , ht= 171 ; ( -13909/2691 , 11164/585 ), ( -161/171 , 1426/285 )
**u= 25/84 , ht= 31 ; ( -17/31 , 130/31 ), ( 5029/3335 , 110849/16675 )
**u= 25/86 , ht= 1375 ; ( 1375/257 , 25816/1285 ), ( -254/1853 , -13805/3706 )
**u= 25/87 , ht= 769 ; ( -769/217 , 424/31 ), ( 1981/4591 , 92946/22955 )
**u= 25/89 , ht= 7255 ; ( -8657/955 , -163538/4775 ), ( -6273/7255 , 180832/36275 )
**u= 25/92 , ht= 691 ; ( 2019/179 , 38706/895 ), ( -689/691 , 18719/3455 )
**u= 25/93 , ht= 2287 ; ( 1991/2287 , 11704/2287 ), ( 2563361/393481 , -4566768/178855 )
**u= 25/98 , ht= 1 ; ( -1 , 28/5 ), ( -1 , -28/5 )
**u= 26/7 , ht= 32 ; ( -4/45 , 77/45 ), ( -17/32 , -7/32 )
**u= 26/15 , ht= 712 ; ( -1996/719 , -2717/719 ), ( 67/712 , -863/712 )
**u= 26/19 , ht= 18 ; ( -18 , -27 ), ( 1/18 , -3/2 )
**u= 26/27 , ht= 142 ; ( -142/5 , -817/15 ), ( 5/142 , 817/426 )
**u= 26/43 , ht= 563 ; ( 647/1077 , 3266/1077 ), ( -563/383 , -1766/383 )
**u= 26/55 , ht= 21 ; ( -1/21 , -62/21 ), ( -1459499/67356 , -1436239/22452 )
**u= 26/75 , ht= 5 ; ( -74/475 , 8223/2375 ), ( 1/5 , -86/25 )
**u= 26/87 , ht= 4061 ; ( -19948/2063 , 72967/2063 ), ( 1/4061 , -14774/4061 )
**u= 32 , ht= 257 ; ( -257/255 , 1/510 ), ( -255/257 , -1/514 )
**u= 32/9 , ht= 5 ; ( -3/5 , -49/30 ), ( -455/417 , -4189/2502 )
**u= 32/25 , ht= 365 ; ( -365/331 , -8623/3310 ), ( -1607/1505 , -39181/15050 )
**u= 32/33 , ht= 167 ; ( -167/31 , 21/2 ), ( -31/167 , 651/334 )
**u= 32/49 , ht= 891 ; ( 891/755 , -40287/10570 ), ( -1885/1683 , 29273/7854 )
**u= 32/81 , ht= 1 ; ( -13191/12865 , -1055093/231570 ), ( -1 , -9/2 )
**u= 32/97 , ht= 629 ; ( 629/1117 , -8959/2234 ), ( -147/629 , -263/74 )
**u= 34/5 , ht= 39 ; ( -31/39 , -58/39 ), ( 82/97 , 19/97 )
**u= 34/21 , ht= 523 ; ( 523/509 , 1142/509 ), ( 202/671 , 939/671 )
**u= 34/25 , ht= 63 ; ( 99/124 , 9/4 ), ( -31/63 , -62/35 )
**u= 34/33 , ht= 151 ; ( 151/25 , 282/25 ), ( -25/151 , 282/151 )
**u= 34/41 , ht= 60 ; ( 19/60 , -143/60 ), ( -13800/919 , -31643/919 )
**u= 34/45 , ht= 261 ; ( 261/61 , -1790/183 ), ( -586/219 , 4433/657 )
**u= 34/61 , ht= 1299 ; ( 1299/958 , 4299/958 ), ( 2525/6747 , 6338/2249 )
**u= 34/65 , ht= 209 ; ( 343/1552 , 4463/1552 ), ( -116/209 , -655/209 )
**u= 34/69 , ht= 3739 ; ( -5777/226 , 16221/226 ), ( -3739/2563 , -12986/2563 )
**u= 36/7 , ht= 27 ; ( -22/27 , 119/81 ), ( 55/6 , -301/18 )
**u= 36/23 , ht= 52 ; ( -17/52 , -293/156 ), ( -160 , -877/3 )
**u= 36/47 , ht= 965 ; ( 2158/1735 , 18833/5205 ), ( 965/166 , -6941/498 )
**u= 36/79 , ht= 103 ; ( -4911/1172 , -44353/3516 ), ( 24/103 , -929/309 )
**u= 36/95 , ht= 332 ; ( 2686/691 , 26819/2073 ), ( -332/143 , 3545/429 )
**u= 40/17 , ht= 587 ; ( 587/9 , 425/9 ), ( 1941/3607 , 4201/3607 )
**u= 40/33 , ht= 43 ; ( 4573/1801 , -8307/1801 ), ( -43 , -85 )
**u= 40/49 , ht= 51 ; ( -459/377 , -9129/2639 ), ( -7/51 , 15/7 )
**u= 40/57 , ht= 631 ; ( -53/631 , -1557/631 ), ( 1439/4403 , -10781/4403 )
**u= 40/81 , ht= 41 ; ( 41/3 , -1037/27 ), ( -493/169 , -1501/169 )
**u= 40/89 , ht= 1139 ; ( -463/1139 , 3701/1139 ), ( -291/2023 , 6023/2023 )
**u= 41/3 , ht= 215 ; ( -131/215 , -466/215 ), ( -925/449 , -2186/449 )
**u= 41/4 , ht= 55 ; ( -3/55 , -129/55 ), ( 517/471 , -217/157 )
**u= 41/5 , ht= 783 ; ( -783/743 , 564/743 ), ( -1499/531 , 1016/177 )
**u= 41/7 , ht= 417 ; ( 347/417 , -598/417 ), ( 5927/1027 , 11158/1027 )
**u= 41/10 , ht= 351 ; ( -289/351 , -532/351 ), ( 1891/701 , -3208/701 )
**u= 41/12 , ht= 43 ; ( 43/23 , 1 ), ( -23/43 , 23/43 )
**u= 41/13 , ht= 297 ; ( 837/317 , 264/317 ), ( 115/297 , 34/99 )
**u= 41/14 , ht= 57 ; ( 3363/829 , 912/829 ), ( 20/57 , 1/2 )
**u= 41/16 , ht= 33 ; ( 2263/441 , -1409/441 ), ( -33/31 , 58/31 )
**u= 41/18 , ht= 9 ; ( 469/4139 , -7052/4139 ), ( -9 , -46/3 )
**u= 41/21 , ht= 379 ; ( 379/295 , -646/295 ), ( -461/3067 , 3276/3067 )
**u= 41/23 , ht= 1017 ; ( -1017/133 , -1206/133 ), ( -11963/3033 , 2386/337 )
**u= 41/24 , ht= 182 ; ( -5363/1261 , 6918/1261 ), ( 182/121 , 1429/484 )
**u= 41/27 , ht= 89 ; ( 4513/355 , -6214/355 ), ( 89/19 , -166/19 )
**u= 41/31 , ht= 1473 ; ( -1473/109 , -2274/109 ), ( 72835/11733 , -46996/3911 )
**u= 41/37 , ht= 6193 ; ( -1207/6193 , 12812/6193 ), ( 6773/113 , -13822/113 )
**u= 41/40 , ht= 118 ; ( 118/37 , -923/148 ), ( -40069/2106 , -112375/2808 )
**u= 41/45 , ht= 3881 ; ( -3881/1069 , 8050/1069 ), ( -114639/36791 , -788092/110373 )
**u= 41/48 , ht= 49 ; ( 21097/2585 , 43913/2585 ), ( -31/49 , 123/49 )
**u= 41/51 , ht= 1915 ; ( 7/5311 , 12316/5311 ), ( 1915/589 , -4616/589 )
**u= 41/54 , ht= 190 ; ( 24881/9950 , 119759/19900 ), ( -190/53 , 933/106 )
**u= 41/55 , ht= 93 ; ( -17/93 , -226/93 ), ( -10039/11411 , -35050/11411 )
**u= 41/58 , ht= 342 ; ( 342/325 , 2241/650 ), ( 325/342 , -249/76 )
**u= 41/67 , ht= 3643 ; ( -3643/3375 , -12674/3375 ), ( 11405/41391 , -35846/13797 )
**u= 41/68 , ht= 25357 ; ( -35673/12437 , -95559/12437 ), ( 25357/5583 , -22747/1861 )
**u= 41/69 , ht= 7819 ; ( -7313/7819 , 3974/1117 ), ( 2284625/1202819 , -6782062/1202819 )
**u= 41/75 , ht= 97 ; ( 16645/36563 , 549972/182815 ), ( 97/53 , -302/53 )
**u= 41/79 , ht= 8029 ; ( 1867/8029 , 23228/8029 ), ( -1093991/63043 , 3092638/63043 )
**u= 41/81 , ht= 151 ; ( 151/47 , 28/3 ), ( -47/151 , 1316/453 )
**u= 41/84 , ht= 8531 ; ( -8531/1367 , -24381/1367 ), ( -90491/4597 , 263198/4597 )
**u= 41/86 , ht= 18082 ; ( 113571/4765 , -324564/4765 ), ( 18082/10341 , -40513/6894 )
**u= 41/88 , ht= 1425 ; ( 1831/1391 , -6713/1391 ), ( -1425/49 , 4234/49 )
**u= 41/91 , ht= 5087 ; ( 5087/355 , -15002/355 ), ( -13645201/667643 , 41216726/667643 )
**u= 41/96 , ht= 277 ; ( -41731/2810 , 1012377/22480 ), ( -277/131 , 1889/262 )
**u= 41/97 , ht= 3211 ; ( 3211/281 , -9806/281 ), ( -242391/36581 , -762106/36581 )
**u= 41/100 , ht= 357 ; ( 149/357 , -358/105 ), ( -357/149 , 6086/745 )
**u= 49/2 , ht= 1 ; ( -1 , 4/7 ), ( -1 , -4/7 )
**u= 49/4 , ht= 35 ; ( 33/35 , -279/245 ), ( -65/63 , 152/147 )
**u= 49/5 , ht= 153 ; ( -153/157 , -1122/1099 ), ( 143/153 , 8/21 )
**u= 49/6 , ht= 134 ; ( -73/134 , -3513/1876 ), ( -134/73 , -3513/1022 )
**u= 49/8 , ht= 1 ; ( -304/315 , 10699/8820 ), ( -1 , -8/7 )
**u= 49/9 , ht= 85 ; ( 77/85 , 804/595 ), ( -111/77 , 3686/1617 )
**u= 49/11 , ht= 1145 ; ( 1393/909 , -374/6363 ), ( 1145/991 , -11696/6937 )
**u= 49/15 , ht= 293 ; ( -781/329 , 1714/2303 ), ( -127/293 , -734/2051 )
**u= 49/17 , ht= 59 ; ( -59/9 , 46/63 ), ( -27/143 , -218/1001 )
**u= 49/18 , ht= 1 ; ( -24/95 , -6721/3990 ), ( -1 , -12/7 )
**u= 49/27 , ht= 181 ; ( 181/31 , 4478/651 ), ( 145/199 , 2392/1393 )
**u= 49/30 , ht= 143 ; ( -7/143 , -1810/1001 ), ( -671 , -8488/7 )
**u= 49/32 , ht= 207 ; ( -207/175 , 2991/1225 ), ( -1639/1764 , 71861/32928 )
**u= 49/33 , ht= 563 ; ( 563/145 , -5818/1015 ), ( 877/1273 , 16866/8911 )
**u= 49/36 , ht= 113 ; ( -113/49 , 1356/343 ), ( -211/113 , -27/7 )
**u= 49/38 , ht= 191 ; ( -383/225 , 5192/1575 ), ( 191 , -2588/7 )
**u= 49/41 , ht= 411 ; ( 411/83 , 4908/581 ), ( -355/1257 , -5134/2933 )
**u= 49/47 , ht= 257 ; ( -257/25 , 3296/175 ), ( -25/257 , -3296/1799 )
**u= 49/50 , ht= 3732 ; ( -6405/5861 , 610632/205135 ), ( -3535/3732 , 241533/87080 )
**u= 49/51 , ht= 805 ; ( 13537/889 , -182344/6223 ), ( -805/317 , -12868/2219 )
**u= 49/53 , ht= 667 ; ( -287/667 , 10954/4669 ), ( -667/287 , -10954/2009 )
**u= 49/59 , ht= 421 ; ( 19323/4691 , 293532/32837 ), ( 421/67 , -6812/469 )
**u= 49/62 , ht= 2929 ; ( 4093/1117 , -64528/7819 ), ( -2929/1721 , -54506/12047 )
**u= 49/68 , ht= 269 ; ( -271 , -4322/7 ), ( 71/269 , 4457/1883 )
**u= 49/69 , ht= 5059 ; ( -8513/27781 , 495158/194467 ), ( 5059/2713 , -97018/18991 )
**u= 49/72 , ht= 264 ; ( 1887/193 , 93794/4053 ), ( -49/264 , -53183/22176 )
**u= 49/74 , ht= 242 ; ( -451/67 , 7636/469 ), ( -91/242 , 8711/3388 )
**u= 49/76 , ht= 141 ; ( 141/65 , 2661/455 ), ( 121/183 , -1262/427 )
**u= 49/85 , ht= 1267 ; ( -297/1267 , -24438/8869 ), ( 1267/297 , -8146/693 )
**u= 49/86 , ht= 351 ; ( -351/7 , -6378/49 ), ( -7/351 , 2126/819 )
**u= 49/93 , ht= 91 ; ( 29/91 , 1868/637 ), ( -45445/10913 , 915258/76391 )
**u= 49/96 , ht= 46 ; ( 1751/905 , 38259/6335 ), ( 46/13 , 7597/728 )
**u= 49/99 , ht= 207 ; ( -301/697 , -898/287 ), ( -119/207 , 14144/4347 )
**u= 50 , ht= 1 ; ( -1 , 2/5 ), ( -1 , -2/5 )
**u= 50/9 , ht= 1 ; ( -1 , 6/5 ), ( -1 , -6/5 )
**u= 50/33 , ht= 760 ; ( -1615/112 , 11127/560 ), ( -760/329 , -7359/1645 )
**u= 50/49 , ht= 1 ; ( 1 , -14/5 ), ( 404/297 , 787/231 )
**u= 50/73 , ht= 137 ; ( 2255/576 , 27389/2880 ), ( -9/137 , 322/137 )
**u= 50/81 , ht= 1 ; ( -1 , 18/5 ), ( -1 , -18/5 )
**u= 50/93 , ht= 599 ; ( 599/382 , 1923/382 ), ( 7474/5335 , -125969/26675 )
**u= 52/7 , ht= 34 ; ( -963/850 , -21/50 ), ( -34/27 , 17/9 )
**u= 52/15 , ht= 112 ; ( 112/53 , -29/53 ), ( -418/367 , -653/367 )
**u= 52/23 , ht= 17 ; ( 17/6 , 17/6 ), ( 6/17 , 1 )
**u= 52/47 , ht= 24 ; ( 1956/875 , 3861/875 ), ( 23/24 , 21/8 )
**u= 52/63 , ht= 764 ; ( 764/323 , -1771/323 ), ( 2189/6928 , 15427/6928 )
**u= 52/71 , ht= 70 ; ( -6209/2950 , 15721/2950 ), ( -70 , 169 )
**u= 52/87 , ht= 2476 ; ( -10074208/545215 , 25513683/545215 ), ( 997/2476 , 6793/2476 )
**u= 58/3 , ht= 79 ; ( 323/329 , -286/329 ), ( 79/38 , 219/38 )
**u= 58/11 , ht= 25 ; ( 19/25 , 38/25 ), ( 25/19 , 2 )
**u= 58/23 , ht= 179 ; ( -179/143 , -262/143 ), ( 79/792 , -157/264 )
**u= 58/27 , ht= 77 ; ( -4114/1063 , 4081/1063 ), ( 25/77 , -22/21 )
**u= 58/31 , ht= 588 ; ( 588/125 , 681/125 ), ( 553/1803 , -734/601 )
**u= 58/35 , ht= 143 ; ( 1033/683 , -1790/683 ), ( 143/53 , -266/53 )
**u= 58/47 , ht= 5495 ; ( 