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Integer Points on A^4+3*B^4=C^4+3*D^4

[2026.05.16]A^4+3*B^4=C^4+3*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=3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639個のuが抽出される。
[MAGMAによる計算]
> PP(3,1,100);
**u= 1 , ht= 1 ; ( -1/2 , 3/2 ), ( 0 , -1 )
**u= 1/3 , ht= 5 ; ( -1/5 , -11/5 ), ( 35/11 , 79/11 )
**u= 1/4 , ht= 7 ; ( -7/5 , -21/5 ), ( -9/7 , -4 )
**u= 1/8 , ht= 4 ; ( 4 , -57/4 ), ( -1/4 , 57/16 )
**u= 1/9 , ht= 1 ; ( -11/2 , 41/2 ), ( 0 , -11/3 )
**u= 1/12 , ht= 1009 ; ( 1009/859 , -5621/859 ), ( -2743/2597 , -16027/2597 )
**u= 1/16 , ht= 35 ; ( 13/35 , -183/35 ), ( 15/37 , 391/74 )
**u= 1/17 , ht= 26 ; ( -327/98 , 1723/98 ), ( -26/9 , -139/9 )
**u= 1/19 , ht= 43 ; ( 127/279 , -1637/279 ), ( 5/43 , 231/43 )
**u= 1/20 , ht= 47 ; ( -47/13 , -267/13 ), ( 361/311 , -2610/311 )
**u= 1/27 , ht= 545 ; ( 467/545 , -13703/1635 ), ( 1259/1091 , 31807/3273 )
**u= 1/36 , ht= 31 ; ( 31/13 , -19 ), ( -13/31 , 247/31 )
**u= 1/40 , ht= 163 ; ( 602/97 , 18891/388 ), ( 163/58 , -5361/232 )
**u= 1/44 , ht= 511 ; ( 1139/1007 , -12351/1007 ), ( -511/127 , 4278/127 )
**u= 1/48 , ht= 14401 ; ( 14401/13819 , -338711/27638 ), ( 14405/13823 , 338809/27646 )
**u= 1/49 , ht= 58 ; ( 11/58 , -3543/406 ), ( 10/59 , -513/59 )
**u= 1/51 , ht= 61 ; ( -61/7 , -537/7 ), ( -355/461 , -5089/461 )
**u= 1/52 , ht= 331 ; ( 2633/1385 , -26274/1385 ), ( -85/331 , 3018/331 )
**u= 1/59 , ht= 65 ; ( -53/65 , 789/65 ), ( 2765/2883 , 37579/2883 )
**u= 1/64 , ht= 135 ; ( 83/135 , 6211/540 ), ( -135/83 , -6211/332 )
**u= 1/65 , ht= 233 ; ( 233/38 , -2331/38 ), ( -2092/1353 , -24601/1353 )
**u= 1/73 , ht= 127 ; ( -127/92 , -1641/92 ), ( -232/863 , -9351/863 )
**u= 1/75 , ht= 569 ; ( 569/305 , -34237/1525 ), ( 631/295 , 36941/1475 )
**u= 1/83 , ht= 929 ; ( 929/265 , -10779/265 ), ( 1291/129 , 14477/129 )
**u= 1/91 , ht= 1641 ; ( -22759/1993 , -266913/1993 ), ( 1641/647 , -20609/647 )
**u= 1/97 , ht= 150 ; ( 53/150 , -1919/150 ), ( 150/53 , -1919/53 )
**u= 2/3 , ht= 5 ; ( -2/5 , 17/10 ), ( 17/14 , 67/28 )
**u= 2/7 , ht= 4 ; ( 3/4 , 23/8 ), ( 4/3 , 23/6 )
**u= 2/11 , ht= 114 ; ( -29/114 , -679/228 ), ( 138/17 , 803/34 )
**u= 2/19 , ht= 17 ; ( -93/2 , -701/4 ), ( -2/17 , -129/34 )
**u= 2/23 , ht= 133 ; ( 64/133 , 1227/266 ), ( 640/203 , 5583/406 )
**u= 2/31 , ht= 207 ; ( 64/281 , -2781/562 ), ( 52/207 , 2057/414 )
**u= 2/51 , ht= 10 ; ( -7/10 , 151/20 ), ( -97/158 , -2293/316 )
**u= 2/55 , ht= 323 ; ( -1224/811 , 18859/1622 ), ( 268/323 , 5391/646 )
**u= 2/75 , ht= 73 ; ( 73/46 , 6471/460 ), ( -163/94 , 14113/940 )
**u= 2/79 , ht= 160 ; ( -1348/4577 , -73461/9154 ), ( 160/81 , 2761/162 )
**u= 2/83 , ht= 105 ; ( 473/1054 , -18231/2108 ), ( 105/22 , 1693/44 )
**u= 2/91 , ht= 163 ; ( -163/110 , 3249/220 ), ( 1286/2651 , 48681/5302 )
**u= 3 , ht= 1 ; ( 1 , -1 ), ( 1 , 1 )
**u= 3/2 , ht= 1 ; ( 3 , -2 ), ( 0 , 1/2 )
**u= 3/4 , ht= 1 ; ( -1 , -2 ), ( -1 , 2 )
**u= 3/14 , ht= 17 ; ( 3/17 , -46/17 ), ( 15/29 , 86/29 )
**u= 3/16 , ht= 53 ; ( -571/505 , -4309/1010 ), ( -53/47 , -401/94 )
**u= 3/23 , ht= 26 ; ( 107/84 , -461/84 ), ( 26/3 , 89/3 )
**u= 3/25 , ht= 1 ; ( -1 , 5 ), ( -1 , -5 )
**u= 3/38 , ht= 178 ; ( -178/69 , -1663/138 ), ( 215/6 , 1877/12 )
**u= 3/43 , ht= 83 ; ( 83/40 , 427/40 ), ( 418/289 , 2357/289 )
**u= 3/49 , ht= 1 ; ( -551/599 , -28201/4193 ), ( -1 , -7 )
**u= 3/62 , ht= 69 ; ( 69/7 , -386/7 ), ( 7/69 , -386/69 )
**u= 3/64 , ht= 1 ; ( -1 , -8 ), ( -1 , 8 )
**u= 3/67 , ht= 746 ; ( 277/746 , 4607/746 ), ( -1268/541 , -7981/541 )
**u= 3/74 , ht= 218 ; ( 33/218 , 2683/436 ), ( -345/173 , -2348/173 )
**u= 3/86 , ht= 65 ; ( 21/65 , -448/65 ), ( -1581/721 , 1628/103 )
**u= 3/88 , ht= 181 ; ( 536/635 , 22049/2540 ), ( -181/83 , 1321/83 )
**u= 3/95 , ht= 48 ; ( -47/48 , 463/48 ), ( -1068/4537 , 32119/4537 )
**u= 3/100 , ht= 695 ; ( -695/167 , -25268/835 ), ( 3265/2851 , 153253/14255 )
**u= 4/3 , ht= 61 ; ( 61/19 , -97/38 ), ( 125/83 , 353/166 )
**u= 4/7 , ht= 73 ; ( -83/15 , -259/30 ), ( 45/73 , 271/146 )
**u= 4/11 , ht= 43 ; ( -211/163 , -1077/326 ), ( 43/21 , 197/42 )
**u= 4/15 , ht= 11 ; ( 1/11 , -53/22 ), ( 221/281 , 1691/562 )
**u= 4/27 , ht= 4669 ; ( 4669/4339 , -121657/26034 ), ( 4733/4403 , 123449/26418 )
**u= 4/35 , ht= 87 ; ( -36711/7529 , 270989/15058 ), ( 17/87 , -641/174 )
**u= 4/43 , ht= 551 ; ( 551/465 , -5789/930 ), ( 465/551 , -5789/1102 )
**u= 4/55 , ht= 57 ; ( -53/57 , -707/114 ), ( -57/53 , 707/106 )
**u= 4/59 , ht= 985 ; ( -313/985 , -9729/1970 ), ( 3775/1887 , -39721/3774 )
**u= 4/71 , ht= 3737 ; ( -3737/949 , 39771/1898 ), ( -949/3737 , -39771/7474 )
**u= 4/75 , ht= 1 ; ( 1 , -15/2 ), ( -2401/2201 , 34549/4402 )
**u= 4/95 , ht= 1169 ; ( -1149/1169 , -19567/2338 ), ( -16751/2629 , -202467/5258 )
**u= 5 , ht= 2 ; ( 1/2 , -3/2 ), ( 4/3 , 5/3 )
**u= 5/2 , ht= 3 ; ( -1/3 , 4/3 ), ( 