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Integer Points on A^4+12*B^4=C^4+12*D^4

[2026.05.27]A^4+12*B^4=C^4+12*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=12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662個のuが抽出される。
[MAGMAによる計算]
> PP(12,1,100);
**u= 1 , ht= 5 ; ( 5 , 12 ), ( 3/5 , 14/5 )
**u= 1/2 , ht= 1 ; ( 0 , 7/2 ), ( 23/7 , -12 )
**u= 1/3 , ht= 211 ; ( 145/211 , 1088/211 ), ( -583/653 , 3712/653 )
**u= 1/4 , ht= 19 ; ( 5/21 , 106/21 ), ( 7/19 , 99/19 )
**u= 1/6 , ht= 19 ; ( 307/61 , 1876/61 ), ( 2/19 , 229/38 )
**u= 1/11 , ht= 17 ; ( 75/53 , 746/53 ), ( -17/5 , 144/5 )
**u= 1/12 , ht= 7 ; ( -335/271 , 3656/271 ), ( -7/5 , 73/5 )
**u= 1/13 , ht= 33 ; ( -43/113 , 1068/113 ), ( 33/7 , -298/7 )
**u= 1/14 , ht= 99 ; ( -99/83 , 1184/83 ), ( 313/145 , 3162/145 )
**u= 1/17 , ht= 201 ; ( -201/139 , 2468/139 ), ( 37/275 , 2802/275 )
**u= 1/18 , ht= 13 ; ( -28/143 , -233/22 ), ( 13/3 , 416/9 )
**u= 1/19 , ht= 719 ; ( -719/3 , 7676/3 ), ( -849/785 , -12346/785 )
**u= 1/20 , ht= 117 ; ( 227/167 , -3087/167 ), ( 17/117 , 1295/117 )
**u= 1/24 , ht= 38 ; ( 109/178 , -10019/712 ), ( -38/11 , -1899/44 )
**u= 1/25 , ht= 121 ; ( 19/123 , 7622/615 ), ( 17/121 , -7482/605 )
**u= 1/27 , ht= 17491 ; ( 16849/17491 , 927344/52473 ), ( -51847/52493 , 2817232/157479 )
**u= 1/29 , ht= 1165 ; ( -1165/1097 , -21108/1097 ), ( -737/2859 , 38944/2859 )
**u= 1/39 , ht= 277 ; ( 121/277 , -4624/277 ), ( 277/121 , -4624/121 )
**u= 1/41 , ht= 3631 ; ( 597/3631 , -57716/3631 ), ( -4423/1645 , 74016/1645 )
**u= 1/42 , ht= 1357 ; ( -323/1357 , -22144/1357 ), ( -3221/251 , -51288/251 )
**u= 1/46 , ht= 77 ; ( -61/77 , -1632/77 ), ( -5609/2303 , -100734/2303 )
**u= 1/48 , ht= 167035 ; ( -167035/165881 , -7990031/331762 ), ( -167047/165893 , -7990609/331786 )
**u= 1/49 , ht= 5 ; ( -8813/1179 , 1067186/8253 ), ( -1/5 , 612/35 )
**u= 1/51 , ht= 2597 ; ( -3479/23 , -2646 ), ( -1679/2597 , 1104/53 )
**u= 1/53 , ht= 293 ; ( -1955/59 , -34878/59 ), ( -293/25 , 5244/25 )
**u= 1/55 , ht= 4759 ; ( 12653/3 , 229850/3 ), ( 4759/1261 , -89436/1261 )
**u= 1/56 , ht= 11 ; ( -3/11 , -19 ), ( 11/3 , -209/3 )
**u= 1/61 , ht= 1533 ; ( 1045/1533 , 35494/1533 ), ( 1533/1045 , 35494/1045 )
**u= 1/66 , ht= 157 ; ( -6073/1649 , 125226/1649 ), ( -157/80 , 7013/160 )
**u= 1/68 , ht= 279 ; ( -271/749 , -16089/749 ), ( 41/279 , -5696/279 )
**u= 1/75 , ht= 1 ; ( 8879/9635 , 1389708/48175 ), ( -1 , 30 )
**u= 1/77 , ht= 65 ; ( -65/23 , -1482/23 ), ( -23/65 , 114/5 )
**u= 1/78 , ht= 187 ; ( -187/112 , 9431/224 ), ( -401/4201 , 91294/4201 )
**u= 1/80 , ht= 1409 ; ( 12223/1663 , -270258/1663 ), ( 1409/639 , 33896/639 )
**u= 1/82 , ht= 2080 ; ( -2080/749 , 98073/1498 ), ( 6369/5836 , 383221/11672 )
**u= 1/84 , ht= 401 ; ( -401/47 , 9064/47 ), ( 2221/877 , 53608/877 )
**u= 1/90 , ht= 314 ; ( -314/111 , 1255/18 ), ( -333/1178 , 170681/7068 )
**u= 1/91 , ht= 2949 ; ( 4889/1067 , -116928/1067 ), ( 2949/37 , 68914/37 )
**u= 1/94 , ht= 1781 ; ( 989/2963 , -74184/2963 ), ( -1368/1781 , -106667/3562 )
**u= 1/95 , ht= 5227 ; ( -6107/3583 , -169044/3583 ), ( 5227/4087 , 158412/4087 )
**u= 2 , ht= 1 ; ( 0 , -2 ), ( -1/4 , 3/2 )
**u= 2/5 , ht= 3 ; ( -2/3 , -14/3 ), ( -3/2 , 7 )
**u= 2/17 , ht= 44 ; ( 44/7 , -318/7 ), ( -7/44 , -159/22 )
**u= 2/21 , ht= 103 ; ( -1618/517 , -13478/517 ), ( -103/2 , -409 )
**u= 2/37 , ht= 374 ; ( -374/107 , -4098/107 ), ( -487/450 , -3493/225 )
**u= 2/65 , ht= 1444 ; ( -2456/2289 , -46882/2289 ), ( -1119/1444 , -12755/722 )
**u= 2/69 , ht= 2063 ; ( -2242/1507 , -38866/1507 ), ( -2063/950 , -16339/475 )
**u= 2/85 , ht= 366 ; ( -366/179 , 6506/179 ), ( 28037/378 , 223883/189 )
**u= 3 , ht= 5 ; ( -11 , -8 ), ( 1/5 , -4/5 )
**u= 3/4 , ht= 1 ; ( -1 , 4 ), ( -103/133 , 473/133 )
**u= 3/7 , ht= 17 ; ( 11/17 , 76/17 ), ( -37/61 , -266/61 )
**u= 3/10 , ht= 4 ; ( 1/4 , -37/8 ), ( -4 , 37/2 )
**u= 3/16 , ht= 1 ; ( -1 , -8 ), ( 1 , 8 )
**u= 3/22 , ht= 7 ; ( -4/7 , -107/14 ), ( -7/4 , -107/8 )
**u= 3/25 , ht= 1 ; ( -1 , -10 ), ( -1 , 10 )
**u= 3/31 , ht= 137 ; ( 2125/203 , -16802/203 ), ( 137/13 , -1084/13 )
**u= 3/43 , ht= 199 ; ( -199/113 , 2122/113 ), ( -409/1183 , 1658/169 )
**u= 3/46 , ht= 197 ; ( -875/827 , -11548/827 ), ( 197/13 , 1894/13 )
**u= 3/49 , ht= 1 ; ( -1 , 14 ), ( -1 , -14 )
**u= 3/55 , ht= 119 ; ( 973/2503 , -28168/2503 ), ( 79/119 , 214/17 )
**u= 3/58 , ht= 172 ; ( 85/172 , 4133/344 ), ( -1097/121 , 11888/121 )
**u= 3/64 , ht= 1 ; ( 3893/4663 , -274903/18652 ), ( -1 , 16 )
**u= 3/70 , ht= 11 ; ( 4/11 , 277/22 ), ( -11/4 , 277/8 )
**u= 3/73 , ht= 917 ; ( 1895/581 , -23948/581 ), ( 917/41 , 11092/41 )
**u= 3/94 , ht= 953 ; ( 1382/583 , 41131/1166 ), ( -953/457 , -14492/457 )
**u= 3/100 , ht= 433 ; ( -433/85 , 31201/425 ), ( -80807/80005 , -8040703/400025 )
**u= 4/3 , ht= 85 ; ( 85/43 , -193/43 ), ( 101/59 , 257/59 )
**u= 4/11 , ht= 163 ; ( -163/101 , -777/101 ), ( 469/117 , -1973/117 )
**u= 4/27 , ht= 25 ; ( 851/371 , -17713/1113 ), ( -17/25 , -577/75 )
**u= 4/31 , ht= 723 ; ( -2369/1723 , 19971/1723 ), ( -723/455 , -5827/455 )
**u= 4/35 , ht= 127 ; ( 57/127 , 1009/127 ), ( -2389/61 , 17325/61 )
**u= 4/55 , ht= 1963 ; ( -1963/1447 , 22149/1447 ), ( -2033/1157 , 21249/1157 )
**u= 4/63 , ht= 443 ; ( -443/33 , -12953/99 ), ( -5389/281 , 52467/281 )
**u= 4/67 , ht= 215 ; ( 215/81 , -2303/81 ), ( 19029/6299 , -200969/6299 )
**u= 4/75 , ht= 34645 ; ( 34645/33739 , -2564617/168695 ), ( 34661/33755 , 2565833/168775 )
**u= 4/83 , ht= 795 ; ( 795/629 , -11311/629 ), ( -997/1493 , 20031/1493 )
**u= 6 , ht= 1 ; ( 0 , -2 ), ( 3/4 , -1/2 )
**u= 6/37 , ht= 51 ; ( 14/51 , 322/51 ), ( 633/70 , -277/5 )
**u= 6/83 , ht= 769 ; ( -520/769 , -8458/769 ), ( 3649/584 , 16837/292 )
**u= 6/85 , ht= 201 ; ( -2218/4887 , -49486/4887 ), ( 201/14 , -929/7 )
**u= 6/97 , ht= 576 ; ( 576/479 , 7378/479 ), ( 1029/1784 , -10141/892 )
**u= 10/3 , ht= 2 ; ( -2 , 2 ), ( -29/2 , 27 )
**u= 10/7 , ht= 29 ; ( 8/29 , -66/29 ), ( 3/136 , 127/68 )
**u= 10/19 , ht= 34 ; ( 34/23 , -6 ), ( 23/34 , 69/17 )
**u= 10/71 , ht= 961 ; ( 748/961 , 7950/961 ), ( 4231/468 , -13903/234 )
**u= 10/87 , ht= 79 ; ( -8/79 , -574/79 ), ( 427/124 , 1607/62 )
**u= 10/99 , ht= 614 ; ( 614/277 , 5190/277 ), ( 821/282 , 10039/423 )
**u= 12 , ht= 25 ; ( -25/23 , 1/23 ), ( -23/25 , -1/25 )
**u= 12/13 , ht= 35 ; ( 185/29 , 461/29 ), ( -17/35 , 97/35 )
**u= 12/25 , ht= 1 ; ( 1 , -5 ), ( 1 , -5 )
**u= 12/49 , ht= 4975 ; ( -4975/4777 , -238949/33439 ), ( -5023/4825 , -241349/33775 )
**u= 12/61 , ht= 1087 ; ( 965/1087 , 8029/1087 ), ( -413/1783 , 10093/1783 )
**u= 12/85 , ht= 61 ; ( -689/1691 , 11911/1691 ), ( -41/61 , -479/61 )
**u= 12/97 , ht= 241 ; ( 2945/1393 , 22679/1393 ), ( 241/47 , 1711/47 )
**u= 13 , ht= 77 ; ( 49/141 , 352/141 ), ( -73/77 , -36/77 )
**u= 13/2 , ht= 11 ; ( 3/11 , -2 ), ( 11/3 , -22/3 )
**u= 13/3 , ht= 35 ; ( 343/181 , -156/181 ), ( -19/35 , 18/35 )
**u= 13/4 , ht= 5 ; ( 5 , -3 ), ( 1/5 , -3/5 )
**u= 13/6 , ht= 16 ; ( -431/16 , 1121/32 ), ( -16/11 , -69/22 )
**u= 13/7 , ht= 173 ; ( 19/173 , -354/173 ), ( 179/193 , -468/193 )
**u= 13/8 , ht= 45 ; ( -170/47 , -1221/188 ), ( -43/45 , -119/45 )
**u= 13/10 , ht= 37 ; ( -148/287 , 1443/574 ), ( 3/37 , 2 )
**u= 13/11 , ht= 29 ; ( -15/29 , 76/29 ), ( -163/563 , 1254/563 )
**u= 13/14 , ht= 615 ; ( 3075/3194 , 22571/6388 ), ( -358/615 , 3553/1230 )
**u= 13/16 , ht= 5 ; ( 5 , 27/2 ), ( 1/5 , 27/10 )
**u= 13/17 , ht= 679 ; ( 1549/9097 , 26436/9097 ), ( -613/679 , -2556/679 )
**u= 13/18 , ht= 17 ; ( 6/17 , 317/102 ), ( 17/6 , 317/36 )
**u= 13/22 , ht= 7 ; ( -7/4 , -51/8 ), ( 1387/4621 , -15186/4621 )
**u= 13/23 , ht= 63 ; ( 13587/1271 , -43876/1271 ), ( 29/63 , 32/9 )
**u= 13/25 , ht= 31 ; ( -25/31 , -678/155 ), ( 31/25 , -678/125 )
**u= 13/30 , ht= 274 ; ( -1187/838 , 10785/1676 ), ( 274/109 , 2207/218 )
**u= 13/32 , ht= 43 ; ( -19/43 , -363/86 ), ( 89/562 , -17379/4496 )
**u= 13/35 , ht= 327 ; ( -327/23 , 1310/23 ), ( -43217/8373 , -177868/8373 )
**u= 13/38 , ht= 2648 ; ( -1505/2732 , -26193/5464 ), ( -2648/449 , 22599/898 )
**u= 13/44 , ht= 619 ; ( -3151/1249 , -15234/1249 ), ( -619/415 , -3363/415 )
**u= 13/46 , ht= 1647 ; ( -2969/2666 , -36759/5332 ), ( -1647/865 , 8588/865 )
**u= 13/48 , ht= 2585 ; ( 2585/1511 , -14072/1511 ), ( -3451/815 , -33471/1630 )
**u= 13/50 , ht= 438 ; ( -386/5389 , 52053/10778 ), ( 137/438 , 4399/876 )
**u= 13/51 , ht= 485 ; ( -485/331 , -2846/331 ), ( -3431/2995 , 22104/2995 )
**u= 13/55 , ht= 27011 ; ( -27011/3811 , 137130/3811 ), ( 40957/18737 , -227268/18737 )
**u= 13/57 , ht= 805 ; ( -47555/30649 , -289924/30649 ), ( 83/805 , 4142/805 )
**u= 13/58 , ht= 53 ; ( 53888/6879 , -561007/13758 ), ( 53/5 , 276/5 )
**u= 13/59 , ht= 2203 ; ( 2203/1967 , -15408/1967 ), ( -1159/2499 , -14356/2499 )
**u= 13/62 , ht= 789 ; ( -15189/4837 , 85144/4837 ), ( 293/789 , -4496/789 )
**u= 13/63 , ht= 2029 ; ( -2029/1737 , -43196/5211 ), ( -6333/289 , 102736/867 )
**u= 13/66 , ht= 1603 ; ( 1603/877 , 10076/877 ), ( -3647/13282 , -663/116 )
**u= 13/67 , ht= 677 ; ( -395/677 , 4362/677 ), ( 2463/32881 , 183074/32881 )
**u= 13/69 , ht= 8747 ; ( -6295/8747 , 60844/8747 ), ( -11441/1945 , -65582/1945 )
**u= 13/70 , ht= 883 ; ( 883/853 , -6978/853 ), ( -38203/4427 , 218904/4427 )
**u= 13/74 , ht= 1289 ; ( 5373/5077 , 43198/5077 ), ( 1289/660 , 16939/1320 )
**u= 13/76 , ht= 1879 ; ( 1879/1309 , 13557/1309 ), ( 3673/8537 , 54996/8537 )
**u= 13/77 , ht= 765 ; ( 765/523 , 5522/523 ), ( -17453/719 , 104256/719 )
**u= 13/79 , ht= 4881 ; ( 4091/4881 , 38464/4881 ), ( 21389/6221 , 134634/6221 )
**u= 13/80 , ht= 2657 ; ( 123/2773 , -16885/2773 ), ( 2657/1377 , -18196/1377 )
**u= 13/81 , ht= 379 ; ( 55/379 , -2344/379 ), ( 2481/55 , -136706/495 )
**u= 13/83 , ht= 6183 ; ( 6183/5851 , -52684/5851 ), ( 8419/1613 , -53106/1613 )
**u= 13/85 , ht= 2911 ; ( -2911/2631 , 24574/2631 ), ( 54531/49601 , -461752/49601 )
**u= 13/87 , ht= 9977 ; ( 1835/12971 , -83086/12971 ), ( -9977/4871 , 70394/4871 )
**u= 13/90 , ht= 201 ; ( -4/201 , 7781/1206 ), ( 201/4 , 7781/24 )
**u= 13/92 , ht= 33671 ; ( -33671/3037 , -220119/3037 ), ( -250223/8481 , 1632806/8481 )
**u= 13/93 , ht= 1831 ; ( 29075/4151 , -192268/4151 ), ( 1831/653 , 12744/653 )
**u= 13/95 , ht= 2423 ; ( -4077/103 , -26984/103 ), ( 1093/2423 , -17592/2423 )
**u= 13/98 , ht= 301 ; ( -6507/12211 , 93094/12211 ), ( 301/218 , 35001/3052 )
**u= 13/99 , ht= 409 ; ( -9/409 , 8302/1227 ), ( -409/9 , -8302/27 )
**u= 13/100 , ht= 777 ; ( 421/779 , 6018/779 ), ( -777/275 , -28013/1375 )
**u= 15 , ht= 3 ; ( 11/27 , -70/27 ), ( 3 , 8 )
**u= 15/4 , ht= 3 ; ( -3 , -1 ), ( 87/229 , -109/229 )
**u= 15/7 , ht= 11 ; ( -331/33 , 40/3 ), ( 11/3 , -22/3 )
**u= 15/13 , ht= 489 ; ( 307/489 , -1348/489 ), ( -1347/161 , 3256/161 )
**u= 15/19 , ht= 189 ; ( -189/37 , 14 ), ( -37/189 , -74/27 )
**u= 15/22 , ht= 19 ; ( 262/9 , -1525/18 ), ( -3/19 , 56/19 )
**u= 15/28 , ht= 31 ; ( 39/53 , 221/53 ), ( -3/31 , -103/31 )
**u= 15/31 , ht= 309 ; ( 247/309 , 1396/309 ), ( 57/20579 , 71756/20579 )
**u= 15/46 , ht= 1008 ; ( 196/1737 , 15061/3474 ), ( -901/1008 , -11593/2016 )
**u= 15/76 , ht= 4311 ; ( 4311/587 , -23951/587 ), ( -19287/2221 , 107213/2221 )
**u= 15/82 , ht= 2151 ; ( 63477/11899 , 369392/11899 ), ( 388/2151 , -25003/4302 )
**u= 15/91 , ht= 333 ; ( -6087/19211 , 121696/19211 ), ( -333/31 , 2020/31 )
**u= 16/3 , ht= 1 ; ( 1 , -3/2 ), ( 11/5 , 41/10 )
**u= 16/7 , ht= 85 ; ( 85 , 207/2 ), ( 1/85 , 207/170 )
**u= 16/23 , ht= 133 ; ( 87/133 , 941/266 ), ( 301/25 , 1809/50 )
**u= 16/27 , ht= 353 ; ( 4573/4243 , -118969/25458 ), ( -353/95 , -7067/570 )
**u= 16/31 , ht= 251 ; ( -289/83 , -2031/166 ), ( -183/251 , -2111/502 )
**u= 16/35 , ht= 69 ; ( -69/29 , -539/58 ), ( 467/117 , 3515/234 )
**u= 16/43 , ht= 63 ; ( 223/3663 , -29641/7326 ), ( -63/23 , -541/46 )
**u= 16/47 , ht= 485 ; ( -485/79 , 4107/158 ), ( -79/485 , -4107/970 )
**u= 16/71 , ht= 105 ; ( -105/37 , -31/2 ), ( 1813/961 , 21201/1922 )
**u= 16/75 , ht= 34525 ; ( 34525/33619 , -2555497/336190 ), ( 34781/33875 , 2574953/338750 )
**u= 16/79 , ht= 3189 ; ( 503/3189 , 35201/6378 ), ( 2791/5477 , -66849/10954 )
**u= 16/87 , ht= 1727 ; ( 1027/2593 , -31893/5186 ), ( -1727/13 , 19757/26 )
**u= 16/91 , ht= 893 ; ( -893/381 , 11333/762 ), ( -669/2243 , 27317/4486 )
**u= 24 , ht= 1 ; ( 0 , 7/2 ), ( -24/7 , 23/2 )
**u= 24/25 , ht= 21 ; ( 21/4 , 103/8 ), ( 12/29 , 763/290 )
**u= 24/37 , ht= 22 ; ( 282/23 , 1691/46 ), ( -22/15 , -163/30 )
**u= 24/61 , ht= 31 ; ( -30/31 , 337/62 ), ( 3138/3809 , -38497/7618 )
**u= 24/85 , ht= 138 ; ( -83/138 , -1487/276 ), ( 3486/1021 , -33583/2042 )
**u= 25 , ht= 15 ; ( -7/15 , 238/75 ), ( 57/35 , -116/25 )
**u= 25/2 , ht= 43 ; ( -24/43 , -191/86 ), ( -43/24 , -191/48 )
**u= 25/3 , ht= 113 ; ( -199/293 , -532/293 ), ( -113/91 , 184/91 )
**u= 25/6 , ht= 11 ; ( 11/5 , 16/25 ), ( -5/11 , 16/55 )
**u= 25/7 , ht= 17 ; ( -17/3 , 4/15 ), ( 3/17 , 4/85 )
**u= 25/9 , ht= 63 ; ( 55/63 , 1922/945 ), ( -63/55 , -1922/825 )
**u= 25/11 , ht= 51 ; ( 51/7 , 64/7 ), ( 7/51 , 64/51 )
**u= 25/12 , ht= 2497 ; ( 2497/2203 , -27611/11015 ), ( -14407/14117 , -171781/70585 )
**u= 25/13 , ht= 79 ; ( -79/51 , 776/255 ), ( -13401/431 , -135406/2155 )
**u= 25/14 , ht= 4 ; ( -3/4 , 19/8 ), ( 4/3 , -19/6 )
**u= 25/16 , ht= 85 ; ( -85/63 , 2011/630 ), ( -37/85 , -843/425 )
**u= 25/17 , ht= 117 ; ( 321/253 , 808/253 ), ( -19/117 , -218/117 )
**u= 25/24 , ht= 46 ; ( 5/46 , -2317/920 ), ( -46/5 , 2317/100 )
**u= 25/26 , ht= 225 ; ( 225/199 , -3734/995 ), ( -571/3147 , 7696/3147 )
**u= 25/27 , ht= 1513 ; ( 1513/427 , 58172/6405 ), ( -55591/56237 , 603632/168711 )
**u= 25/28 , ht= 269 ; ( -269/81 , 3539/405 ), ( 487/473 , 8808/2365 )
**u= 25/37 , ht= 103 ; ( 463/537 , -10594/2685 ), ( 103/29 , -324/29 )
**u= 25/38 , ht= 86 ; ( -9/86 , -2657/860 ), ( 317/469 , -1698/469 )
**u= 25/39 , ht= 12583 ; ( -12583/1769 , 38230/1769 ), ( 13205/517 , -205574/2585 )
**u= 25/46 , ht= 145 ; ( 145/7 , 2382/35 ), ( -1719/3295 , -61306/16475 )
**u= 25/48 , ht= 1 ; ( 1 , 24/5 ), ( -11/49 , 241/70 )
**u= 25/49 , ht= 153 ; ( -65/153 , -20102/5355 ), ( 83/181 , 678/181 )
**u= 25/52 , ht= 737 ; ( 607/737 , -1536/335 ), ( -2497/79 , 8910/79 )
**u= 25/53 , ht= 863 ; ( -1193/1457 , -33642/7285 ), ( -863/409 , -3426/409 )
**u= 25/54 , ht= 1949 ; ( 10435/6061 , 216252/30305 ), ( -1949/907 , 23350/2721 )
**u= 25/58 , ht= 433 ; ( -31/433 , -1632/433 ), ( -1041/1337 , 6310/1337 )
**u= 25/59 , ht= 2905 ; ( 2905/1607 , 62214/8035 ), ( 24935/2279 , -474558/11395 )
**u= 25/62 , ht= 2543 ; ( 993/2543 , 52916/12715 ), ( 8636/173 , 67089/346 )
**u= 25/66 , ht= 2756 ; ( -2756/1117 , -23569/2234 ), ( 3503/1040 , 11243/800 )
**u= 25/67 , ht= 2073 ; ( -473/2073 , 42854/10365 ), ( -3037/5807 , -130956/29035 )
**u= 25/68 , ht= 159 ; ( -1467/3973 , 85912/19865 ), ( -79/159 , 3574/795 )
**u= 25/72 , ht= 203 ; ( -132/203 , -60517/12180 ), ( -203/132 , 60517/7920 )
**u= 25/81 , ht= 1225 ; ( -85/2607 , -519572/117315 ), ( 1225/433 , -86194/6495 )
**u= 25/84 , ht= 2315 ; ( -7079/1895 , -163992/9475 ), ( -2315/2011 , -68878/10055 )
**u= 25/87 , ht= 1927 ; ( 1927/187 , -44084/935 ), ( -17743/5573 , 426104/27865 )
**u= 25/94 , ht= 215 ; ( -15437/10084 , -174951/20168 ), ( 215/121 , 5868/605 )
**u= 25/97 , ht= 649 ; ( 649/321 , -17438/1605 ), ( -19683/46277 , 172982/33055 )
**u= 26 , ht= 11 ; ( 102/101 , -46/101 ), ( -11/10 , -9/5 )
**u= 26/33 , ht= 1130 ; ( -5602/3193 , -17558/3193 ), ( 421/1130 , 1631/565 )
**u= 26/77 , ht= 1487 ; ( 26800/4099 , 113766/4099 ), ( 1487/1340 , 63/10 )
**u= 26/81 , ht= 46 ; ( 46/17 , -634/51 ), ( 17/46 , -317/69 )
**u= 26/93 , ht= 68 ; ( 68/37 , 358/37 ), ( -122027/232148 , 606371/116074 )
**u= 26/97 , ht= 1213 ; ( 4130/929 , 19974/929 ), ( 1213/630 , -3239/315 )
**u= 27 , ht= 1 ; ( 25/43 , -392/129 ), ( -1 , -2/3 )
**u= 27/4 , ht= 389 ; ( -389/359 , 1249/1077 ), ( -583/613 , 2153/1839 )
**u= 27/7 , ht= 61 ; ( -61/23 , -64/69 ), ( -23/61 , 64/183 )
**u= 27/10 , ht= 133 ; ( -133/107 , 712/321 ), ( 28/163 , 967/978 )
**u= 27/13 , ht= 19 ; ( -19/7 , 88/21 ), ( -47/395 , -1636/1185 )
**u= 27/16 , ht= 7 ; ( -281/131 , -3229/786 ), ( -7/5 , 101/30 )
**u= 27/22 , ht= 73 ; ( 40/73 , -1141/438 ), ( -493/200 , 7369/1200 )
**u= 27/25 , ht= 1 ; ( 1 , -10/3 ), ( 1/5 , 172/75 )
**u= 27/46 , ht= 199 ; ( -673/17 , -6364/51 ), ( -199/31 , -1958/93 )
**u= 27/49 , ht= 1 ; ( 1 , -14/3 ), ( 7/13 , -1016/273 )
**u= 27/64 , ht= 1 ; ( -3175/2461 , -181457/29532 ), ( -1 , 16/3 )
**u= 27/67 , ht= 12601 ; ( 14953/2491 , 174352/7473 ), ( -12601/2273 , -149158/6819 )
**u= 27/79 , ht= 293 ; ( -23/293 , -3712/879 ), ( -4073/193 , -51502/579 )
**u= 27/85 , ht= 761 ; ( 31063/8453 , -418216/25359 ), ( -761/259 , -10516/777 )
**u= 27/97 , ht= 6241 ; ( 1003/6241 , 88312/18723 ), ( 6241/1003 , 88312/3009 )
**u= 27/100 , ht= 1 ; ( 263/125 , -20551/1875 ), ( 1 , 20/3 )
**u= 30/11 , ht= 79 ; ( -42/79 , 154/79 ), ( -419/1638 , 121/117 )
**u= 30/37 , ht= 152 ; ( 152/41 , -418/41 ), ( -181/152 , 17/4 )
**u= 30/73 , ht= 47 ; ( -26/47 , -206/47 ), ( -101851/11518 , 197165/5759 )
**u= 34/7 , ht= 11 ; ( 122/73 , -42/73 ), ( -11/10 , 9/5 )
**u= 34/11 , ht= 127 ; ( -276/67 , -214/67 ), ( 127/104 , 123/52 )
**u= 34/23 , ht= 79 ; ( -822/3203 , -7174/3203 ), ( 79/10 , 87/5 )
**u= 34/39 , ht= 283 ; ( 2806/557 , 7282/557 ), ( -283/10 , 383/5 )
**u= 34/59 , ht= 349 ; ( -172/349 , -1266/349 ), ( -349/172 , 633/86 )
**u= 34/71 , ht= 419 ; ( 5790/12251 , 48254/12251 ), ( -419/78 , -761/39 )
**u= 37/2 , ht= 69 ; ( 450/1037 , 5827/2074 ), ( -67/69 , 22/69 )
**u= 37/5 , ht= 109 ; ( 319/269 , 180/269 ), ( 109/49 , -216/49 )
**u= 37/6 , ht= 232 ; ( 305/353 , -556/353 ), ( -232/97 , 891/194 )
**u= 37/8 , ht= 17 ; ( -9/17 , 31/17 ), ( 811/18 , -6163/72 )
**u= 37/9 , ht= 101 ; ( -101/51 , -166/153 ), ( -1373/1615 , -2252/1615 )
**u= 37/10 , ht= 136 ; ( 136/39 , 5/6 ), ( -871/3376 , -351/6752 )
**u= 37/13 , ht= 17 ; ( 17/9 , -22/9 ), ( -8753/11189 , -18876/11189 )
**u= 37/14 , ht= 183 ; ( 183/62 , 427/124 ), ( -3506/4941 , -269/162 )
**u= 37/15 , ht= 979 ; ( 749/979 , -186/89 ), ( 6149/37819 , 43054/37819 )
**u= 37/17 , ht= 215 ; ( 215/79 , -318/79 ), ( -14761/67587 , -91976/67587 )
**u= 37/19 , ht= 635 ; ( 14251 , 20688 ), ( 369/635 , 1184/635 )
**u= 37/21 , ht= 685 ; ( 685/293 , -1246/293 ), ( -293/685 , 1246/685 )
**u= 37/22 , ht= 245 ; ( 3802519/602223 , -6404998/602223 ), ( 212/245 , 1203/490 )
**u= 37/25 , ht= 3771 ; ( 61537/45363 , -149716/45363 ), ( 3771/2275 , 6598/1625 )
**u= 37/26 , ht= 1932 ; ( -6489/4561 , -15814/4561 ), ( 307/1932 , 7361/3864 )
**u= 37/27 , ht= 673 ; ( 79/673 , -1522/673 ), ( -3323/661 , 22754/1983 )
**u= 37/28 , ht= 541 ; ( 541/47 , -1071/47 ), ( 3009/10177 , -21188/10177 )
**u= 37/33 , ht= 5021 ; ( -2767/5021 , 13628/5021 ), ( 46511/2615 , -113234/2615 )
**u= 37/34 , ht= 1791 ; ( 1570/1791 , 11257/3582 ), ( -80953/2942 , 398703/5884 )
**u= 37/35 , ht= 881 ; ( 69/881 , 2200/881 ), ( -3013/7977 , 19592/7977 )
**u= 37/38 , ht= 825 ; ( 825/8 , -3931/16 ), ( -33287/13871 , 91974/13871 )
**u= 37/41 , ht= 1171 ; ( -1171/551 , -3264/551 ), ( 26357/79657 , 210372/79657 )
**u= 37/45 , ht= 1637 ; ( -1011/2329 , -20968/6987 ), ( 17/1637 , -4296/1637 )
**u= 37/48 , ht= 949 ; ( 949/263 , 5377/526 ), ( 18659/7223 , 113669/14446 )
**u= 37/52 , ht= 5503 ; ( 17525/10401 , 58577/10401 ), ( -4611/5503 , 20771/5503 )
**u= 37/54 , ht= 3413 ; ( -10891/596 , 189857/3576 ), ( 3413/1907 , -35062/5721 )
**u= 37/56 , ht= 3429 ; ( -11965/1934 , 143553/7736 ), ( -701/3429 , -10369/3429 )
**u= 37/57 , ht= 1225 ; ( -407561/180337 , 1338882/180337 ), ( -827/1225 , -4464/1225 )
**u= 37/58 , ht= 77096 ; ( 5153837/2324771 , -17143812/2324771 ), ( -28377/77096 , -497347/154192 )
**u= 37/59 , ht= 49121 ; ( -128451/27191 , 399992/27191 ), ( 49121/2865 , -154654/2865 )
**u= 37/61 , ht= 7169 ; ( -5641/7169 , -28794/7169 ), ( 17793/6203 , 59972/6203 )
**u= 37/64 , ht= 775 ; ( 775/123 , -9977/492 ), ( -3143/3133 , 57189/12532 )
**u= 37/65 , ht= 1179 ; ( 1179/659 , -4354/659 ), ( 491401/27171 , -1619128/27171 )
**u= 37/71 , ht= 12037 ; ( -5145/12037 , -44764/12037 ), ( -11353/21165 , 80984/21165 )
**u= 37/76 , ht= 6277 ; ( -6277/1513 , 22467/1513 ), ( -919/10139 , 35391/10139 )
**u= 37/77 , ht= 10205 ; ( -10205/6581 , -42738/6581 ), ( 60997/8791 , 219768/8791 )
**u= 37/79 , ht= 20027 ; ( 9125/20027 , -79242/20027 ), ( -465617/91333 , 1712616/91333 )
**u= 37/80 , ht= 201 ; ( -12587/6443 , 50664/6443 ), ( -89/201 , 787/201 )
**u= 37/82 , ht= 710 ; ( -18046/6267 , -138391/12534 ), ( -69/710 , -5159/1420 )
**u= 37/83 , ht= 107227 ; ( 126765/84601 , 557344/84601 ), ( -107227/74709 , -480826/74709 )
**u= 37/85 , ht= 10071 ; ( -720811/504209 , 3257004/504209 ), ( 10071/2659 , -38936/2659 )
**u= 37/86 , ht= 46564 ; ( -102517/11003 , -382128/11003 ), ( -46564/2069 , 350787/4138 )
**u= 37/87 , ht= 154003 ; ( -154003/77939 , -645406/77939 ), ( 8625817/209785 , -32651846/209785 )
**u= 37/89 , ht= 5305 ; ( 583/5567 , 21414/5567 ), ( -3061/5305 , -23184/5305 )
**u= 37/91 , ht= 8149 ; ( -25225/3609 , 97238/3609 ), ( -8049/8149 , -43996/8149 )
**u= 37/92 , ht= 62819 ; ( -141903/48629 , -576299/48629 ), ( 62819/46995 , -303598/46995 )
**u= 37/97 , ht= 120763 ; ( -120763/35601 , -496784/35601 ), ( -844017/350635 , 3640276/350635 )
**u= 37/98 , ht= 1603 ; ( -403/2011 , 57546/14077 ), ( 331/1603 , 45426/11221 )
**u= 37/99 , ht= 7371 ; ( -7371/529 , -88316/1587 ), ( 15239/7319 , 67982/7319 )
**u= 39/4 , ht= 301 ; ( 185/301 , 593/301 ), ( 401/367 , 556/367 )
**u= 39/10 , ht= 271 ; ( -503/202 , 391/404 ), ( -94/271 , 49/542 )
**u= 39/19 , ht= 1279 ; ( -4195/631 , 5912/631 ), ( 1279/25 , -2542/25 )
**u= 39/28 , ht= 3989 ; ( -9391/2159 , 18488/2159 ), ( -3989/491 , 8972/491 )
**u= 39/31 , ht= 841 ; ( 841/97 , 1726/97 ), ( -13141/4157 , -31682/4157 )
**u= 39/37 , ht= 119 ; ( 959/1181 , 3664/1181 ), ( -119/29 , -304/29 )
**u= 39/55 , ht= 179 ; ( -179/41 , -524/41 ), ( -289/41 , 866/41 )
**u= 39/58 , ht= 1411 ; ( -1411/706 , -9319/1412 ), ( -629/1691 , 5312/1691 )
**u= 39/61 , ht= 619 ; ( -235/619 , 2054/619 ), ( 947/1069 , -4366/1069 )
**u= 39/70 , ht= 281 ; ( 9941/4621 , 35674/4621 ), ( 281 , 934 )
**u= 39/79 , ht= 7973 ; ( 7973/1021 , -27746/1021 ), ( -17629/23135 , -101128/23135 )
**u= 39/88 , ht= 353 ; ( 353/23 , -1291/23 ), ( -1453/6203 , -23269/6203 )
**u= 39/94 , ht= 257 ; ( 4648/841 , -35683/1682 ), ( -257/119 , 1082/119 )
**u= 40/7 , ht= 4 ; ( -248/179 , -237/358 ), ( -3/4 , -5/8 )
**u= 40/11 , ht= 34 ; ( 34/23 , -81/46 ), ( 206/877 , 267/1754 )
**u= 40/19 , ht= 1283 ; ( 1381/602 , -4401/1204 ), ( 314/1283 , -3657/2566 )
**u= 40/23 , ht= 223 ; ( -223/76 , 783/152 ), ( -556/663 , 3143/1326 )
**u= 40/27 , ht= 53 ; ( -53/14 , -607/84 ), ( -82/161 , -687/322 )
**u= 40/43 , ht= 782 ; ( 782/309 , 4157/618 ), ( 16521/1342 , 87137/2684 )
**u= 40/51 , ht= 19 ; ( 19/2 , 103/4 ), ( 33986/15677 , -210455/31354 )
**u= 40/71 , ht= 4421 ; ( 33744/26857 , -280649/53714 ), ( 4421/1368 , -1607/144 )
**u= 40/91 , ht= 3771 ; ( 2074/7443 , -57485/14886 ), ( -3771/3418 , 37637/6836 )
**u= 48 , ht= 385 ; ( -385/383 , 1/766 ), ( -383/385 , -1/770 )
**u= 48/13 , ht= 19 ; ( -19/7 , -19/14 ), ( -137/475 , 13/50 )
**u= 48/25 , ht= 1 ; ( -1 , -5/2 ), ( -1735/1633 , -42509/16330 )
**u= 48/37 , ht= 679 ; ( 605/679 , -3943/1358 ), ( 679/605 , -3943/1210 )
**u= 48/49 , ht= 1 ; ( -1 , -7/2 ), ( 47/55 , -2491/770 )
**u= 49/3 , ht= 1 ; ( -1 , 6/7 ), ( -1 , 6/7 )
**u= 49/4 , ht= 143 ; ( 143/141 , -19/21 ), ( 141/143 , 893/1001 )
**u= 49/8 , ht= 24 ; ( 24/19 , 451/532 ), ( 79/55 , 963/385 )
**u= 49/9 , ht= 55 ; ( -55/37 , 128/259 ), ( -159/55 , -6362/1155 )
**u= 49/10 , ht= 62 ; ( -62/43 , 639/602 ), ( 117/187 , -604/1309 )
**u= 49/12 , ht= 1349 ; ( 1055/1349 , -16789/9443 ), ( 7973/8255 , -95003/57785 )
**u= 49/16 , ht= 3 ; ( -145/223 , -2991/1561 ), ( -3 , -79/14 )
**u= 49/18 , ht= 377 ; ( 377/73 , 2608/511 ), ( -1381/780 , -113573/32760 )
**u= 49/19 , ht= 275 ; ( 275/9 , 1964/63 ), ( 791/563 , 11268/3941 )
**u= 49/22 , ht= 89 ; ( 89/34 , 1821/476 ), ( -34/89 , -1821/1246 )
**u= 49/23 , ht= 281 ; ( -10181/4171 , -110466/29197 ), ( 281/51 , 3904/357 )
**u= 49/25 , ht= 207 ; ( -89/207 , 15242/7245 ), ( 207/89 , 15242/3115 )
**u= 49/26 , ht= 191 ; ( -82/191 , 5697/2674 ), ( 761/33 , 10826/231 )
**u= 49/27 , ht= 3451 ; ( -2825/3451 , -174848/72471 ), ( -66247/66893 , 3590032/1404753 )
**u= 49/29 , ht= 193 ; ( -155/193 , -3348/1351 ), ( -1153/2139 , 964/483 )
**u= 49/30 , ht= 89 ; ( -281/124 , -7599/1736 ), ( -41/89 , -1216/623 )
**u= 49/31 , ht= 717 ; ( -21497/4791 , 270586/33537 ), ( 717/221 , -11078/1547 )
**u= 49/34 , ht= 79 ; ( -79/23 , -1086/161 ), ( -1741/520 , -55533/7280 )
**u= 49/36 , ht= 109 ; ( 901/197 , 12568/1379 ), ( 57/109 , -5179/2289 )
**u= 49/37 , ht= 4619 ; ( 4619/2769 , -5956/1491 ), ( 4719/535 , -75634/3745 )
**u= 49/40 , ht= 211 ; ( -211/21 , -3077/147 ), ( -401/439 , 9159/3073 )
**u= 49/44 , ht= 673 ; ( 853/149 , 13368/1043 ), ( -31/673 , 