22769/17787 , 7258/2541 ), ( -5495/3764 , 12397/3764 )
**u= 58/51 , ht= 178 ; ( 35323/13937 , -9554/1991 ), ( 77/178 , -343/178 )
**u= 58/59 , ht= 1581 ; ( 714/3841 , -8313/3841 ), ( -1369/1581 , -82/31 )
**u= 58/95 , ht= 6392 ; ( -6392/5917 , 22255/5917 ), ( -60209/11341 , 160190/11341 )
**u= 58/99 , ht= 5845 ; ( -5463/5845 , -62806/17535 ), ( 25705/12483 , -226954/37449 )
**u= 64/9 , ht= 9 ; ( -8/9 , -143/108 ), ( 24/23 , 337/276 )
**u= 64/17 , ht= 81 ; ( 81/64 , 327/256 ), ( -64/81 , -109/108 )
**u= 64/33 , ht= 52 ; ( -52/47 , 393/188 ), ( 80/41 , -583/164 )
**u= 64/65 , ht= 463 ; ( 8/463 , 3953/1852 ), ( 463/8 , 3953/32 )
**u= 64/73 , ht= 717 ; ( 672/1445 , -14031/5780 ), ( -232/717 , -2061/956 )
**u= 65/3 , ht= 19 ; ( 7879/8533 , 11882/8533 ), ( -19/17 , -30/17 )
**u= 65/6 , ht= 487 ; ( 191/487 , -1090/487 ), ( 1633 , 3928 )
**u= 65/7 , ht= 943 ; ( -9/943 , 2124/943 ), ( 13919/6597 , 3142/733 )
**u= 65/9 , ht= 343 ; ( 129/343 , -284/147 ), ( -343 , 700 )
**u= 65/11 , ht= 41 ; ( 3909/5713 , 9150/5713 ), ( 41/3 , 26 )
**u= 65/22 , ht= 86 ; ( -53/86 , -287/172 ), ( 189/893 , -52/893 )
**u= 65/24 , ht= 2299 ; ( -2299/322 , 3883/1288 ), ( 1721/2497 , -275/227 )
**u= 65/27 , ht= 139 ; ( 116171/5267 , -237328/15801 ), ( -139/7 , -706/21 )
**u= 65/28 , ht= 509 ; ( -509/387 , -760/387 ), ( -17007/9761 , 29794/9761 )
**u= 65/29 , ht= 433 ; ( 201/433 , -756/433 ), ( 21733/15039 , 13006/5013 )
**u= 65/31 , ht= 43 ; ( 5947/14751 , 25954/14751 ), ( -43/19 , 4 )
**u= 65/34 , ht= 44 ; ( -44/27 , 133/54 ), ( 659/123 , 386/41 )
**u= 65/36 , ht= 629 ; ( -629/287 , -881/287 ), ( 287/629 , -881/629 )
**u= 65/37 , ht= 279 ; ( 17879/13423 , 1028/433 ), ( -279/17 , -496/17 )
**u= 65/42 , ht= 1733 ; ( -2617/71 , -3524/71 ), ( 1733/46 , 6353/92 )
**u= 65/43 , ht= 3681 ; ( -7713/5219 , -14322/5219 ), ( -413/3681 , 1706/1227 )
**u= 65/48 , ht= 3397 ; ( -3397/71 , 10255/142 ), ( -18943/9389 , -77553/18778 )
**u= 65/49 , ht= 41 ; ( -41/3 , -442/21 ), ( -6303/2909 , -12886/2909 )
**u= 65/51 , ht= 5981 ; ( -35267/38581 , 93440/38581 ), ( 983/5981 , -9650/5981 )
**u= 65/63 , ht= 23899 ; ( 23899/3613 , -44464/3613 ), ( -42533/5319 , 269030/15957 )
**u= 65/74 , ht= 1769 ; ( 918/1769 , 8739/3538 ), ( 23203/15174 , -13399/3372 )
**u= 65/76 , ht= 13553 ; ( -9551/13553 , 36404/13553 ), ( 43061/2287 , 97403/2287 )
**u= 65/81 , ht= 38751 ; ( 420309/369053 , 657676/195381 ), ( -31093/38751 , 989570/348759 )
**u= 65/83 , ht= 7497 ; ( 7497/1189 , 16518/1189 ), ( 518369/44937 , -406504/14979 )
**u= 65/92 , ht= 723 ; ( 723/79 , 1677/79 ), ( -349279/7773 , -285599/2591 )
**u= 65/93 , ht= 16987 ; ( 10849/16987 , -48834/16987 ), ( 255313/108931 , 677938/108931 )
**u= 65/94 , ht= 16288 ; ( 229487/48584 , -1097095/97168 ), ( -16288/7259 , 87475/14518 )
**u= 65/97 , ht= 35431 ; ( 35431/5503 , -85228/5503 ), ( 2519483/389169 , -193730/11793 )
**u= 65/99 , ht= 31357 ; ( -175333/47219 , -437620/47219 ), ( -6219/31357 , 230744/94071 )
**u= 68/15 , ht= 518 ; ( -518/457 , 535/457 ), ( -83701/20474 , 146565/20474 )
**u= 68/31 , ht= 18 ; ( 3546/725 , 3267/725 ), ( -13/18 , 3/2 )
**u= 68/55 , ht= 314 ; ( -314/49 , 517/49 ), ( -11256/569 , 22115/569 )
**u= 68/63 , ht= 479 ; ( -418/499 , 1269/499 ), ( -479/318 , 3413/954 )
**u= 68/87 , ht= 16 ; ( -35638/4121 , 78083/4121 ), ( -16/13 , -47/13 )
**u= 72 , ht= 1297 ; ( -1297/1295 , 1/3885 ), ( -1295/1297 , -1/3891 )
**u= 72/17 , ht= 21 ; ( -21/13 , -1/39 ), ( 565/123 , 2939/369 )
**u= 72/25 , ht= 55 ; ( 55/47 , -1171/705 ), ( -47/145 , 1079/2175 )
**u= 72/49 , ht= 1 ; ( -1 , 7/3 ), ( -5903/6097 , 292753/128037 )
**u= 72/97 , ht= 295 ; ( -287/295 , 2869/885 ), ( 2617/1841 , -22541/5523 )
**u= 73/2 , ht= 425 ; ( 424/425 , 469/850 ), ( -425/424 , 469/848 )
**u= 73/3 , ht= 127 ; ( -271/275 , 226/275 ), ( 127/71 , 372/71 )
**u= 73/11 , ht= 2023 ; ( 2809/4011 , 922/573 ), ( -2023/1297 , 3388/1297 )
**u= 73/12 , ht= 1255 ; ( 571/1255 , 2247/1255 ), ( -1883/2011 , -1864/2011 )
**u= 73/14 , ht= 965 ; ( 517/965 , 1628/965 ), ( 4264/3949 , -1039/718 )
**u= 73/15 , ht= 4297 ; ( 4297/3307 , -2418/3307 ), ( -3307/4297 , -2418/4297 )
**u= 73/17 , ht= 6873 ; ( 7617/38489 , 67008/38489 ), ( -5105/6873 , 1592/2291 )
**u= 73/20 , ht= 727 ; ( -727/373 , 145/373 ), ( -28121/6211 , -47767/6211 )
**u= 73/23 , ht= 313 ; ( 313/129 , -124/129 ), ( 45849/9187 , 77186/9187 )
**u= 73/26 , ht= 491 ; ( 3414/125 , -1071/250 ), ( -29/491 , 86/491 )
**u= 73/27 , ht= 107 ; ( -107/31 , -194/93 ), ( 31/107 , -194/321 )
**u= 73/29 , ht= 126779 ; ( -126779/1981 , 72952/1981 ), ( 62545/347267 , 225802/347267 )
**u= 73/36 , ht= 833 ; ( 833/445 , -1121/445 ), ( 921/631 , 5122/1893 )
**u= 73/42 , ht= 527 ; ( 527/359 , 898/359 ), ( -27025/21422 , -109047/42844 )
**u= 73/44 , ht= 215 ; ( -181/215 , 449/215 ), ( -5/1887 , -791/629 )
**u= 73/45 , ht= 841 ; ( -369/841 , -4784/2523 ), ( -349/2841 , -11126/8523 )
**u= 73/48 , ht= 377 ; ( -4303/5729 , 12079/5729 ), ( 41/377 , 521/377 )
**u= 73/56 , ht= 807 ; ( 58939/10100 , -374693/40400 ), ( -807/257 , -1607/257 )
**u= 73/57 , ht= 9467 ; ( 9467/1715 , -15284/1715 ), ( -64205/15253 , -126856/15253 )
**u= 73/59 , ht= 583 ; ( 5219/2505 , -9766/2505 ), ( 465/583 , 1312/583 )
**u= 73/60 , ht= 439 ; ( -439/119 , -108/17 ), ( 61117/11017 , 121975/11017 )
**u= 73/66 , ht= 533 ; ( -83/533 , -1098/533 ), ( -7111/1964 , 29843/3928 )
**u= 73/68 , ht= 739 ; ( 739/535 , -1721/535 ), ( -8137/3389 , -17857/3389 )
**u= 73/75 , ht= 187 ; ( -187/61 , 1896/305 ), ( 61/187 , 1896/935 )
**u= 73/78 , ht= 176 ; ( 12941/5285 , -27746/5285 ), ( 176/7 , -767/14 )
**u= 73/80 , ht= 411 ; ( 411 , 1629/2 ), ( 130337/75567 , 107829/25189 )
**u= 73/82 , ht= 5519 ; ( 4101/15827 , 36126/15827 ), ( -5519/312 , -8187/208 )
**u= 73/83 , ht= 1193 ; ( 4999/3135 , 12316/3135 ), ( 1193/529 , -2872/529 )
**u= 73/87 , ht= 673 ; ( 288647/134309 , 13778/2741 ), ( -371/673 , 1638/673 )
**u= 73/90 , ht= 1193 ; ( -383/1193 , 2874/1193 ), ( -1681/1114 , 9099/2228 )
**u= 73/93 , ht= 11447 ; ( -19321/62333 , -151922/62333 ), ( 11447/5681 , -29512/5681 )
**u= 73/99 , ht= 7189 ; ( 56033/53767 , 180616/53767 ), ( 7189/1821 , 53344/5463 )
**u= 74/11 , ht= 387 ; ( 387/361 , -318/361 ), ( 2191/603 , -1414/201 )
**u= 74/27 , ht= 7 ; ( -7/5 , -26/15 ), ( 23/158 , -61/158 )
**u= 74/47 , ht= 1977 ; ( -5556/1259 , -7707/1259 ), ( 755/1977 , 