27/44 , 19/88 )
**u= 5/9 , ht= 61 ; ( 61/2 , -95/2 ), ( -68/59 , -149/59 )
**u= 5/14 , ht= 307 ; ( 2531/417 , -5150/417 ), ( 111/307 , -658/307 )
**u= 5/17 , ht= 73 ; ( 73/56 , -207/56 ), ( -1092/251 , 2563/251 )
**u= 5/18 , ht= 8 ; ( 8 , -37/2 ), ( 1/8 , -37/16 )
**u= 5/22 , ht= 94 ; ( -87/221 , -614/221 ), ( 94/83 , -645/166 )
**u= 5/26 , ht= 107 ; ( -49/107 , 330/107 ), ( -51/118 , 715/236 )
**u= 5/33 , ht= 26 ; ( -7/26 , 85/26 ), ( -1666/127 , 5277/127 )
**u= 5/46 , ht= 1432 ; ( -3693/424 , -27565/848 ), ( -1432/661 , 11733/1322 )
**u= 5/57 , ht= 109 ; ( 109/52 , 499/52 ), ( -272/241 , 1503/241 )
**u= 5/66 , ht= 344 ; ( 227/1679 , 7546/1679 ), ( 47/344 , 3087/688 )
**u= 5/78 , ht= 19 ; ( -5789/2188 , 59843/4376 ), ( 19/7 , 14 )
**u= 5/89 , ht= 2012 ; ( -569/2012 , 10809/2012 ), ( -488/2449 , -12897/2449 )
**u= 6/5 , ht= 8 ; ( -17/8 , -35/16 ), ( -7/8 , 23/16 )
**u= 6/29 , ht= 121 ; ( -197/64 , -1109/128 ), ( 92/121 , -817/242 )
**u= 6/41 , ht= 242 ; ( 242/119 , -1723/238 ), ( -415/118 , 2771/236 )
**u= 6/65 , ht= 34 ; ( -31/34 , -371/68 ), ( -34/31 , 371/62 )
**u= 12 , ht= 17 ; ( -7/17 , -77/34 ), ( -25/7 , -17/2 )
**u= 12/25 , ht= 1 ; ( -1 , -5/2 ), ( -1 , -5/2 )
**u= 12/49 , ht= 1 ; ( -1 , -7/2 ), ( -1 , 7/2 )
**u= 12/61 , ht= 569 ; ( 569/325 , -3607/650 ), ( -1021/199 , -5779/398 )
**u= 12/73 , ht= 529 ; ( -529/55 , -3199/110 ), ( 4063/1289 , 25847/2578 )
**u= 12/85 , ht= 733 ; ( -733/457 , -5623/914 ), ( 1607/757 , -11605/1514 )
**u= 13/4 , ht= 33 ; ( -17/33 , 4/3 ), ( 99 , -143 )
**u= 13/5 , ht= 3 ; ( -3/2 , 1/2 ), ( -2/3 , -1/3 )
**u= 13/7 , ht= 45 ; ( -181/57 , 41/57 ), ( 45/17 , 59/17 )
**u= 13/8 , ht= 38 ; ( -57/25 , 38/25 ), ( 31/38 , 9/8 )
**u= 13/12 , ht= 5 ; ( -401/673 , -1004/673 ), ( -5 , -7 )
**u= 13/16 , ht= 19 ; ( -37/75 , 121/75 ), ( 9/19 , 53/38 )
**u= 13/20 , ht= 111 ; ( 703/83 , 999/83 ), ( 61/111 , -5/3 )
**u= 13/21 , ht= 11 ; ( -163/82 , 273/82 ), ( -10/11 , -23/11 )
**u= 13/23 , ht= 257 ; ( 119/257 , -477/257 ), ( -985/2511 , 4219/2511 )
**u= 13/29 , ht= 2437 ; ( 12087/22150 , 46987/22150 ), ( 958/2437 , -4671/2437 )
**u= 13/32 , ht= 85 ; ( 958/19 , -14319/152 ), ( -85/43 , -186/43 )
**u= 13/36 , ht= 353 ; ( -353/3 , -2111/9 ), ( -1039/31 , -2164/31 )
**u= 13/40 , ht= 17 ; ( -17/7 , 39/7 ), ( 387/752 , 7193/3008 )
**u= 13/47 , ht= 187 ; ( -937/1789 , 4737/1789 ), ( -187 , 441 )
**u= 13/53 , ht= 178 ; ( -6039/2812 , 16367/2812 ), ( 178/19 , -447/19 )
**u= 13/56 , ht= 843 ; ( 837/1117 , -3557/1117 ), ( -718/843 , -11243/3372 )
**u= 13/60 , ht= 337 ; ( 2953/1063 , -8208/1063 ), ( -337/47 , 902/47 )
**u= 13/63 , ht= 1675 ; ( 3209/3 , -25771/9 ), ( 1675/107 , -4557/107 )
**u= 13/64 , ht= 67 ; ( -317/573 , 1786/573 ), ( -3/67 , 181/67 )
**u= 13/68 , ht= 863 ; ( 12501/3355 , -36064/3355 ), ( -259/863 , -2511/863 )
**u= 13/69 , ht= 3157 ; ( -9197/2648 , -26869/2648 ), ( 1570/3157 , 9913/3157 )
**u= 13/71 , ht= 1889 ; ( -11321/2311 , -32901/2311 ), ( -241/1889 , -5421/1889 )
**u= 13/76 , ht= 14223 ; ( -31757/3507 , -94162/3507 ), ( 14223/3203 , 43363/3203 )
**u= 13/77 , ht= 67 ; ( -67/10 , -201/10 ), ( -10/67 , 3 )
**u= 13/79 , ht= 4559 ; ( 4559/3295 , -16959/3295 ), ( -55955/2663 , 169887/2663 )
**u= 13/85 , ht= 458 ; ( 1297/362 , 4203/362 ), ( 458/67 , 1455/67 )
**u= 13/88 , ht= 1613 ; ( -943/2929 , -9834/2929 ), ( -499/1613 , -5364/1613 )
**u= 13/93 , ht= 8240 ; ( -4501/8240 , -30811/8240 ), ( 17140/1733 , -56613/1733 )
**u= 13/95 , ht= 97 ; ( -7/373 , -1239/373 ), ( -97/17 , -327/17 )
**u= 13/96 , ht= 64 ; ( 31/64 , -1897/512 ), ( 5576/2309 , 161037/18472 )
**u= 13/100 , ht= 369 ; ( 32921/3015 , 559918/15075 ), ( 369/155 , 6811/775 )
**u= 15/2 , ht= 2 ; ( 1/2 , -7/4 ), ( -49/37 , 68/37 )
**u= 15/7 , ht= 13 ; ( 9/58 , -77/58 ), ( 6/13 , -1/13 )
**u= 15/11 , ht= 119 ; ( 119/2 , -77/2 ), ( 2/119 , -11/17 )
**u= 15/13 , ht= 17 ; ( 29/3 , 25/3 ), ( -9/17 , 19/17 )
**u= 15/14 , ht= 17 ; ( -17/11 , -2 ), ( -11/38 , -77/76 )
**u= 15/16 , ht= 3 ; ( 1381/357 , -1556/357 ), ( -1/3 , -7/6 )
**u= 15/26 , ht= 19 ; ( -7/19 , -34/19 ), ( -19/7 , 34/7 )
**u= 15/28 , ht= 79 ; ( -33/79 , -148/79 ), ( 327/89 , -590/89 )
**u= 15/31 , ht= 764 ; ( 237/764 , -1453/764 ), ( 2768/381 , 5101/381 )
**u= 15/37 , ht= 453 ; ( 57/1201 , 2375/1201 ), ( -453/361 , -59/19 )
**u= 15/38 , ht= 56 ; ( 191/37 , 10 ), ( -37/56 , 37/16 )
**u= 15/59 , ht= 178 ; ( 1499/968 , -4315/968 ), ( 178 , -437 )
**u= 15/62 , ht= 218 ; ( 218 , -1075/2 ), ( 949/298 , -4993/596 )
**u= 15/76 , ht= 3559 ; ( -2103/3559 , -11432/3559 ), ( -28317/2881 , 78965/2881 )
**u= 15/79 , ht= 51 ; ( -51/28 , 163/28 ), ( -134/4527 , 12653/4527 )
**u= 15/83 , ht= 832 ; ( 5129/398 , 14741/398 ), ( 832/251 , 2515/251 )
**u= 15/98 , ht= 179 ; ( 119/253 , -6142/1771 ), ( -179/167 , -5366/1169 )
**u= 16/3 , ht= 461 ; ( 419/461 , -1823/1844 ), ( 605/563 , 2273/2252 )