10368/4711 )
**u= 49/45 , ht= 6199 ; ( -1281/6199 , -325882/130179 ), ( 42709/6489 , -2227696/136269 )
**u= 49/47 , ht= 1291 ; ( -871/1291 , 26592/9037 ), ( 39233/975 , 688202/6825 )
**u= 49/48 , ht= 51649 ; ( 51649/50491 , -2449103/706874 ), ( 61253/60095 , 2914897/841330 )
**u= 49/51 , ht= 349 ; ( 1429/1633 , 38152/11431 ), ( 349/145 , 6788/1015 )
**u= 49/52 , ht= 1165 ; ( 67375/5469 , 1149571/38283 ), ( -291/1165 , -20506/8155 )
**u= 49/53 , ht= 397 ; ( 2013/4027 , -81982/28189 ), ( -397/27 , -7342/189 )
**u= 49/57 , ht= 383 ; ( 11485/2251 , 210174/15757 ), ( 383/73 , 7414/511 )
**u= 49/59 , ht= 5173 ; ( -12587/629 , 230238/4403 ), ( -5173/2379 , -109114/16653 )
**u= 49/60 , ht= 17 ; ( -13/17 , -409/119 ), ( -1723/47 , -33599/329 )
**u= 49/65 , ht= 1069 ; ( -881/1069 , 27480/7483 ), ( 4579/2629 , 105510/18403 )
**u= 49/66 , ht= 169 ; ( 169/7 , -3288/49 ), ( -4789/1448 , 202893/20272 )
**u= 49/71 , ht= 411 ; ( 411/127 , 8732/889 ), ( 2591/177 , -54646/1239 )
**u= 49/73 , ht= 4985 ; ( 4985/21 , -102352/147 ), ( -5903/8843 , -220908/61901 )
**u= 49/74 , ht= 270 ; ( -270/211 , 14377/2954 ), ( 7061/4323 , 175898/30261 )
**u= 49/75 , ht= 4129 ; ( -4129/575 , 434542/20125 ), ( 138005/139799 , -20830948/4892965 )
**u= 49/81 , ht= 83 ; ( 79/297 , 61778/18711 ), ( -83/79 , -32/7 )
**u= 49/82 , ht= 728 ; ( -728/297 , -3137/378 ), ( -4543/3887 , -132924/27209 )
**u= 49/85 , ht= 579 ; ( 1243/1073 , 37008/7511 ), ( 579/239 , -14284/1673 )
**u= 49/86 , ht= 205 ; ( -205/139 , 5598/973 ), ( -521/167 , 12564/1169 )
**u= 49/87 , ht= 9443 ; ( 9443/1613 , -216126/11291 ), ( 12593/65 , 291498/455 )
**u= 49/88 , ht= 103 ; ( 103/18 , -9493/504 ), ( 10546/1801 , -995259/50428 )
**u= 49/96 , ht= 2945 ; ( 2945/257 , 70186/1799 ), ( 367/8911 , -11143/3283 )
**u= 49/100 , ht= 491 ; ( -779/321 , -20491/2247 ), ( 159/491 , 12539/3437 )
**u= 50/13 , ht= 70 ; ( -174/125 , 1042/625 ), ( -31/70 , -99/175 )
**u= 50/41 , ht= 423 ; ( -136/423 , -22/9 ), ( -47/568 , 2961/1420 )
**u= 50/57 , ht= 328 ; ( -4316/1295 , 57326/6475 ), ( 185/328 , 2421/820 )
**u= 50/61 , ht= 383 ; ( 78/383 , 5422/1915 ), ( -4387/666 , -6161/333 )
**u= 50/73 , ht= 13189 ; ( -26480/2939 , -386682/14695 ), ( -13189/2360 , 100941/5900 )
**u= 50/77 , ht= 202 ; ( 11086/1377 , -33374/1377 ), ( -125/202 , -1791/505 )
**u= 51 , ht= 29 ; ( -3/29 , -146/29 ), ( -63/31 , -278/31 )
**u= 51/7 , ht= 433 ; ( -393/433 , -634/433 ), ( 571/249 , 1132/249 )
**u= 51/10 , ht= 121 ; ( 951/616 , 829/1232 ), ( 121/111 , 194/111 )
**u= 51/13 , ht= 101 ; ( -101/81 , 136/81 ), ( -293/747 , -242/747 )
**u= 51/19 , ht= 597 ; ( 597/179 , 658/179 ), ( 101/633 , 628/633 )
**u= 51/22 , ht= 4536 ; ( -59244/5249 , -143099/10498 ), ( -3769/4536 , 18181/9072 )
**u= 51/28 , ht= 1233 ; ( 283/2523 , -5192/2523 ), ( 1233/1205 , -3139/1205 )
**u= 51/40 , ht= 288 ; ( -13/288 , 2665/1152 ), ( -419/741 , -137/57 )
**u= 51/43 , ht= 197 ; ( 31005/48377 , -132364/48377 ), ( 197/147 , -562/147 )
**u= 51/52 , ht= 205 ; ( -3975/2317 , 11153/2317 ), ( -81/205 , -529/205 )
**u= 51/55 , ht= 937 ; ( -937/603 , 2792/603 ), ( -313/963 , 2500/963 )
**u= 51/58 , ht= 49 ; ( 9/49 , 134/49 ), ( -9239/2361 , 25604/2361 )
**u= 51/64 , ht= 1431 ; ( 2199/877 , 25481/3508 ), ( 743/1431 , -8713/2862 )
**u= 51/73 , ht= 2187 ; ( -783/4433 , -94/31 ), ( 143/2187 , -6292/2187 )
**u= 51/76 , ht= 7535 ; ( 4875/16693 , -52816/16693 ), ( 543/7535 , 22166/7535 )
**u= 51/79 , ht= 747 ; ( 115/747 , -2342/747 ), ( 30569/24723 , -120292/24723 )
**u= 51/82 , ht= 8094 ; ( -2525/8094 , 53363/16188 ), ( 22418/10815 , 156161/21630 )
**u= 51/100 , ht= 4261 ; ( -4689/3439 , -19880/3439 ), ( -4261/1761 , -15935/1761 )
**u= 52/7 , ht= 85 ; ( 487/529 , -33/23 ), ( -69/85 , 23/85 )
**u= 52/11 , ht= 509 ; ( 1505/1571 , 2547/1571 ), ( 289/509 , -141/509 )
**u= 52/15 , ht= 31 ; ( -31 , 1 ), ( 1/31 , 1/31 )
**u= 52/19 , ht= 85 ; ( -89/101 , -207/101 ), ( 9/85 , -79/85 )
**u= 52/31 , ht= 71 ; ( 71/9 , 119/9 ), ( -3629/20179 , -34251/20179 )
**u= 52/35 , ht= 81 ; ( -1509/1711 , 4637/1711 ), ( 11/81 , 149/81 )
**u= 52/43 , ht= 385 ; ( 385/131 , 861/131 ), ( 131/385 , 123/55 )
**u= 52/55 , ht= 371 ; ( 371/69 , 917/69 ), ( -2461/371 , 927/53 )
**u= 52/59 , ht= 2963 ; ( -23605/77521 , 217041/77521 ), ( -2963/577 , -8109/577 )
**u= 52/63 , ht= 323 ; ( -2029/2283 , -24787/6849 ), ( 309/323 , 3611/969 )
**u= 52/67 , ht= 28309 ; ( -28309/233 , -76713/233 ), ( 96437/96839 , -379953/96839 )
**u= 52/95 , ht= 5191 ; ( 2151/5191 , 18767/5191 ), ( 87613/21493 , -301959/21493 )
**u= 54/11 , ht= 17 ; ( 20/13 , 34/39 ), ( 17/16 , -41/24 )
**u= 54/13 , ht= 42 ; ( -42/23 , 82/69 ), ( 23/42 , -41/63 )
**u= 54/25 , ht= 16 ; ( -16/9 , -82/27 ), ( 1/24 , -47/36 )
**u= 54/59 , ht= 140 ; ( 140/37 , -1078/111 ), ( -1283/280 , -743/60 )
**u= 54/61 , ht= 51 ; ( -10/51 , 418/153 ), ( -67/6 , 271/9 )
**u= 54/71 , ht= 1154 ; ( -1154/337 , -9922/1011 ), ( -3889/442 , 16873/663 )
**u= 58/15 , ht= 16 ; ( -16/11 , 18/11 ), ( 2267/868 , -2099/434 )
**u= 58/27 , ht= 514 ; ( 11986/1007 , -47518/3021 ), ( 107/514 , 353/257 )
**u= 58/31 , ht= 392 ; ( -392/197 , 714/197 ), ( -3201/364 , 467/26 )
**u= 58/47 , ht= 463 ; ( 21956/16363 , -59274/16363 ), ( -463/336 , -643/168 )
**u= 58/59 , ht= 737 ; ( 5462/497 , 13002/497 ), ( 737/30 , 947/15 )
**u= 58/75 , ht= 341 ; ( 142/341 , 5234/1705 ), ( 31357/5066 , 226359/12665 )
**u= 58/91 , ht= 310 ; ( 310/37 , 942/37 ), ( 116629/14086 , -183159/7043 )
**u= 60/13 , ht= 233 ; ( 1049/1023 , -1633/1023 ), ( 129/233 , -73/233 )
**u= 60/37 , ht= 101 ; ( -511/303 , 1081/303 ), ( 27/101 , -181/101 )
**u= 60/73 , ht= 107 ; ( -107/39 , 301/39 ), ( -22487/5691 , -64205/5691 )
**u= 61 , ht= 31 ; ( -25/31 , -102/31 ), ( -1257/1261 , 68/1261 )
**u= 61/2 , ht= 241 ; ( 89/241 , -882/241 ), ( 383/362 , -1083/724 )
**u= 61/4 , ht= 85 ; ( 2791/2727 , 1816/2727 ), ( 81/85 , -19/85 )
**u= 61/8 , ht= 391 ; ( -335/391 , -603/391 ), ( 463/335 , -12/5 )
**u= 61/9 , ht= 297 ; ( 133/1951 , -4028/1951 ), ( 297/95 , 94/15 )
**u= 61/11 , ht= 1293 ; ( 1293/1271 , 1844/1271 ), ( 4993/2661 , 9172/2661 )
**u= 61/12 , ht= 329 ; ( 325/329 , 509/329 ), ( 3583/3043 , 5931/3043 )
**u= 61/13 , ht= 529 ; ( -529/329 , -306/329 ), ( -9441/3689 , 17566/3689 )