986/659 )
**u= 74/51 , ht= 1069 ; ( 9311/4925 , -16142/4925 ), ( 457/1069 , -1746/1069 )
**u= 74/55 , ht= 881 ; ( 4753/1563 , 7802/1563 ), ( 661/881 , 1838/881 )
**u= 74/67 , ht= 867 ; ( -607/867 , -2066/867 ), ( -6565/2279 , 13994/2279 )
**u= 74/79 , ht= 3265 ; ( -1053/3265 , 7398/3265 ), ( -72265/62469 , 22106/6941 )
**u= 74/91 , ht= 193 ; ( 11/193 , 446/193 ), ( -193/11 , 446/11 )
**u= 80/9 , ht= 23 ; ( -14/23 , 85/46 ), ( -127/46 , 531/92 )
**u= 80/17 , ht= 99 ; ( -99/82 , 165/164 ), ( 122/99 , -11/6 )
**u= 80/33 , ht= 221 ; ( -122/221 , 45/26 ), ( 697/766 , -2567/1532 )
**u= 80/73 , ht= 62 ; ( 11/62 , 257/124 ), ( 206/5973 , 7019/3982 )
**u= 81 , ht= 69 ; ( -93/229 , 11996/2061 ), ( 69/67 , -964/603 )
**u= 81/2 , ht= 3265 ; ( -3256/3265 , 32641/58770 ), ( -3305/3296 , 32969/59328 )
**u= 81/4 , ht= 37 ; ( -97/225 , 5896/2025 ), ( -37/9 , -1039/81 )
**u= 81/5 , ht= 83 ; ( 73/83 , -1126/747 ), ( -391/389 , 2666/3501 )
**u= 81/8 , ht= 1 ; ( -1 , 8/9 ), ( -1 , -8/9 )
**u= 81/16 , ht= 129 ; ( -409/297 , -373/2673 ), ( 129/127 , -1484/1143 )
**u= 81/19 , ht= 93 ; ( 3/155 , -2444/1395 ), ( -59/93 , -172/837 )
**u= 81/20 , ht= 49 ; ( 49/29 , -59/261 ), ( -29/49 , -59/441 )
**u= 81/22 , ht= 2237 ; ( 2237/1170 , 8663/21060 ), ( -2370/611 , -72361/10998 )
**u= 81/26 , ht= 643 ; ( 866/307 , -4501/5526 ), ( 643/266 , -19349/4788 )
**u= 81/31 , ht= 1435 ; ( 3693/469 , 2468/603 ), ( -1435/921 , -22106/8289 )
**u= 81/32 , ht= 1 ; ( -688/815 , -102817/58680 ), ( -1 , -16/9 )
**u= 81/34 , ht= 33 ; ( 33 , 208/9 ), ( 6475/627 , -98806/5643 )
**u= 81/46 , ht= 55 ; ( 1089/2768 , -91429/49824 ), ( 55/9 , 884/81 )
**u= 81/49 , ht= 309 ; ( -173/309 , 37676/19467 ), ( -335/147 , -39782/9261 )
**u= 81/50 , ht= 1 ; ( 1 , 20/9 ), ( -1 , -20/9 )
**u= 81/53 , ht= 1201 ; ( 1201/565 , -17434/5085 ), ( 7375/167 , -122072/1503 )
**u= 81/65 , ht= 271 ; ( -271/119 , 4448/1071 ), ( -349/41 , -6178/369 )
**u= 81/67 , ht= 105 ; ( 29/105 , -274/135 ), ( -37289/4191 , -666764/37719 )
**u= 81/70 , ht= 219 ; ( 3249/1079 , 53306/9711 ), ( -219/19 , -3970/171 )
**u= 81/71 , ht= 535 ; ( 1481/913 , -28228/8217 ), ( 535/109 , 9866/981 )
**u= 81/77 , ht= 331 ; ( 10567/95 , 172156/855 ), ( -131/331 , -5924/2979 )
**u= 81/82 , ht= 679 ; ( 16/1779 , 68257/32022 ), ( 351/679 , 13346/6111 )
**u= 81/83 , ht= 4415 ; ( -4217/4415 , -111548/39735 ), ( 11965/7483 , -263752/67347 )
**u= 81/85 , ht= 1439 ; ( 1439/249 , 25448/2241 ), ( -5673/107 , 110398/963 )
**u= 81/89 , ht= 3287 ; ( 9961/3197 , 188894/28773 ), ( -3287/1519 , -70576/13671 )
**u= 81/92 , ht= 1517 ; ( 323/1517 , -31042/13653 ), ( 421/1649 , -1837/873 )
**u= 81/97 , ht= 283 ; ( -283/105 , -5746/945 ), ( 6187/5217 , -160582/46953 )
**u= 81/98 , ht= 1 ; ( 424/905 , 284489/114030 ), ( 1 , 28/9 )
**u= 81/100 , ht= 401 ; ( 1647/461 , -164927/20745 ), ( -51/401 , -7757/3609 )
**u= 82/5 , ht= 9 ; ( 1/9 , -26/9 ), ( 526966/59141 , -1522165/59141 )
**u= 82/17 , ht= 59 ; ( 129/95 , 54/95 ), ( 43/59 , -22/59 )
**u= 82/29 , ht= 2441 ; ( 2441/34 , -91/34 ), ( -102/5437 , -233/5437 )
**u= 82/57 , ht= 457 ; ( 151/457 , 882/457 ), ( 154985/41783 , -296082/41783 )
**u= 82/77 , ht= 14690 ; ( -15469/1163 , -27878/1163 ), ( -14690/5099 , 31769/5099 )
**u= 89/3 , ht= 677 ; ( 323/677 , 2308/677 ), ( 14705/4037 , 54746/4037 )
**u= 89/4 , ht= 615 ; ( 173/615 , -1987/615 ), ( -1743/1679 , -1868/1679 )
**u= 89/13 , ht= 1171 ; ( -859/1171 , 1844/1171 ), ( 13045/12239 , 16022/12239 )
**u= 89/15 , ht= 4411 ; ( 33523/37243 , -49736/37243 ), ( -3551/4411 , 1142/4411 )
**u= 89/18 , ht= 219 ; ( -59/219 , 16/9 ), ( -1387/1792 , 1971/3584 )
**u= 89/19 , ht= 239 ; ( 27/239 , 426/239 ), ( -1605/2327 , -418/2327 )
**u= 89/21 , ht= 2435 ; ( 3959/2797 , 2354/2797 ), ( -2435/1177 , -38/11 )
**u= 89/22 , ht= 738 ; ( 617/738 , 2233/1476 ), ( -2125/427 , 3662/427 )
**u= 89/26 , ht= 2523 ; ( 2523/1171 , -702/1171 ), ( -22111/12807 , -12106/4269 )
**u= 89/31 , ht= 11737 ; ( -23111/4635 , -6182/4635 ), ( 1531/11737 , -82/1067 )
**u= 89/33 , ht= 439 ; ( 241/439 , 744/439 ), ( 2873/1553 , 4868/1553 )
**u= 89/35 , ht= 7149 ; ( -8997/1073 , 5262/1073 ), ( 6949/7149 , 4120/2383 )
**u= 89/36 , ht= 99 ; ( -99/73 , 413/219 ), ( 73/99 , 413/297 )
**u= 89/43 , ht= 615 ; ( 529/615 , 1172/615 ), ( -1904151/240431 , -3288056/240431 )
**u= 89/46 , ht= 639 ; ( -127881/72127 , -183786/72127 ), ( 455/639 , -346/213 )
**u= 89/49 , ht= 10111 ; ( -14533/2931 , -121252/20517 ), ( -10111/1213 , -125198/8491 )
**u= 89/51 , ht= 1423 ; ( 1423/301 , -254/43 ), ( -22925/12181 , -43288/12181 )
**u= 89/52 , ht= 179 ; ( -1643/2907 , -5566/2907 ), ( -55/179 , -239/179 )
**u= 89/54 , ht= 1351 ; ( 38773/2389 , -147832/7167 ), ( -1058/1351 , -15373/8106 )
**u= 89/60 , ht= 2681 ; ( -10687/1217 , -15112/1217 ), ( -2029/2681 , 5311/2681 )
**u= 89/62 , ht= 8704 ; ( -2859/8704 , -1977/1024 ), ( -952/10797 , -10421/7198 )
**u= 89/66 , ht= 461 ; ( 2036/463 , 6413/926 ), ( -461/188 , -1849/376 )
**u= 89/76 , ht= 1645 ; ( -407/1645 , 3361/1645 ), ( 17827/5401 , -36799/5401 )
**u= 89/80 , ht= 13 ; ( -3/13 , -27/13 ), ( 13/3 , -9 )
**u= 89/82 , ht= 2593 ; ( 2593/215 , -4618/215 ), ( -215/2593 , 4618/2593 )
**u= 89/84 , ht= 14149 ; ( -20683/6085 , -39342/6085 ), ( -9413/14149 , -32112/14149 )
**u= 89/85 , ht= 7639 ; ( 7639/5939 , -18590/5939 ), ( -9453/8567 , -25102/8567 )
**u= 89/87 , ht= 15227 ; ( 7183/16637 , 37406/16637 ), ( 3743/15227 , -29154/15227 )
**u= 89/90 , ht= 9883 ; ( 26581/9571 , 54078/9571 ), ( 9883/2717 , -1970/247 )
**u= 89/100 , ht= 3703 ; ( -2169/3703 , 6654/2645 ), ( 3703/2169 , 15526/3615 )
**u= 90/11 , ht= 557 ; ( 271/557 , 3226/1671 ), ( -569/157 , 3542/471 )
**u= 90/23 , ht= 29 ; ( -29/17 , -26/51 ), ( -113/44 , 571/132 )
**u= 90/31 , ht= 53 ; ( 10833/1781 , -1670/5343 ), ( -9/53 , -14/159 )
**u= 90/79 , ht= 51 ; ( 444/523 , 3907/1569 ), ( -51/28 , -341/84 )
**u= 90/83 , ht= 1621 ; ( 1621/127 , -8662/381 ), ( -6833/2629 , -44426/7887 )
**u= 90/91 , ht= 1679 ; ( 33261/137 , -188006/411 ), ( 1679/678 , 11395/2034 )
**u= 97/2 , ht= 747 ; ( -56/747 , -7349/1494 ), ( -3277/411 , -5346/137 )
**u= 97/5 , ht= 783 ; ( -137/783 , 2428/783 ), ( 2861/2921 , -286/2921 )
**u= 97/6 , ht= 5690 ; ( 4997/5690 , 17211/11380 ), ( -20371/12341 , 47578/12341 )
**u= 97/7 , ht= 5875 ; ( 2277/5875 , -14652/5875 ), ( -7955/1089 , -214/11 )
**u= 97/9 , ht= 