**u= 16/11 , ht= 19 ; ( -3/19 , 101/76 ), ( -19/3 , -101/12 )
**u= 16/15 , ht= 77 ; ( 7/157 , 875/628 ), ( 23/77 , 45/44 )
**u= 16/27 , ht= 4189 ; ( 4189/3859 , -108217/46308 ), ( 5213/4883 , 136889/58596 )
**u= 16/35 , ht= 121 ; ( -121/89 , 1077/356 ), ( 551/2161 , 15657/8644 )
**u= 16/47 , ht= 249 ; ( -249/221 , -2789/884 ), ( 2465/491 , 21477/1964 )
**u= 16/51 , ht= 791 ; ( -967/359 , 8909/1436 ), ( -127/791 , 6897/3164 )
**u= 16/55 , ht= 131 ; ( 307/153 , 3089/612 ), ( -89/131 , 1431/524 )
**u= 16/67 , ht= 6247 ; ( 6247/1825 , 64719/7300 ), ( -857/7183 , -71847/28732 )
**u= 16/75 , ht= 241 ; ( -241/215 , -17113/4300 ), ( 1937/815 , 112037/16300 )
**u= 16/79 , ht= 187 ; ( -493/455 , 7299/1820 ), ( -129/187 , 2467/748 )
**u= 16/83 , ht= 103 ; ( -63/103 , 1351/412 ), ( 37/203 , -327/116 )
**u= 17/2 , ht= 25 ; ( -19/25 , -36/25 ), ( 61/62 , -57/124 )
**u= 17/6 , ht= 31 ; ( 31/29 , -28/29 ), ( -41057/10123 , -56476/10123 )
**u= 17/10 , ht= 4 ; ( -4 , 3/2 ), ( -1/4 , -3/8 )
**u= 17/13 , ht= 38 ; ( 251/628 , 861/628 ), ( 38 , 51 )
**u= 17/14 , ht= 53 ; ( 53/25 , 54/25 ), ( 25/53 , 54/53 )
**u= 17/22 , ht= 184 ; ( 1193/379 , 1596/379 ), ( 184/63 , -583/126 )
**u= 17/29 , ht= 97 ; ( -101/914 , 1551/914 ), ( 48/97 , 167/97 )
**u= 17/30 , ht= 59 ; ( -49/59 , -126/59 ), ( 593/77 , 146/11 )
**u= 17/37 , ht= 89 ; ( 89/52 , -183/52 ), ( -1024/2787 , 5219/2787 )
**u= 17/38 , ht= 56 ; ( -1033/69 , -1832/69 ), ( 56/37 , 249/74 )
**u= 17/42 , ht= 199 ; ( 19/383 , 758/383 ), ( 199/88 , 853/176 )
**u= 17/50 , ht= 9 ; ( -146/771 , 3355/1542 ), ( -9/2 , 197/20 )
**u= 17/53 , ht= 1091 ; ( -5521/5662 , -17109/5662 ), ( -22/1091 , -2319/1091 )
**u= 17/58 , ht= 477 ; ( -2279/157 , 5094/157 ), ( 361/477 , 1348/477 )
**u= 17/62 , ht= 7917 ; ( 27451/8975 , 66864/8975 ), ( 7917/1159 , 18938/1159 )
**u= 17/66 , ht= 25 ; ( 722/257 , -3667/514 ), ( 25/19 , 4 )
**u= 17/69 , ht= 6500 ; ( 2659/6500 , 17453/6500 ), ( -12634/131 , 31489/131 )
**u= 17/77 , ht= 150 ; ( -9601/14808 , 46153/14808 ), ( 150/73 , 437/73 )
**u= 17/93 , ht= 4888 ; ( -971/4888 , -14349/4888 ), ( -6370/3023 , -20269/3023 )
**u= 17/98 , ht= 265 ; ( -62/945 , 39181/13230 ), ( -265/227 , 7188/1589 )
**u= 24 , ht= 4 ; ( 3/4 , -37/16 ), ( 60/59 , -173/236 )
**u= 24/13 , ht= 19 ; ( -38/27 , -133/108 ), ( -6/19 , -1/4 )
**u= 24/49 , ht= 1 ; ( 1296/47 , -1297/28 ), ( 0 , 47/28 )
**u= 24/73 , ht= 49 ; ( -49/24 , -461/96 ), ( 685/1488 , 13831/5952 )
**u= 24/85 , ht= 26 ; ( -11942/2253 , 110647/9012 ), ( 9/26 , -251/104 )
**u= 24/97 , ht= 1473 ; ( -1473/952 , -17201/3808 ), ( 167/2592 , -25321/10368 )
**u= 25 , ht= 8 ; ( -5/8 , -111/40 ), ( 30/17 , 439/85 )
**u= 25/3 , ht= 157 ; ( 157/151 , -29/151 ), ( 347/323 , 1639/1615 )
**u= 25/4 , ht= 61 ; ( 107/101 , 36/101 ), ( -61/35 , -474/175 )
**u= 25/8 , ht= 8 ; ( -8/7 , 111/140 ), ( -31/33 , -28/33 )
**u= 25/12 , ht= 1301 ; ( -1301/1159 , 6749/5795 ), ( 2261/2111 , 13759/10555 )
**u= 25/17 , ht= 129 ; ( -333/44 , -17/4 ), ( 44/129 , 451/645 )
**u= 25/19 , ht= 25 ; ( 313/85 , -1227/425 ), ( -13/25 , 123/125 )
**u= 25/27 , ht= 389 ; ( 311/389 , -9803/5835 ), ( 1571/1403 , 8171/4209 )
**u= 25/32 , ht= 461 ; ( 5103/1223 , 65789/12230 ), ( -77/461 , 582/461 )
**u= 25/33 , ht= 104 ; ( 223/14 , -283/14 ), ( -104/61 , 887/305 )
**u= 25/36 , ht= 271 ; ( -293/95 , 2111/475 ), ( -271/53 , -435/53 )
**u= 25/48 , ht= 13153 ; ( 13153/12571 , -61627/25142 ), ( -45799/45221 , -1092889/452210 )
**u= 25/52 , ht= 577 ; ( 1507/701 , -2862/701 ), ( -577/463 , 6588/2315 )
**u= 25/56 , ht= 2180 ; ( -1173/2180 , -92401/43600 ), ( -3896/3335 , -188973/66700 )
**u= 25/57 , ht= 653 ; ( 653/178 , 6083/890 ), ( -676/1357 , -2749/1357 )
**u= 25/59 , ht= 2515 ; ( 2547/895 , -24797/4475 ), ( -1261/2515 , -25989/12575 )
**u= 25/64 , ht= 175 ; ( -197/529 , -4509/2116 ), ( -175/101 , -8019/2020 )
**u= 25/73 , ht= 181 ; ( -8557/2028 , 90401/10140 ), ( 108/181 , 437/181 )
**u= 25/81 , ht= 5 ; ( 5/4 , 211/60 ), ( -4/5 , -211/75 )
**u= 25/84 , ht= 173 ; ( 173/89 , -433/89 ), ( 205/4909 , 54334/24545 )
**u= 25/91 , ht= 129 ; ( 25/129 , 1553/645 ), ( -129/25 , 1553/125 )
**u= 25/92 , ht= 179 ; ( -4027/6139 , -17334/6139 ), ( -179/41 , -2181/205 )
**u= 25/99 , ht= 15 ; ( -15/7 , 601/105 ), ( 7/15 , -601/225 )
**u= 26/3 , ht= 68 ; ( -67/68 , -93/136 ), ( 565/316 , -2023/632 )
**u= 26/7 , ht= 23 ; ( 279/230 , -7/20 ), ( 23/2 , -69/4 )
**u= 26/11 , ht= 7 ; ( 1652/915 , 331/1830 ), ( 4/7 , -3/14 )
**u= 26/15 , ht= 773 ; ( -773/22 , -17/44 ), ( -161/1054 , 417/2108 )
**u= 26/31 , ht= 954 ; ( -954/937 , 3569/1874 ), ( 6074/4617 , 21037/9234 )
**u= 26/43 , ht= 1904 ; ( -10773/1528 , 32207/3056 ), ( 719/1904 , -873/544 )
**u= 26/51 , ht= 67 ; ( 67/16 , -227/32 ), ( -15760/83843 , 280611/167686 )
**u= 26/59 , ht= 5648 ; ( -5648/1491 , -20939/2982 ), ( -7188/9391 , 43267/18782 )
**u= 26/67 , ht= 5444 ; ( 42161/4976 , 162807/9952 ), ( -1055/5444 , 21291/10888 )
**u= 26/71 , ht= 211 ; ( -986/647 , -4731/1294 ), ( 211/2 , -873/4 )
**u= 26/79 , ht= 3989 ; ( 921834/314087 , -4098313/628174 ), ( 3910/3989 , 23841/7978 )
**u= 26/87 , ht= 2735 ; ( -55514/3269 , -245463/6538 ), ( 2735/38 , 12437/76 )
**u= 26/95 , ht= 1409 ; ( 1409/1314 , 9013/2628 ), ( 1314/1409 , 9013/2818 )
**u= 27 , ht= 1 ; ( 1/11 , -11/3 ), ( 1 , -1/3 )
**u= 27/4 , ht= 1 ; ( -1 , 2/3 ), ( -1 , -2/3 )
**u= 27/14 , ht= 9 ; ( 213/73 , 86/219 ), ( -5/9 , 16/27 )
**u= 27/16 , ht= 37 ; ( -91/25 , -229/150 ), ( 25/37 , -203/222 )
**u= 27/19 , ht= 47 ; ( 47/28 , 139/84 ), ( -34/61 , -173/183 )
**u= 27/23 , ht= 32 ; ( 13/36 , -151/108 ), ( 32/9 , -133/27 )
**u= 27/25 , ht= 95 ; ( -71/95 , -2209/1425 ), ( -485/433 , -11743/6495 )
**u= 27/26 , ht= 307 ; ( 1047/163 , -3098/489 ), ( 307/177 , 1388/531 )
**u= 27/47 , ht= 33 ; ( -33/14 , 167/42 ), ( -1188/269 , 6191/807 )
**u= 27/49 , ht= 1 ; ( -1 , 7/3 ), ( 35/97 , -3431/2037 )
**u= 27/62 , ht= 115 ; ( -9/115 , 662/345 ), ( 945/356 , -11509/2136 )
**u= 27/64 , ht= 1 ; ( -1 , -8/3 ), ( -1 , 8/3 )
**u= 27/67 , ht= 143 ; ( -143/76 , -923/228 ), ( -46/247 , -109/57 )
**u= 27/71 , ht= 93 ; ( 93/22 , -557/66 ), ( -6/349 , -2029/1047 )
**u= 27/74 , ht= 102 ; ( -713/582 , -11147/3492 ), ( -102/83 , 1607/498 )
**u= 27/88 , ht= 16 ; ( -133/155 , -1358/465 ), ( -5/16 , 439/192 )
**u= 27/91 , ht= 430 ; ( -389/430 , 3917/1290 ), ( -430/389 , -3917/1167 )
**u= 27/95 , ht= 1896 ; ( 6129/1664 , -43259/4992 ), ( -1896/289 , -13387/867 )
**u= 27/100 , ht= 19915 ; ( -19915/19513 , -985507/292695 ), ( -20887/20485 , -1034593/307275 )
**u= 29 , ht= 38 ; ( 21/38 , 121/38 ), ( 782/41 , -2979/41 )
**u= 29/2 , ht= 385 ; ( -368/385 , 699/770 ), ( -12433/1528 , 66933/3056 )
**u= 29/9 , ht= 70 ; ( -1053/1094 , 3307/3282 ), ( -70/57 , 241/171 )
**u= 29/10 , ht= 18 ; ( -18/13 , -11/26 ), ( 1777/1122 , 4489/2244 )
**u= 29/17 , ht= 127 ; ( 127/42 , 59/42 ), ( -334/499 , -447/499 )
**u= 29/18 , ht= 7 ; ( -7/3 , 14/9 ), ( 1727/3752 , 753/1072 )
**u= 29/22 , ht= 82 ; ( 27/82 , 223/164 ), ( 194/601 , -981/1202 )
**u= 29/26 , ht= 166 ; ( -166/107 , -417/214 ), ( 243/854 , -1657/1708 )
**u= 29/30 , ht= 988 ; ( -11329/1976 , 1853/304 ), ( -988/53 , -2821/106 )
**u= 29/38 , ht= 233 ; ( 157/233 , -408/233 ), ( -23562/26123 , -97651/52246 )
**u= 29/41 , ht= 10091 ; ( -10091/2250 , 13873/2250 ), ( -29042/5521 , 46269/5521 )
**u= 29/42 , ht= 337 ; ( -337/13 , 456/13 ), ( 38597/4577 , -61554/4577 )
**u= 29/49 , ht= 527 ; ( 527/404 , -1041/404 ), ( 800/231 , -9727/1617 )
**u= 29/54 , ht= 131 ; ( -13379/3658 , 44401/7316 ), ( -23/131 , -212/131 )
**u= 29/62 , ht= 327 ; ( -107/327 , 634/327 ), ( 327/107 , 634/107 )
**u= 29/66 , ht= 1739 ; ( -10013/17641 , -38092/17641 ), ( -1739/1583 , 4358/1583 )
**u= 29/70 , ht= 7522 ; ( 2307/7522 , -30643/15044 ), ( -36529/36901 , -98772/36901 )
**u= 29/73 , ht= 7899 ; ( 7899/6074 , -19231/6074 ), ( 18800/1499 , 37587/1499 )
**u= 29/78 , ht= 11848 ; ( -175159/123035 , -426572/123035 ), ( 11848/707 , -48759/1414 )
**u= 29/82 , ht= 1388 ; ( -2464/365 , -10053/730 ), ( 445/1388 , -5901/2776 )
**u= 29/86 , ht= 977 ; ( 5783/494 , -24021/988 ), ( -485/977 , -2274/977 )
**u= 29/90 , ht= 54 ; ( 54/19 , -731/114 ), ( -3481/534 , -46337/3204 )
**u= 29/94 , ht= 496 ; ( 92181/26945 , -208916/26945 ), ( 496/167 , -2337/334 )
**u= 29/97 , ht= 5889 ; ( 31239/5524 , 70057/5524 ), ( 1136/5889 , -1019/453 )
**u= 30/41 , ht= 44 ; ( 3/44 , -137/88 ), ( -756/433 , -2605/866 )
**u= 30/53 , ht= 71 ; ( 71/18 , 227/36 ), ( 167/114 , 671/228 )
**u= 30/77 , ht= 558 ; ( 558/19 , 2135/38 ), ( 1318/471 , -5599/942 )
**u= 30/89 , ht= 1048 ; ( -1048/999 , -6103/1998 ), ( -529/2052 , 8791/4104 )
**u= 37/5 , ht= 513 ; ( -513/488 , 95/488 ), ( -3796/1311 , -373/69 )
**u= 37/7 , ht= 51 ; ( -461/435 , -229/435 ), ( -51/47 , -49/47 )
**u= 37/12 , ht= 59 ; ( -53/59 , 64/59 ), ( 811/943 , 629/943 )
**u= 37/13 , ht= 277 ; ( -277/256 , 243/256 ), ( 1194/1753 , -163/1753 )
**u= 37/16 , ht= 135 ; ( -41/135 , 179/135 ), ( 307/563 , 96/563 )
**u= 37/20 , ht= 377 ; ( -153/377 , -38/29 ), ( -221/771 , -130/771 )
**u= 37/21 , ht= 127 ; ( 317/170 , -217/170 ), ( 22/127 , 19/127 )
**u= 37/23 , ht= 695 ; ( -2247/421 , -977/421 ), ( 695/603 , 941/603 )
**u= 37/24 , ht= 47 ; ( -47/23 , 37/23 ), ( 16/89 , 181/356 )
**u= 37/28 , ht= 4745 ; ( -16897/405 , -11651/405 ), ( -4745/4087 , 114/67 )
**u= 37/31 , ht= 247 ; ( 941/755 , 1281/755 ), ( -247/187 , -369/187 )
**u= 37/32 , ht= 605 ; ( -477/605 , -922/605 ), ( 605/477 , -922/477 )
**u= 37/39 , ht= 5441 ; ( 5441/839 , 5849/839 ), ( -331/6611 , -6971/6611 )
**u= 37/44 , ht= 12773 ; ( -12773/7115 , -18276/7115 ), ( -27655/7237 , -41919/7237 )
**u= 37/45 , ht= 82 ; ( -63/82 , -431/246 ), ( -508/717 , -3425/2151 )
**u= 37/47 , ht= 2199 ; ( 2199/977 , 3083/977 ), ( 10205/5693 , 16983/5693 )
**u= 37/55 , ht= 7151 ; ( 7151/791 , 9921/791 ), ( 123921/28331 , 202301/28331 )
**u= 37/60 , ht= 607 ; ( -607/53 , 889/53 ), ( 653/263 , -1147/263 )
**u= 37/68 , ht= 919 ; ( 919/443 , 1641/443 ), ( -7895/5671 , 16392/5671 )
**u= 37/72 , ht= 329 ; ( 198/329 , 8033/3948 ), ( 28963/1269 , 154969/3807 )
**u= 37/76 , ht= 3079 ; ( -23053/4193 , -39579/4193 ), ( 381/3079 , -5233/3079 )
**u= 37/77 , ht= 124 ; ( -107/124 , 291/124 ), ( 2704/537 , -5041/537 )
**u= 37/79 , ht= 109 ; ( -49/109 , -219/109 ), ( 109/49 , 219/49 )
**u= 37/80 , ht= 193 ; ( 3017/2137 , 6579/2137 ), ( 193/63 , 376/63 )
**u= 37/93 , ht= 7987 ; ( 7987/4174 , 17233/4174 ), ( 29204/18323 , 67707/18323 )
**u= 39/4 , ht= 5 ; ( 1/5 , -11/5 ), ( 5 , -11 )
**u= 39/7 , ht= 32 ; ( 2299/2096 , -347/2096 ), ( -32/17 , -49/17 )
**u= 39/11 , ht= 69 ; ( -371/498 , -599/498 ), ( 58/69 , -31/69 )
**u= 39/14 , ht= 43 ; ( 6/43 , 119/86 ), ( -285/226 , -673/452 )
**u= 39/28 , ht= 223 ; ( 1189/251 , 806/251 ), ( -127/223 , -218/223 )
**u= 39/31 , ht= 71 ; ( -71/22 , 61/22 ), ( 596/1433 , 1343/1433 )
**u= 39/35 , ht= 93 ; ( -93/68 , 125/68 ), ( 54/439 , -397/439 )
**u= 39/38 , ht= 96 ; ( -96 , -187/2 ), ( 1/96 , 187/192 )
**u= 39/40 , ht= 23 ; ( -23/2 , -95/8 ), ( 1558/1687 , -1607/964 )
**u= 39/55 , ht= 824 ; ( -1069/856 , -1955/856 ), ( -824 , -1295 )
**u= 39/59 , ht= 381 ; ( -381/32 , 533/32 ), ( -620/501 , -1219/501 )
**u= 39/61 , ht= 485 ; ( 731/2927 , 4891/2927 ), ( 485/241 , -863/241 )
**u= 39/74 , ht= 2533 ; ( 4961777/597516 , 16086373/1195032 ), ( 2533/1719 , -5252/1719 )
**u= 39/76 , ht= 1819 ; ( -4417/1055 , -7454/1055 ), ( 1819/1477 , -4042/1477 )
**u= 39/83 , ht= 301 ; ( -8643/2252 , 15437/2252 ), ( 114/301 , -13/7 )
**u= 39/85 , ht= 919 ; ( 7489/3749 , 14821/3749 ), ( 919/409 , -1861/409 )
**u= 39/88 , ht= 337 ; ( -24481/4735 , 44458/4735 ), ( 337/73 , -653/73 )
**u= 39/98 , ht= 3 ; ( 3/2 , 97/28 ), ( 3417/653 , -48424/4571 )
**u= 40/3 , ht= 22 ; ( 502/539 , 205/196 ), ( -22/19 , 121/76 )
**u= 40/7 , ht= 48 ; ( -29/48 , 283/192 ), ( -649/212 , 4377/848 )
**u= 40/19 , ht= 679 ; ( 679/318 , 589/1272 ), ( 1061/1498 , 4425/5992 )
**u= 40/23 , ht= 172 ; ( 172/11 , -3/44 ), ( -76/223 , -393/892 )
**u= 40/31 , ht= 1327 ; ( -1327/316 , -4179/1264 ), ( 4951/2192 , -27363/8768 )
**u= 40/39 , ht= 23 ; ( -23/16 , -127/64 ), ( -16/23 , 127/92 )
**u= 40/43 , ht= 866 ; ( -1046/373 , 4971/1492 ), ( -866/467 , 5379/1868 )
**u= 40/51 , ht= 62 ; ( 62/11 , 313/44 ), ( -5038/19289 , -99933/77156 )
**u= 40/71 , ht= 127 ; ( 127/56 , 873/224 ), ( 2041/2112 , -19135/8448 )
**u= 40/79 , ht= 361 ; ( 2264/3133 , 26967/12532 ), ( -192/361 , 2747/1444 )
**u= 41/2 , ht= 155 ; ( 769/771 , 344/771 ), ( 155 , 498 )
**u= 41/6 , ht= 385 ; ( -61/385 , -66/35 ), ( 616/647 , 407/1294 )
**u= 41/13 , ht= 243 ; ( 243/178 , -1/178 ), ( -334/69 , -473/69 )
**u= 41/14 , ht= 54 ; ( 47/54 , 121/108 ), ( 135/149 , 122/149 )
**u= 41/18 , ht= 38 ; ( -38/27 , 149/162 ), ( -859/1166 , 1641/2332 )
**u= 41/21 , ht= 109 ; ( -1159/598 , -587/598 ), ( -38/109 , -7/109 )
**u= 41/26 , ht= 2857 ; ( 2857/621 , -1414/621 ), ( 83559/23707 , 110626/23707 )
**u= 41/29 , ht= 107 ; ( 107/20 , 69/20 ), ( -4488/239 , 5969/239 )
**u= 41/37 , ht= 479 ; ( -479/76 , 441/76 ), ( -1222/2737 , -2973/2737 )
**u= 41/46 , ht= 191 ; ( -191/85 , 246/85 ), ( 3586718/688281 , 10577209/1376562 )
**u= 41/50 , ht= 23 ; ( -1067/5175 , -1706/1125 ), ( 23/2 , -69/4 )
**u= 41/53 , ht= 34 ; ( 115121/215556 , -358663/215556 ), ( 34/19 , 3 )
**u= 41/58 , ht= 1955 ; ( -1728/1955 , 7679/3910 ), ( 755/2408 , 6831/4816 )
**u= 41/61 , ht= 170 ; ( -13/170 , 273/170 ), ( -43928/10179 , 5521/783 )
**u= 41/70 , ht= 52 ; ( 2983/8 , 8985/16 ), ( 52/17 , 183/34 )
**u= 41/77 , ht= 83 ; ( -82387/15172 , 134253/15172 ), ( -50/83 , -159/83 )
**u= 41/85 , ht= 62 ; ( 12553/6008 , -23937/6008 ), ( -62/57 , 149/57 )
**u= 41/86 , ht= 4739 ; ( 4739/1815 , 8744/1815 ), ( 20735/4739 , 38994/4739 )
**u= 41/90 , ht= 704 ; ( -704/511 , -3123/1022 ), ( 31/2616 , -27481/15696 )
**u= 41/98 , ht= 3074 ; ( -3074/665 , 81129/9310 ), ( 6755/2329 , 96858/16303 )
**u= 48 , ht= 1537 ; ( -1537/1535 , 1/6140 ), ( -1535/1537 , -1/6148 )
**u= 48/13 , ht= 19 ; ( 7/19 , -109/76 ), ( -19/7 , 109/28 )
**u= 48/25 , ht= 13 ; ( 13/5 , -73/100 ), ( -35/109 , -61/2180 )
**u= 48/37 , ht= 173 ; ( -8233/869 , -23923/3476 ), ( 25/173 , -511/692 )
**u= 48/49 , ht= 1 ; ( -3463/3265 , -163349/91420 ), ( -1 , -7/4 )
**u= 48/61 , ht= 67 ; ( 67/55 , -469/220 ), ( -55/67 , 7/4 )
**u= 48/97 , ht= 803 ; ( -749/803 , -7669/3212 ), ( -691/1637 , 12017/6548 )
**u= 49 , ht= 20 ; ( 29/30 , 271/210 ), ( -20/19 , 219/133 )
**u= 49/3 , ht= 601 ; ( 601/595 , -589/4165 ), ( 1235/1211 , 6079/8477 )
**u= 49/8 , ht= 47 ; ( -48/47 , -17/28 ), ( 47/17 , 564/119 )
**u= 49/9 , ht= 30 ; ( -41/40 , 183/280 ), ( -30/17 , 937/357 )
**u= 49/11 , ht= 47 ; ( 303/325 , -2183/2275 ), ( -41/47 , 81/329 )
**u= 49/12 , ht= 3941 ; ( 3791/3941 , -25579/27587 ), ( -17143/16997 , -104827/118979 )
**u= 49/16 , ht= 43 ; ( -43/31 , -57/434 ), ( -165/43 , -1613/301 )
**u= 49/24 , ht= 29 ; ( 319/130 , -657/3640 ), ( -22/29 , -691/812 )
**u= 49/25 , ht= 17 ; ( -17/6 , 31/210 ), ( -32/43 , -261/301 )
**u= 49/27 , ht= 133 ; ( 133/55 , -1297/1155 ), ( 2459/2291 , 66607/48111 )
**u= 49/40 , ht= 14 ; ( 14 , -309/28 ), ( 97/217 , 1503/1519 )
**u= 49/41 , ht= 350 ; ( -43671/3848 , -250583/26936 ), ( -350/347 , 3873/2429 )
**u= 49/43 , ht= 53 ; ( -53/33 , -41/21 ), ( 935/419 , 9339/2933 )
**u= 49/44 , ht= 239 ; ( 2647/425 , 16974/2975 ), ( 239/173 , 2547/1211 )
**u= 49/48 , ht= 9601 ; ( 9601/9019 , -221111/126266 ), ( 19205/18623 , 456409/260722 )
**u= 49/52 , ht= 1317 ; ( -763/1317 , 14398/9219 ), ( 4981/1159 , -50811/8113 )
**u= 49/57 , ht= 520 ; ( 463/520 , 6529/3640 ), ( -682/743 , 9227/5201 )
**u= 49/59 , ht= 1547 ; ( 25233/4739 , 214097/33173 ), ( -839/1547 , -1191/833 )
**u= 49/64 , ht= 163 ; ( 53/261 , 5669/3654 ), ( 151/163 , 8643/4564 )
**u= 49/67 , ht= 797 ; ( 3463/783 , 32603/5481 ), ( 663/797 , 10223/5579 )
**u= 49/68 , ht= 641 ; ( 539/641 , 8583/4487 ), ( -1355/549 , -15656/3843 )
**u= 49/73 , ht= 443 ; ( -443/68 , 4341/476 ), ( -1304/959 , 17313/6713 )
**u= 49/75 , ht= 11081 ; ( -11081/10855 , 821437/379925 ), ( -28465/28913 , 2149349/1011955 )
**u= 49/81 , ht= 101 ; ( 491/58 , -15349/1218 ), ( 78/101 , 12463/6363 )
**u= 49/89 , ht= 146 ; ( 2841/2218 , 41131/15526 ), ( 146/57 , -1877/399 )
**u= 49/92 , ht= 807 ; ( 807/25 , 9026/175 ), ( -4369/2991 , 63254/20937 )
**u= 49/96 , ht= 2303 ; ( -2303/214 , -212353/11984 ), ( 4669/997 , 119037/13958 )
**u= 49/100 , ht= 159 ; ( -17/159 , -10168/5565 ), ( 527/673 , -10368/4711 )
**u= 50/3 , ht= 41 ; ( -38/41 , -95/82 ), ( -65/38 , 81/20 )
**u= 50/7 , ht= 68 ; ( -65/68 , 573/680 ), ( -68/65 , -573/650 )
**u= 50/19 , ht= 53 ; ( -42/53 , -631/530 ), ( -53/42 , 631/420 )
**u= 50/31 , ht= 24 ; ( 97/4 , -69/8 ), ( -7/24 , 25/48 )
**u= 50/47 , ht= 841 ; ( 3440/291 , 32489/2910 ), ( -240/841 , 8561/8410 )
**u= 50/51 , ht= 694 ; ( -683/694 , 2415/1388 ), ( 3458/2081 , -10707/4162 )
**u= 50/59 , ht= 91 ; ( 58/117 , -143/90 ), ( -91/86 , -39/20 )
**u= 50/71 , ht= 164 ; ( -247/108 , 3707/1080 ), ( -49/164 , 2319/1640 )
**u= 50/91 , ht= 1906 ; ( -369/1906 , -33541/19060 ), ( -3853/1698 , -71893/16980 )
**u= 50/99 , ht= 86 ; ( 86/31 , 1523/310 ), ( 273/86 , -3061/516 )
**u= 51/16 , ht= 107 ; ( -107/105 , 199/210 ), ( -385/129 , 536/129 )
**u= 51/19 , ht= 90 ; ( 71/90 , 107/90 ), ( -86/123 , 47/123 )
**u= 51/23 , ht= 1457 ; ( -1457/728 , -13/56 ), ( 2366/641 , -3133/641 )
**u= 51/25 , ht= 27 ; ( -27/11 , 1/5 ), ( -231/419 , 209/419 )
**u= 51/26 , ht= 1409 ; ( -1409/502 , 209/1004 ), ( -5113/3571 , -6548/3571 )
**u= 51/38 , ht= 14 ; ( -14/5 , 23/10 ), ( -5837/262 , -15617/524 )
**u= 51/40 , ht= 489 ; ( -489 , -359 ), ( 516/401 , 3019/1604 )
**u= 51/49 , ht= 29 ; ( -79/27 , 593/189 ), ( -3/29 , -197/203 )
**u= 51/52 , ht= 63 ; ( -32429/21885 , -45388/21885 ), ( 37/63 , 83/63 )
**u= 51/62 , ht= 488 ; ( -488/473 , 1829/946 ), ( 439/811 , -1166/811 )
**u= 51/74 , ht= 1322 ; ( 17063/8146 , 52939/16292 ), ( 1322/101 , -4207/202 )
**u= 51/86 , ht= 274 ; ( 274/145 , -953/290 ), ( -1429/7042 , 21587/14084 )
**u= 51/95 , ht= 4034 ; ( 139/4034 , 7063/4034 ), ( 12386/4159 , 22661/4159 )
**u= 51/97 , ht= 2127 ; ( -2127/977 , 3833/977 ), ( 4567/687 , -8137/687 )
**u= 51/98 , ht= 580 ; ( 1243/145 , -14204/1015 ), ( -580/433 , 2489/866 )
**u= 51/100 , ht= 177 ; ( -907/1401 , 14563/7005 ), ( -91/177 , 1663/885 )
**u= 52/3 , ht= 5 ; ( -1/5 , -29/10 ), ( -32029/5423 , -186839/10846 )
**u= 52/7 , ht= 521 ; ( -473/521 , 1053/1042 ), ( 1215/823 , 3689/1646 )
**u= 52/11 , ht= 523 ; ( 523/457 , -27/914 ), ( 6869/6759 , -11491/13518 )
**u= 52/23 , ht= 427 ; ( -1097/1249 , 2991/2498 ), ( -427/309 , -1063/618 )
**u= 52/31 , ht= 2293 ; ( 2293/805 , 2379/1610 ), ( 2553/11543 , -8647/23086 )
**u= 52/35 , ht= 7779 ; ( 18537/827 , 19273/1654 ), ( 2809/7779 , -10949/15558 )
**u= 52/43 , ht= 217 ; ( -911/723 , -2443/1446 ), ( 45/217 , 367/434 )
**u= 52/55 , ht= 131 ; ( -131/21 , 283/42 ), ( 6323/493 , 18177/986 )
**u= 52/59 , ht= 3837 ; ( -19609/4109 , -45657/8218 ), ( 887/3837 , 9001/7674 )
**u= 52/63 , ht= 2543 ; ( -3301/1123 , -8513/2246 ), ( -1041/2543 , 20347/15258 )
**u= 52/67 , ht= 635 ; ( 3287/3019 , -12303/6038 ), ( 635/367 , 2139/734 )
**u= 52/79 , ht= 8055 ; ( -13807/255 , 38567/510 ), ( -8055/5849 , -30719/11698 )
**u= 53/10 , ht= 67 ; ( 57/67 , 74/67 ), ( -817/82 , 2787/164 )
**u= 53/17 , ht= 29 ; ( 29/22 , 9/22 ), ( 1068/1459 , -181/1459 )
**u= 53/22 , ht= 2001 ; ( 2001/1253 , 