**u= 61/14 , ht= 2571 ; ( 16865/8066 , -99/436 ), ( 1258/2571 , -1147/5142 )
**u= 61/15 , ht= 299 ; ( -4133/1787 , 1294/1787 ), ( 239/299 , -388/299 )
**u= 61/18 , ht= 306 ; ( -306/11 , -521/66 ), ( 187/4043 , 1168/4043 )
**u= 61/21 , ht= 6763 ; ( -6763/461 , 5366/461 ), ( 4857187/452731 , -9117138/452731 )
**u= 61/22 , ht= 263 ; ( -1739/219 , -1588/219 ), ( 87/263 , 284/263 )
**u= 61/23 , ht= 3475 ; ( 3475/2279 , -5472/2279 ), ( 362989/285295 , -740952/285295 )
**u= 61/25 , ht= 21 ; ( 49/51 , -112/51 ), ( 21/13 , 214/65 )
**u= 61/26 , ht= 240 ; ( 240/227 , -1043/454 ), ( 3743/1039 , -7332/1039 )
**u= 61/30 , ht= 3391 ; ( 3391/946 , -10157/1892 ), ( -22037/10363 , 46200/10363 )
**u= 61/35 , ht= 27131 ; ( 604591/16799 , 970086/16799 ), ( 2591/27131 , -3372/2087 )
**u= 61/36 , ht= 85 ; ( -85/81 , -659/243 ), ( -7779/47 , -48887/141 )
**u= 61/40 , ht= 1093 ; ( -26989/4811 , -49185/4811 ), ( -1093/677 , -2658/677 )
**u= 61/41 , ht= 36505 ; ( 124219/20243 , 229578/20243 ), ( -35327/36505 , -101706/36505 )
**u= 61/43 , ht= 351 ; ( 31805/1017 , 59716/1017 ), ( -351/79 , 794/79 )
**u= 61/50 , ht= 1374 ; ( 1374/1051 , -7547/2102 ), ( 2315/7109 , 78636/35545 )
**u= 61/51 , ht= 44801 ; ( 44801/26543 , -113242/26543 ), ( 47699/2717 , -113166/2717 )
**u= 61/53 , ht= 897 ; ( -237561/52879 , -527174/52879 ), ( 473/897 , 2242/897 )
**u= 61/54 , ht= 127 ; ( 23615/24617 , -236374/73851 ), ( 116/127 , 789/254 )
**u= 61/56 , ht= 363 ; ( -363/211 , -961/211 ), ( 2101/2997 , -8444/2997 )
**u= 61/57 , ht= 1075 ; ( 1075/619 , -2866/619 ), ( 14177/1067 , -35216/1067 )
**u= 61/62 , ht= 2067 ; ( -11585/1714 , -55563/3428 ), ( -382/2067 , -9983/4134 )
**u= 61/63 , ht= 1627 ; ( -1627/207 , -512/27 ), ( -207/1627 , 11776/4881 )
**u= 61/67 , ht= 13669 ; ( 130145/11421 , -323776/11421 ), ( -13669/3349 , 37218/3349 )
**u= 61/68 , ht= 3821 ; ( 39059/10771 , 101724/10771 ), ( -3821/429 , 10264/429 )
**u= 61/69 , ht= 25247 ; ( 25247/15449 , -75958/15449 ), ( 130579/52885 , 375506/52885 )
**u= 61/70 , ht= 1917 ; ( -1483/1917 , 6410/1917 ), ( -6152/3683 , 38187/7366 )
**u= 61/74 , ht= 588 ; ( -588/181 , 3241/362 ), ( 15259/12040 , -15081/3440 )
**u= 61/76 , ht= 8953 ; ( -24379/8187 , -68792/8187 ), ( -8953/395 , -25149/395 )
**u= 61/77 , ht= 1501 ; ( -325/1501 , 4326/1501 ), ( 22747/20745 , -84928/20745 )
**u= 61/78 , ht= 1385 ; ( 1385/46 , 7479/92 ), ( -11239/1385 , 116/5 )
**u= 61/81 , ht= 21 ; ( -4721/4473 , -165002/40257 ), ( 21/13 , 634/117 )
**u= 61/83 , ht= 70997 ; ( 74191/34849 , 230814/34849 ), ( -70997/28105 , 221724/28105 )
**u= 61/88 , ht= 733 ; ( 5675/1869 , 17293/1869 ), ( -477/733 , -2552/733 )
**u= 61/89 , ht= 123707 ; ( 123707/84191 , 439548/84191 ), ( 148847/34571 , 459996/34571 )
**u= 61/90 , ht= 2596 ; ( -33424/261 , -585167/1566 ), ( 2021/2596 , -19485/5192 )
**u= 61/91 , ht= 415 ; ( -9047/16069 , -55704/16069 ), ( 51/415 , 1228/415 )
**u= 61/93 , ht= 15917 ; ( 302029/29051 , 81946/2641 ), ( -15917/5873 , -52008/5873 )
**u= 61/95 , ht= 8481 ; ( -8481/449 , 25514/449 ), ( 42181/951 , -131168/951 )
**u= 61/99 , ht= 31357 ; ( 6255/31357 , -303676/94071 ), ( -35887/10335 , 354266/31005 )
**u= 61/100 , ht= 4997 ; ( -4997/4703 , 21501/4703 ), ( 47543/4921 , -760806/24605 )
**u= 64/3 , ht= 6017 ; ( -6017/5983 , 15949/23932 ), ( -6271/6305 , 16819/25220 )
**u= 64/7 , ht= 69 ; ( 349/391 , -2271/1564 ), ( -69 , 631/4 )
**u= 64/15 , ht= 161 ; ( -139/161 , -1115/644 ), ( -217/53 , 1623/212 )
**u= 64/19 , ht= 31 ; ( -455 , 567/4 ), ( -31/7 , -33/4 )
**u= 64/27 , ht= 533 ; ( -533/253 , 9451/3036 ), ( -1171/1469 , 2611/1356 )
**u= 64/31 , ht= 247 ; ( -61/247 , 153/76 ), ( -247/61 , -1989/244 )
**u= 64/43 , ht= 543 ; ( 457/543 , 5789/2172 ), ( 543/457 , 5789/1828 )
**u= 64/59 , ht= 1181 ; ( -117/1181 , -11689/4724 ), ( -131/1285 , 11553/5140 )
**u= 64/63 , ht= 767 ; ( 767/627 , 28643/7524 ), ( 33973/18097 , 383061/72388 )
**u= 64/75 , ht= 1 ; ( -1 , 15/4 ), ( -1177/95 , 64471/1900 )
**u= 64/91 , ht= 111 ; ( -97/111 , 1727/444 ), ( 2777/2959 , -47349/11836 )
**u= 73/2 , ht= 313 ; ( -129/313 , 1226/313 ), ( -617/204 , -5003/408 )
**u= 73/3 , ht= 133 ; ( 59/133 , 422/133 ), ( 449/443 , 404/443 )
**u= 73/5 , ht= 169 ; ( 109/169 , -372/169 ), ( 51953/5803 , 143460/5803 )
**u= 73/8 , ht= 161 ; ( 743/1081 , -1986/1081 ), ( 161/40 , -1437/160 )
**u= 73/9 , ht= 519 ; ( -289/519 , -3022/1557 ), ( -5897/1235 , 12716/1235 )
**u= 73/10 , ht= 459 ; ( -421/459 , -664/459 ), ( -431/481 , 420/481 )
**u= 73/11 , ht= 1203 ; ( -1203/953 , 584/953 ), ( -573227/82967 , 1170804/82967 )
**u= 73/16 , ht= 753 ; ( -475/753 , -2699/1506 ), ( -543/815 , -647/815 )
**u= 73/18 , ht= 919 ; ( 153/919 , 5152/2757 ), ( 12763/18656 , 39011/37312 )
**u= 73/19 , ht= 1149 ; ( -11681/5229 , -6796/5229 ), ( 1149/655 , 2108/655 )
**u= 73/21 , ht= 2753 ; ( 2753/163 , 24/163 ), ( -1343/4327 , 2454/4327 )
**u= 73/23 , ht= 1717 ; ( -16875/5297 , 13564/5297 ), ( -1717/215 , -3204/215 )
**u= 73/24 , ht= 505 ; ( -505/89 , 378/89 ), ( 89/505 , 378/505 )
**u= 73/26 , ht= 77 ; ( -23230/34243 , 12261/6226 ), ( -77/38 , 297/76 )
**u= 73/28 , ht= 1289 ; ( 8837/1641 , -9361/1641 ), ( 1289/55 , 2448/55 )
**u= 73/29 , ht= 49 ; ( -9/49 , 94/49 ), ( 49/9 , -94/9 )
**u= 73/30 , ht= 3937 ; ( -3937/1033 , 4826/1033 ), ( -75629/69469 , 1296/547 )
**u= 73/31 , ht= 53231 ; ( 53231/159 , 62318/159 ), ( 406809/291241 , -856022/291241 )
**u= 73/35 , ht= 153 ; ( 153/43 , 224/43 ), ( -6931/4269 , 14890/4269 )
**u= 73/38 , ht= 45 ; ( -45/26 , -13/4 ), ( -26/45 , 169/90 )
**u= 73/39 , ht= 1019 ; ( 8755/1033 , -13356/1033 ), ( 811/1019 , -2254/1019 )
**u= 73/44 , ht= 571 ; ( 507/1717 , -3718/1717 ), ( 571/351 , 103/27 )
**u= 73/45 , ht= 1643 ; ( -5939/11051 , -25546/11051 ), ( -707/1643 , 3170/1643 )
**u= 73/47 , ht= 13297 ; ( -126715/33931 , -234306/33931 ), ( 9631/13297 , -31242/13297 )
**u= 73/49 , ht= 13835 ; ( 13835/1683 , 25334/1683 ), ( 161687/18253 , -2483316/127771 )
**u= 73/50 , ht= 47 ; ( -3612/1537 , -14903/3074 ), ( 47/28 , -33/8 )
**u= 73/56 , ht= 4929 ; ( -10965/179 , -21799/179 ), ( 4376/4929 , 56087/19716 )
**u= 73/58 , ht= 3761 ; ( 3761/223 , -7668/223 ), ( -7769/9602 , 53187/19204 )
**u= 73/60 , ht= 7003 ; ( 3493/7003 , 18012/7003 ), ( -40709/6661 , 96812/6661 )
**u= 73/63 , ht= 2533 ; ( 2533/939 , 17642/2817 ), ( -939/2533 , -17642/7599 )
**u= 73/67 , ht= 175 ; ( -175/93 , 452/93 ), ( -52599/2743 , 129538/2743 )
**u= 73/71 , ht= 1803 ; ( 9433/15397 , 44484/15397 ), ( 667/1803 , -4486/1803 )
**u= 73/72 , ht= 629 ; ( -120/629 , -19423/7548 ), ( 7801/1263 , -59971/3789 )
**u= 73/75 , ht= 69827 ; ( 100219/13381 , -241282/13381 ), ( -69827/3619 , -901438/18095 )
**u= 73/78 , ht= 14915 ; ( -14915/11059 , 46512/11059 ), ( -91973/21565 , 12994/1135 )
**u= 73/80 , ht= 21797 ; ( -48687/7043 , 243751/14086 ), ( -21797/3803 , -58560/3803 )
**u= 73/84 , ht= 12587 ; ( 10697/12587 , 43623/12587 ), ( 30937/13465 , 90536/13465 )
**u= 73/85 , ht= 61 ; ( -61/41 , -192/41 ), ( 2548863/3744473 , -11836906/3744473 )
**u= 73/86 , ht= 4031 ; ( 67183/11498 , -352017/22996 ), ( 2999/4031 , 13242/4031 )
**u= 73/87 , ht= 16073 ; ( -128887/67559 , 382582/67559 ), ( 1109/16073 , -41812/16073 )
**u= 73/88 , ht= 6815 ; ( 6815/1578 , 73289/6312 ), ( -10519/20105 , -60018/20105 )
**u= 73/91 , ht= 2817 ; ( 2635/2817 , 10568/2817 ), ( 516793/377017 , 1763754/377017 )
**u= 73/94 , ht= 11185 ; ( 40315/18836 , 243351/37672 ), ( -5688/11185 , -68717/22370 )
**u= 73/97 , ht= 5273 ; ( -4109/5273 , 18984/5273 ), ( 1056969/163793 , -441034/23399 )
**u= 73/98 , ht= 2501 ; ( 25380/1031 , 140819/2062 ), ( -2501/363 , 51308/2541 )
**u= 73/99 , ht= 4397 ; ( -201831/27899 , 1705172/83697 ), ( 4397/2065 , -2008/295 )
**u= 73/100 , ht= 857 ; ( 9985/6211 , 166899/31055 ), ( 75/857 , 12059/4285 )
**u= 74/5 , ht= 99 ; ( -1292/1277 , -1026/1277 ), ( -99/76 , -5/2 )
**u= 74/9 , ht= 353 ; ( 250/353 , 626/353 ), ( -1109/142 , -1213/71 )
**u= 74/21 , ht= 1144 ; ( 3740/499 , 186/499 ), ( -277/1144 , 217/572 )
**u= 74/41 , ht= 4857 ; ( 2974/4857 , -11006/4857 ), ( 9177/8726 , -11623/4363 )
**u= 74/73 , ht= 5269 ; ( -11574/12881 , -42322/12881 ), ( 5269/2130 , -7123/1065 )
**u= 75 , ht= 3743 ; ( -3733/3743 , -11224/18715 ), ( -3767/3757 , -11276/18785 )
**u= 75/4 , ht= 1 ; ( -547/529 , -61/529 ), ( -1 , 4/5 )
**u= 75/16 , ht= 1 ; ( 1 , 8/5 ), ( -587/433 , -10243/4330 )
**u= 75/19 , ht= 121 ; ( -89/139 , -1264/695 ), ( -121/31 , -1126/155 )
**u= 75/49 , ht= 1 ; ( -1159/1189 , 115348/41615 ), ( 1 , -14/5 )
**u= 75/58 , ht= 1565 ; ( -1909/1600 , 52997/16000 ), ( 347/1565 , -16138/7825 )
**u= 75/61 , ht= 2579 ; ( -2579/2179 , 36898/10895 ), ( -97069/7729 , -1140488/38645 )
**u= 75/64 , ht= 31 ; ( -31/17 , -155/34 ), ( -17/31 , 5/2 )
**u= 75/88 , ht= 380 ; ( 380/313 , -25973/6260 ), ( -13315/4636 , -40301/4880 )
**u= 75/97 , ht= 3391 ; ( -424073/29111 , -1154522/29111 ), ( 3391/287 , -9710/287 )
**u= 78 , ht= 3 ; ( 2/3 , -14/3 ), ( 3/2 , -7 )
**u= 78/35 , ht= 394 ; ( 394/149 , -574/149 ), ( -641/1046 , 907/523 )
**u= 78/49 , ht= 106 ; ( 150/173 , 3158/1211 ), ( 81/106 , -881/371 )
**u= 78/61 , ht= 40 ; ( 40/21 , 94/21 ), ( -38279/83724 , -95161/41862 )
**u= 78/83 , ht= 430 ; ( -430/151 , 1118/151 ), ( 15881/17458 , 691/203 )
**u= 82/15 , ht= 782 ; ( -1846/1549 , -1902/1549 ), ( -547/782 , 191/391 )
**u= 82/35 , ht= 96 ; ( -96/79 , -190/79 ), ( 109/536 , 333/268 )
**u= 82/51 , ht= 43940 ; ( 43940/3541 , -75562/3541 ), ( 194687/354028 , 367039/177014 )
**u= 82/67 , ht= 3004 ; ( 3004/1083 , 6722/1083 ), ( -20727/9136 , 26111/4568 )
**u= 82/79 , ht= 1486 ; ( 1486/459 , -3598/459 ), ( 459/1486 , 1799/743 )
**u= 82/95 , ht= 619 ; ( -88026/18049 , -230050/18049 ), ( 619/46 , -843/23 )
**u= 85 , ht= 231 ; ( 227/231 , 292/231 ), ( -231/227 , 292/227 )
**u= 85/7 , ht= 807 ; ( 71/6567 , 16826/6567 ), ( 807/359 , -1886/359 )
**u= 85/8 , ht= 127 ; ( -121/127 , -159/127 ), ( -127/121 , 159/121 )
**u= 85/12 , ht= 1759 ; ( -733/1759 , 3483/1759 ), ( 7753/1381 , 16083/1381 )
**u= 85/13 , ht= 2309 ; ( 2309/1763 , -690/1763 ), ( -4911/6503 , -1238/6503 )
**u= 85/19 , ht= 287 ; ( -79357/41331 , 21718/41331 ), ( 287/59 , -540/59 )
**u= 85/21 , ht= 2843 ; ( 7411/13417 , -24524/13417 ), ( 1121/2843 , -172/2843 )
**u= 85/23 , ht= 987 ; ( -987/251 , 14/251 ), ( -3217/11221 , 396/1603 )
**u= 85/24 , ht= 683 ; ( 12473/1856 , 109/7424 ), ( -124/683 , -529/2732 )
**u= 85/26 , ht= 571 ; ( -363/571 , 1076/571 ), ( -223/791 , 78/113 )
**u= 85/27 , ht= 3311 ; ( 27503/1661 , 48428/4983 ), ( -577/3311 , -2190/3311 )
**u= 85/28 , ht= 807 ; ( -328177/21711 , 225920/21711 ), ( 807/359 , 1528/359 )
**u= 85/29 , ht= 163 ; ( -4803/359 , 3724/359 ), ( -11/163 , 126/163 )
**u= 85/32 , ht= 101 ; ( 506/67 , -4035/536 ), ( -101/98 , -1707/784 )
**u= 85/33 , ht= 3187 ; ( 3187/1141 , 3910/1141 ), ( 16669/11983 , -33980/11983 )
**u= 85/38 , ht= 1777 ; ( 1777/1511 , -3690/1511 ), ( -2384/67 , 9297/134 )
**u= 85/41 , ht= 2253 ; ( -1356577/74569 , 168618/6779 ), ( -121/2253 , 3080/2253 )
**u= 85/46 , ht= 481 ; ( 2119/3223 , 7332/3223 ), ( 343/481 , -78/37 )
**u= 85/49 , ht= 631 ; ( 5483/5251 , -98298/36757 ), ( 631/77 , 9228/539 )
**u= 85/53 , ht= 549 ; ( -92311/26303 , -167898/26303 ), ( 337/549 , -1184/549 )
**u= 85/59 , ht= 3271 ; ( 76161/18577 , -147190/18577 ), ( -3271/1213 , 7572/1213 )
**u= 85/62 , ht= 641 ; ( -8787/451 , 16912/451 ), ( 641/207 , -1496/207 )
**u= 85/64 , ht= 299 ; ( 299/283 , 1743/566 ), ( 12279/15943 , 83923/31886 )
**u= 85/71 , ht= 19679 ; ( 13147/19679 , -54180/19679 ), ( 233363/48621 , -562106/48621 )
**u= 85/72 , ht= 12418 ; ( 12418/5659 , 118279/22636 ), ( 13793/959 , 4702/137 )
**u= 85/76 , ht= 27 ; ( -11/27 , -70/27 ), ( 5391/4973 , 17059/4973 )
**u= 85/78 , ht= 8389 ; ( 119803/46054 , 579785/92108 ), ( 8389/2773 , 21540/2773 )
**u= 85/81 , ht= 653 ; ( 227/1659 , 37594/14931 ), ( -329/653 , -1700/653 )
**u= 85/84 , ht= 22303 ; ( 22303/1891 , -52161/1891 ), ( 5731/27277 , 65166/27277 )
**u= 85/88 , ht= 133 ; ( 373/276 , 4573/1104 ), ( -131/133 , -465/133 )
**u= 85/92 , ht= 2611 ; ( -99577/151 , -244566/151 ), ( 2611/1507 , -7818/1507 )
**u= 85/96 , ht= 157 ; ( 1699/1228 , 43207/9824 ), ( -157/101 , -985/202 )
**u= 85/97 , ht= 32737 ; ( 70753/29431 , -195942/29431 ), ( 32737/4219 , -89004/4219 )
**u= 85/98 , ht= 3331 ; ( -41757/7211 , -756778/50477 ), ( 3331/1517 , -68760/10619 )
**u= 87/4 , ht= 61 ; ( -51/61 , 118/61 ), ( -283/177 , -749/177 )
**u= 87/16 , ht= 537 ; ( -765/509 , -212/509 ), ( 537/487 , -848/487 )
**u= 87/22 , ht= 126 ; ( 126/83 , -259/166 ), ( -771/958 , -2573/1916 )
**u= 87/25 , ht= 117 ; ( 117/17 , -28/17 ), ( 83/483 , 142/483 )
**u= 87/28 , ht= 505 ; ( 7005/649 , 4471/649 ), ( 57/505 , -328/505 )
**u= 87/31 , ht= 65 ; ( 65/3 , 56/3 ), ( -3/65 , 56/65 )
**u= 87/37 , ht= 1037 ; ( 1037/519 , -1576/519 ), ( 716463/43951 , -1384198/43951 )
**u= 87/40 , ht= 81 ; ( 81/14 , -433/56 ), ( 346/15771 , 81595/63084 )
**u= 87/43 , ht= 1129 ; ( 3027/361 , -4292/361 ), ( -1129/1107 , 2732/1107 )
**u= 87/52 , ht= 1327 ; ( 1327/1041 , -3104/1041 ), ( -2681/3897 , 8576/3897 )
**u= 87/70 , ht= 9829 ; ( 11359/369 , -23312/369 ), ( 9829/981 , -23032/981 )
**u= 87/73 , ht= 4431 ; ( 3263/4431 , 12562/4431 ), ( -7071/8843 , -25072/8843 )
**u= 87/76 , ht= 369 ; ( -4067/7827 , 20818/7827 ), ( -369/49 , 128/7 )
**u= 87/79 , ht= 25239 ; ( 25831/17931 , 72088/17931 ), ( 25239/19235 , 75056/19235 )
**u= 87/91 , ht= 9879 ; ( -61927/97323 , 293608/97323 ), ( 9879/6239 , -29746/6239 )
**u= 87/94 , ht= 361 ; ( 360/407 , -2779/814 ), ( -361/156 , -2051/312 )
**u= 96/13 , ht= 462 ; ( 163/462 , 3763/1848 ), ( -1302/557 , -10387/2228 )
**u= 96/37 , ht= 31 ; ( -31/6 , -133/24 ), ( -6/31 , -133/124 )
**u= 96/49 , ht= 69 ; ( 69/28 , -3203/784 ), ( 187/36 , 10627/1008 )
**u= 96/61 , ht= 67 ; ( -67/6 , 469/24 ), ( 6/67 , -7/4 )
**u= 96/85 , ht= 79 ; ( 2474/9 , 21563/36 ), ( -6/79 , 691/316 )
**u= 96/97 , ht= 252 ; ( 2820/1559 , -31039/6236 ), ( -252/233 , 3391/932 )
**u= 97 , ht= 261 ; ( -1709/5879 , 39204/5879 ), ( -261/19 , -1814/19 )
**u= 97/6 , ht= 3557 ; ( -3517/3557 , 3402/3557 ), ( -10738/1787 , 61651/3574 )
**u= 97/7 , ht= 691 ; ( 7765/7359 , -2648/7359 ), ( 691/415 , -1548/415 )
**u= 97/8 , ht= 719 ; ( -672/719 , 3743/2876 ), ( -8507/8189 , 10059/8189 )
**u= 97/10 , ht= 749 ; ( 206/749 , -3399/1498 ), ( -1291/1339 , -12/13 )
**u= 97/13 , ht= 4805 ; ( -4805/4191 , 316/381 ), ( -6853/5501 , 11154/5501 )
**u= 97/15 , ht= 469 ; ( 49/3869 , -7892/3869 ), ( -449/469 , -576/469 )
**u= 97/16 , ht= 8457 ; ( 11975/10187 , -22317/20374 ), ( 8457/3193 , 16333/3193 )
**u= 97/18 , ht= 803 ; ( 4056/2627 , 3109/15762 ), ( -803/691 , -1304/691 )
**u= 97/20 , ht= 997 ; ( -1319/791 , -474/791 ), ( 997/667 , 1764/667 )
**u= 97/23 , ht= 3859 ; ( -2305/3859 , 6996/3859 ), ( -1141849/171405 , -2141056/171405 )
**u= 97/29 , ht= 73 ; ( 73/43 , 84/43 ), ( -7087/20405 , -14988/20405 )
**u= 97/32 , ht= 904 ; ( 20256/1247 , 111919/9976 ), ( 197/904 , -5739/7232 )
**u= 97/34 , ht= 535 ; ( 535/161 , -534/161 ), ( -21079/12785 , 40974/12785 )
**u= 97/36 , ht= 1691 ; ( -3701/2949 , -19702/8847 ), ( 1691/919 , -3311/919 )
**u= 97/37 , ht= 5975 ; ( 1723/7199 , 13764/7199 ), ( -3089/5975 , -8322/5975 )
**u= 97/38 , ht= 1845 ; ( -1845/1499 , -3436/1499 ), ( 14758/41111 , 102087/82222 )
**u= 97/42 , ht= 22 ; ( 43795/4861 , -53448/4861 ), ( 1/22 , 53/44 )
**u= 97/44 , ht= 1621 ; ( 25073/1973 , 32139/1973 ), ( 1621/1489 , -3693/1489 )
**u= 97/45 , ht= 5899 ; ( -29467/11663 , 44792/11663 ), ( -5899/4551 , -39080/13653 )
**u= 97/47 , ht= 3319 ; ( -5495/40283 , 80304/40283 ), ( 3319/217 , -942/31 )
**u= 97/50 , ht= 397 ; ( -397/101 , 3066/505 ), ( -2549/4668 , -85237/46680 )
**u= 97/51 , ht= 3535 ; ( 3535/509 , 5348/509 ), ( -35783/8995 , -10532/1285 )
**u= 97/53 , ht= 16403 ; ( -55995/16403 , 1966/349 ), ( -16403/5115 , -34498/5115 )
**u= 97/55 , ht= 13903 ; ( -38887/19083 , -73288/19083 ), ( 13023/13903 , -34838/13903 )
**u= 97/56 , ht= 465 ; ( 465/52 , 233/16 ), ( -52/465 , -3029/1860 )
**u= 97/58 , ht= 463 ; ( -6959/2263 , 12486/2263 ), ( 1/463 , -768/463 )
**u= 97/60 , ht= 2333 ; ( 983/2333 , -5235/2333 ), ( 22117/10357 , -50222/10357 )
**u= 97/63 , ht= 8317 ; ( 7387/11601 , 84784/34803 ), ( -8317/4591 , 19716/4591 )
**u= 97/65 , ht= 6099 ; ( -6099/3131 , -12982/3131 ), ( -38501/51719 , 125796/51719 )
**u= 97/66 , ht= 64261 ; ( -1395296/58189 , 5112197/116378 ), ( -64261/38864 , -315907/77728 )
**u= 97/67 , ht= 2273 ; ( 173255/20763 , 323684/20763 ), ( 531/2273 , -4364/2273 )
**u= 97/70 , ht= 606 ; ( 431/606 , 3175/1212 ), ( 3009/6649 , 14362/6649 )
**u= 97/72 , ht= 13 ; ( -5/13 , -31/13 ), ( -3943/804 , 108769/9648 )
**u= 97/73 , ht= 30433 ; ( -30433/28259 , -87768/28259 ), ( -132459/301015 , 663196/301015 )
**u= 97/76 , ht= 20035 ; ( 58609/38241 , 147526/38241 ), ( 1263/20035 , 40489/20035 )
**u= 97/77 , ht= 479 ; ( -423/479 , 1406/479 ), ( 4327/1709 , 10632/1709 )
**u= 97/80 , ht= 2227 ; ( -2227/397 , 4734/397 ), ( -135067/28203 , 323711/28203 )
**u= 97/83 , ht= 2593 ; ( 4607/1951 , 10878/1951 ), ( -2221/2593 , 7668/2593 )
**u= 97/85 , ht= 1879 ; ( -1879/501 , 4244/501 ), ( 60169/2369 , -145350/2369 )
**u= 97/91 , ht= 8131 ; ( 278237/70393 , -651948/70393 ), ( -8131/5155 , -23298/5155 )
**u= 97/92 , ht= 62505 ; ( 17032083/3370589 , 39604846/3370589 ), ( 27683/62505 , 157919/62505 )
**u= 97/96 , ht= 674 ; ( -595/674 , -17619/5392 ), ( 1177/2527 , -1887/722 )
**u= 98/5 , ht= 23 ; ( -2/23 , -510/161 ), ( -261/226 , 1579/791 )
**u= 98/33 , ht= 356 ; ( 356/7 , 1826/49 ), ( 1537/1400 , -10691/4900 )
**u= 98/37 , ht= 190 ; ( 190/79 , -1662/553 ), ( -4273/1646 , -28881/5761 )
**u= 98/65 , ht= 9 ; ( 4/9 , -146/63 ), ( -112627/48992 , 911247/171472 )
**u= 98/81 , ht= 24 ; ( -3420/253 , -451166/15939 ), ( -19/24 , 2117/756 )
**u= 98/85 , ht= 11 ; ( 1298/113 , 19638/791 ), ( -11/6 , -103/21 )
**u= 100/3 , ht= 1 ; ( -1 , -3/5 ), ( -1 , -3/5 )
**u= 100/27 , ht= 1 ; ( -241/89 , 1777/1335 ), ( 1 , 9/5 )
**u= 100/31 , ht= 89 ; ( -97945/2371 , -244437/11855 ), ( 27/89 , -67/89 )
**u= 100/47 , ht= 1547 ; ( 1547/983 , -14103/4915 ), ( 407/2757 , -18703/13785 )
**u= 100/51 , ht= 251 ; ( 1871/953 , 3309/953 ), ( -251/157 , 553/157 )
**u= 100/59 , ht= 247 ; ( 129/247 , 43/19 ), ( -1885/1419 , 533/165 )
**u= 100/79 , ht= 679 ; ( 205/679 , -8139/3395 ), ( 679/205 , -8139/1025 )
**u= 100/83 , ht= 663 ; ( -2671/2641 , -41883/13205 ), ( -167/663 , -7211/3315 )
**u= 100/87 , ht= 1177 ; ( 811/2321 , -2661/1055 ), ( 1177/1061 , 3641/1061 )
662
>

■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。

[2026.05.28追記]u=3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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