571 ; ( -191/1001 , 2364/1001 ), ( -571/49 , 1366/49 )
**u= 97/11 , ht= 101 ; ( -9/101 , 222/101 ), ( 59747/1953 , -43898/651 )
**u= 97/14 , ht= 869 ; ( 1039/1004 , 1957/2008 ), ( 869/379 , -1624/379 )
**u= 97/17 , ht= 233 ; ( -7471/10617 , 16748/10617 ), ( -209/233 , -196/233 )
**u= 97/20 , ht= 47 ; ( 47/33 , 2/33 ), ( -107/123 , -38/41 )
**u= 97/25 , ht= 3433 ; ( 933/3433 , -5850/3433 ), ( 7123/2277 , 4028/759 )
**u= 97/27 , ht= 329 ; ( 7739/3857 , 662/1653 ), ( -329/157 , -546/157 )
**u= 97/32 , ht= 1523 ; ( -1523/429 , -259/429 ), ( 4385/2567 , 14591/5134 )
**u= 97/39 , ht= 259 ; ( 259/25 , 162/25 ), ( -4103/1123 , 6962/1123 )
**u= 97/42 , ht= 1774 ; ( -6817/1079 , -5494/1079 ), ( 325/1774 , -2913/3548 )
**u= 97/45 , ht= 393 ; ( -393/13 , -1042/39 ), ( 42991/4701 , 221246/14103 )
**u= 97/53 , ht= 511 ; ( -2399/8837 , -15794/8837 ), ( 87/511 , -596/511 )
**u= 97/56 , ht= 4791 ; ( -10057/3295 , -13433/3295 ), ( 2776/4791 , 10119/6388 )
**u= 97/62 , ht= 14639 ; ( 104909/15443 , -142568/15443 ), ( 503/14639 , 19522/14639 )
**u= 97/63 , ht= 461 ; ( -80519/2235 , 326834/6705 ), ( 461/43 , -848/43 )
**u= 97/65 , ht= 4711 ; ( 4711/1409 , 7052/1409 ), ( 1882339/643959 , -1199218/214653 )
**u= 97/66 , ht= 4409 ; ( -28748/9037 , -87733/18074 ), ( 3199/4409 , -8602/4409 )
**u= 97/68 , ht= 6547 ; ( 10567/2169 , -15812/2169 ), ( 5421/6547 , 13892/6547 )
**u= 97/71 , ht= 7245 ; ( -4831/7245 , -2222/1035 ), ( 186753/112583 , 392924/112583 )
**u= 97/75 , ht= 347 ; ( -2890571/238079 , 4544254/238079 ), ( -29/347 , 2728/1735 )
**u= 97/80 , ht= 173 ; ( 1493/14513 , 2503/1262 ), ( -23/173 , -575/346 )
**u= 97/83 , ht= 1779 ; ( -516507/29965 , 872244/29965 ), ( -1779/47 , -3562/47 )
**u= 97/88 , ht= 819 ; ( -738801/164695 , -1339137/164695 ), ( -170/819 , -657/364 )
**u= 97/95 , ht= 347 ; ( -38479/3511 , 71408/3511 ), ( -223/347 , 794/347 )
**u= 97/96 , ht= 12778 ; ( -31177/4178 , -468857/33424 ), ( 12778/6877 , 238957/55016 )
**u= 98 , ht= 1 ; ( 1 , 2/7 ), ( -140/129 , -893/301 )
**u= 98/9 , ht= 1 ; ( -67/112 , -1567/784 ), ( -1 , -6/7 )
**u= 98/17 , ht= 501 ; ( -219/512 , 6387/3584 ), ( 400/501 , 353/1169 )
**u= 98/25 , ht= 1 ; ( -1 , 10/7 ), ( 1 , 10/7 )
**u= 98/37 , ht= 337 ; ( -337/41 , 1126/287 ), ( -145/441 , 722/1029 )
**u= 98/53 , ht= 242 ; ( -999/310 , 8667/2170 ), ( 242/189 , 123/49 )
**u= 98/69 , ht= 599 ; ( -1777/247 , -18342/1729 ), ( 599 , 7874/7 )
**u= 98/73 , ht= 356 ; ( -283/356 , 5633/2492 ), ( 1317/149 , -17686/1043 )
**u= 98/81 , ht= 1 ; ( -1 , 18/7 ), ( 256/399 , 2747/1323 )
**u= 98/89 , ht= 14229 ; ( 61281/37340 , 923547/261380 ), ( -7109/14229 , -67442/33201 )
**u= 98/93 , ht= 398 ; ( 181/398 , 6229/2786 ), ( -539/1693 , -22818/11851 )
**u= 98/97 , ht= 113 ; ( 113/16 , -1489/112 ), ( -121/73 , -2026/511 )
**u= 100/23 , ht= 382 ; ( 382/313 , -1733/1565 ), ( 477/362 , -3673/1810 )
**u= 100/87 , ht= 394 ; ( 394/133 , 723/133 ), ( -728/229 , 1517/229 )
507
>

■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。

[2026.05.23追記]u=2/25.9/98のときの整数解を追加した。
[2026.05.24追記]u=9/50,9/100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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