734/1253 ), ( -6613/1791 , -8834/1791 )
**u= 53/25 , ht= 26 ; ( 15/26 , 167/130 ), ( 26/15 , 167/75 )
**u= 53/26 , ht= 38 ; ( -11/41 , -54/41 ), ( -27/38 , 59/76 )
**u= 53/33 , ht= 12080 ; ( 31717/370 , -11487/370 ), ( 12080/3353 , 2281/479 )
**u= 53/34 , ht= 5352 ; ( 5352/235 , 4619/470 ), ( -9727/4672 , -25983/9344 )
**u= 53/38 , ht= 71 ; ( 18337/3574 , -24477/7148 ), ( 43/71 , 72/71 )
**u= 53/42 , ht= 83 ; ( -83 , -62 ), ( 295/211 , 428/211 )
**u= 53/46 , ht= 7011 ; ( -7011/2465 , 6862/2465 ), ( -33683/27727 , -51852/27727 )
**u= 53/49 , ht= 121 ; ( 121/26 , -9/2 ), ( -2470/259 , 24063/1813 )
**u= 53/50 , ht= 52 ; ( 52/23 , 117/46 ), ( -231/13 , -124/5 )
**u= 53/65 , ht= 19542 ; ( 85379/49834 , -126567/49834 ), ( -19542/5327 , -4283/761 )
**u= 53/66 , ht= 17813 ; ( -17813/2977 , 22032/2977 ), ( 22465/11344 , 73049/22688 )
**u= 53/74 , ht= 1953 ; ( -1562/1953 , 7373/3906 ), ( -96633/6523 , 151574/6523 )
**u= 53/78 , ht= 2701 ; ( 2701/2272 , -10349/4544 ), ( -5491/47 , 8764/47 )
**u= 53/81 , ht= 51 ; ( 51/46 , -929/414 ), ( -4364/7569 , -114721/68121 )
**u= 53/90 , ht= 2563 ; ( 14629/7079 , -24990/7079 ), ( -1577/2563 , -4676/2563 )
**u= 53/97 , ht= 1011 ; ( 1011/332 , 1691/332 ), ( 4810/3161 , 9723/3161 )
**u= 53/98 , ht= 1732 ; ( -1732/669 , -5957/1338 ), ( 251/3767 , 5976/3767 )
**u= 54/5 , ht= 68 ; ( 224/299 , -2885/1794 ), ( -67/68 , -139/408 )
**u= 54/29 , ht= 196 ; ( -49/212 , 1673/1272 ), ( 167/196 , 179/168 )
**u= 54/65 , ht= 34 ; ( -34/31 , 367/186 ), ( -31/34 , -367/204 )
**u= 54/89 , ht= 2947 ; ( -2947/346 , 26269/2076 ), ( -428261/56522 , -4309961/339132 )
**u= 60 , ht= 3 ; ( -1/3 , -31/6 ), ( -189/127 , -1535/254 )
**u= 60/13 , ht= 363 ; ( -259/363 , 925/726 ), ( 729/703 , -35/38 )
**u= 60/61 , ht= 607 ; ( -6273/559 , 12851/1118 ), ( -369/607 , -1619/1214 )
**u= 60/73 , ht= 7601 ; ( -47227/9129 , -115589/18258 ), ( 7601/2637 , -23581/5274 )
**u= 60/97 , ht= 2781 ; ( 9579/2983 , -29561/5966 ), ( 2263/2781 , 107/54 )
**u= 61/4 , ht= 329 ; ( -329/325 , 33/325 ), ( 325/329 , 33/329 )
**u= 61/7 , ht= 145 ; ( 131/145 , 153/145 ), ( 4365/53 , 9289/53 )
**u= 61/15 , ht= 421 ; ( 361/421 , -457/421 ), ( -1022591/95749 , -1580091/95749 )
**u= 61/16 , ht= 909 ; ( -737/909 , -2077/1818 ), ( -909/737 , 31/22 )
**u= 61/24 , ht= 137 ; ( -137/104 , 343/416 ), ( 2401/673 , 3228/673 )
**u= 61/28 , ht= 111 ; ( -1/111 , 4/3 ), ( -5291/1471 , -6993/1471 )
**u= 61/29 , ht= 296 ; ( 5667/2506 , 457/2506 ), ( -296/23 , 393/23 )
**u= 61/32 , ht= 6 ; ( -6/5 , 49/40 ), ( -56725/69058 , -554607/552464 )
**u= 61/36 , ht= 23 ; ( -211/3 , 124/9 ), ( 9/23 , 38/69 )
**u= 61/37 , ht= 641 ; ( 641/86 , 219/86 ), ( -4960/13873 , 7689/13873 )
**u= 61/40 , ht= 289 ; ( 137/567 , -752/567 ), ( 129/289 , 218/289 )
**u= 61/52 , ht= 839 ; ( 2453/7055 , -9849/7055 ), ( -409/839 , 894/839 )
**u= 61/53 , ht= 2350 ; ( -13787/682 , -11799/682 ), ( 2350/1661 , -3519/1661 )
**u= 61/56 , ht= 1119 ; ( 1119/902 , 587/328 ), ( -2662/2259 , 16907/9036 )
**u= 61/63 , ht= 487 ; ( -1973/1695 , -9481/5085 ), ( -255/487 , 1861/1461 )
**u= 61/64 , ht= 645 ; ( 629/645 , -2269/1290 ), ( -83/1889 , 7929/7556 )
**u= 61/71 , ht= 83 ; ( -83/59 , 129/59 ), ( 2275/7671 , -9419/7671 )
**u= 61/72 , ht= 881 ; ( 846/881 , 19597/10572 ), ( -167/922 , 4393/3688 )
**u= 61/80 , ht= 9159 ; ( 9159/1609 , 11795/1609 ), ( 23521/65301 , -179579/130602 )
**u= 61/87 , ht= 2287 ; ( 3581/10019 , -16519/10019 ), ( 25/2287 , 3059/2287 )
**u= 61/88 , ht= 96 ; ( 56297/26113 , 86439/26113 ), ( 96/19 , -617/76 )
**u= 61/92 , ht= 8853 ; ( 9393/8795 , -19261/8795 ), ( -4487/8853 , -14267/8853 )
**u= 61/93 , ht= 11101 ; ( -52657/20162 , -80589/20162 ), ( 8590/11101 , -20841/11101 )
**u= 61/95 , ht= 1931 ; ( -4323/6317 , -11987/6317 ), ( -1931/501 , -3227/501 )
**u= 64/3 , ht= 1163 ; ( 1157/1163 , -4649/9304 ), ( 8285/8243 , 32993/65944 )
**u= 64/23 , ht= 799 ; ( 799/661 , -4377/5288 ), ( 1505/723 , -15853/5784 )
**u= 64/27 , ht= 3821 ; ( 3491/3821 , -106823/91704 ), ( 12893/12563 , 351929/301512 )
**u= 64/31 , ht= 863 ; ( 863/429 , 2479/3432 ), ( 429/863 , 2479/6904 )
**u= 64/43 , ht= 233 ; ( 1759/39 , -7237/312 ), ( -25/233 , 993/1864 )
**u= 64/47 , ht= 527 ; ( -527/37 , -2763/296 ), ( -1489/861 , 16519/6888 )
**u= 64/55 , ht= 153 ; ( 43/153 , -1699/1224 ), ( -261/289 , -3455/2312 )
**u= 64/71 , ht= 307 ; ( 30309/13127 , 307667/105016 ), ( -119/307 , 3039/2456 )
**u= 64/75 , ht= 1 ; ( 31/25 , -2057/1000 ), ( -1 , -15/8 )
**u= 64/79 , ht= 3535 ; ( 3061/3535 , 51693/28280 ), ( 72835/33967 , -934329/271736 )
**u= 64/91 , ht= 12959 ; ( -12959/1055 , -138999/8440 ), ( 24393/1409 , 307957/11272 )
**u= 65/14 , ht= 1987 ; ( -28543/25154 , -13443/50308 ), ( 1791/1987 , 770/1987 )
**u= 65/18 , ht= 19 ; ( -17/19 , 20/19 ), ( 101/2 , -301/4 )
**u= 65/34 , ht= 3318 ; ( -7091/2297 , -756/2297 ), ( -1051/3318 , 55/948 )
**u= 65/37 , ht= 33 ; ( -33/4 , -1/4 ), ( -1042/6321 , 949/6321 )
**u= 65/38 , ht= 413 ; ( -413/24 , -137/48 ), ( -24/413 , 137/826 )
**u= 65/54 , ht= 277 ; ( -277/136 , -1735/816 ), ( -4307/4231 , 20306/12693 )
**u= 65/58 , ht= 4411 ; ( -42288/3619 , -75269/7238 ), ( 4411/1047 , -6148/1047 )
**u= 65/61 , ht= 2308 ; ( -8811/2108 , -8749/2108 ), ( 2308/1881 , -3665/1881 )
**u= 65/62 , ht= 541 ; ( 541/417 , 778/417 ), ( -6198/3629 , -18659/7258 )
**u= 65/74 , ht= 8884 ; ( -8884/1587 , 20545/3174 ), ( 14343/8756 , -46429/17512 )
**u= 65/77 , ht= 7001 ; ( -7001/2162 , -8757/2162 ), ( -12052/3271 , 18273/3271 )
**u= 65/93 , ht= 3052 ; ( 1471/3052 , 5209/3052 ), ( 19598/104321 , 143219/104321 )
**u= 73 , ht= 52 ; ( -17/52 , 297/52 ), ( 3254/3251 , 1071/3251 )
**u= 73/8 , ht= 2855 ; ( -1457/2855 , 5403/2855 ), ( 5603/1704 , 46583/6816 )
**u= 73/9 , ht= 806 ; ( 109/1556 , 3197/1556 ), ( -806/211 , -1607/211 )
**u= 73/11 , ht= 381 ; ( 381/359 , 89/359 ), ( -1139/929 , -1389/929 )
**u= 73/20 , ht= 1357 ; ( -723/1357 , -1838/1357 ), ( -1673/1257 , 2006/1257 )
**u= 73/24 , ht= 1649 ; ( 1649/1526 , -5507/6104 ), ( 4033/4063 , -3974/4063 )
**u= 73/27 , ht= 601 ; ( 179/1183 , -4877/3549 ), ( -601/563 , 1981/1689 )
**u= 73/28 , ht= 27 ; ( -1871/1167 , -98/1167 ), ( -27 , -37 )
**u= 73/35 , ht= 4377 ; ( 10727/4637 , -801/4637 ), ( 2417/4377 , 2023/4377 )
**u= 73/41 , ht= 6959 ; ( 6959/1670 , 1587/1670 ), ( -29440/152149 , -19893/152149 )
**u= 73/43 , ht= 9613 ; ( -9613/621 , 1969/621 ), ( 783/9899 , -2113/9899 )
**u= 73/44 , ht= 565 ; ( 53/1285 , -1692/1285 ), ( -565/513 , 757/513 )
**u= 73/48 , ht= 133 ; ( 3251/323 , 1603/323 ), ( -133/59 , 178/59 )
**u= 73/51 , ht= 497 ; ( -497/13 , 287/13 ), ( -21685/38801 , -5209/5543 )
**u= 73/52 , ht= 911 ; ( -911/77 , 51/7 ), ( 77/911 , 561/911 )
**u= 73/56 , ht= 1163 ; ( -11853/1972 , -35123/7888 ), ( -1163/573 , 1613/573 )
**u= 73/57 , ht= 752 ; ( -521/752 , 23/16 ), ( -376/1163 , -987/1163 )
**u= 73/59 , ht= 1953 ; ( -1953/997 , -2021/997 ), ( -2221/1791 , 3307/1791 )
**u= 73/60 , ht= 367 ; ( 127/367 , -508/367 ), ( 367/127 , -4 )
**u= 73/65 , ht= 58 ; ( -33/58 , -85/58 ), ( 98/383 , 363/383 )
**u= 73/67 , ht= 1819 ; ( -1667/4631 , -6597/4631 ), ( -57/1819 , -1661/1819 )
**u= 73/72 , ht= 59 ; ( -59/13 , -61/13 ), ( 427/141 , 1853/423 )
**u= 73/81 , ht= 346 ; ( -221/346 , -559/346 ), ( -1360/2483 , 779/573 )
**u= 73/88 , ht= 183 ; ( -5239/1262 , -25857/5048 ), ( 183/169 , -2 )
**u= 73/96 , ht= 4319 ; ( 4319/470 , 43981/3760 ), ( -6518/1477 , -11623/1688 )
**u= 73/97 , ht= 1201 ; ( 1201/642 , 17/6 ), ( -642/1201 , 1819/1201 )
**u= 73/100 , ht= 595 ; ( 755/237 , 5242/1185 ), ( 579/595 , 848/425 )
**u= 74/3 , ht= 188 ; ( -167/188 , 619/376 ), ( 2413/248 , -16901/496 )
**u= 74/7 , ht= 146 ; ( -99/146 , 7/4 ), ( 146/99 , -511/198 )
**u= 74/11 , ht= 368 ; ( -49/368 , -1383/736 ), ( 3224/1783 , -10449/3566 )
**u= 74/15 , ht= 346 ; ( -562/893 , 2471/1786 ), ( -329/346 , 405/692 )
**u= 74/19 , ht= 732 ; ( -732/655 , -809/1310 ), ( -38827/30276 , 91309/60552 )
**u= 74/35 , ht= 244 ; ( -244/169 , 345/338 ), ( 169/244 , -345/488 )
**u= 74/39 , ht= 23 ; ( 23/10 , -19/20 ), ( 4999/8782 , -11183/17564 )
**u= 74/51 , ht= 616 ; ( 616/115 , -749/230 ), ( 1133/5264 , -939/1504 )
**u= 74/59 , ht= 3272 ; ( -7373/356 , 11163/712 ), ( -681/3272 , -5267/6544 )
**u= 74/63 , ht= 6 ; ( 6 , -31/6 ), ( 3130/1163 , -8763/2326 )
**u= 74/87 , ht= 2434 ; ( 2434/85 , 5637/170 ), ( -56285/31498 , 181789/62996 )
**u= 74/91 , ht= 35292 ; ( 177418092/8151085 , -426210233/16302170 ), ( -33595/35292 , 131633/70584 )
**u= 75 , ht= 1 ; ( -467/469 , -281/469 ), ( -1 , 1/5 )
**u= 75/2 , ht= 4 ; ( -3/4 , -23/8 ), ( 888/625 , -27401/6250 )
**u= 75/4 , ht= 1 ; ( -1 , 2/5 ), ( -1 , -2/5 )
**u= 75/16 , ht= 3239 ; ( -3173/3239 , -5503/6478 ), ( -4327/4261 , -7247/8522 )
**u= 75/23 , ht= 78 ; ( 127/150 , -839/750 ), ( 78/55 , -479/275 )
**u= 75/26 , ht= 74 ; ( 9/74 , 1033/740 ), ( 153/179 , -128/179 )
**u= 75/28 , ht= 263 ; ( 263/235 , 1129/1175 ), ( -527/635 , -2309/3175 )
**u= 75/38 , ht= 51 ; ( -51/25 , -112/125 ), ( -259/330 , -3041/3300 )
**u= 75/43 , ht= 53 ; ( 91/38 , -49/38 ), ( -10/53 , 59/265 )
**u= 75/47 , ht= 35 ; ( -35/12 , -103/60 ), ( -250/267 , -1723/1335 )
**u= 75/49 , ht= 1 ; ( -1 , 7/5 ), ( -1 , -7/5 )
**u= 75/64 , ht= 1 ; ( 1 , -8/5 ), ( 1 , -8/5 )
**u= 75/67 , ht= 125 ; ( -125/76 , -761/380 ), ( 520/2597 , -12017/12985 )
**u= 75/74 , ht= 1331 ; ( 1331/258 , -2725/516 ), ( 17694/4411 , -253169/44110 )
**u= 75/88 , ht= 1277 ; ( 8596/4955 , 246767/99100 ), ( 1277/320 , -38479/6400 )
**u= 75/91 , ht= 1481 ; ( -2777/1318 , 19193/6590 ), ( -794/1481 , 10601/7405 )
**u= 78/5 , ht= 298 ; ( 287/382 , 1439/764 ), ( 298/73 , -1625/146 )
**u= 78/41 , ht= 368 ; ( 368/163 , 311/326 ), ( 976/1919 , -2057/3838 )
**u= 85/8 , ht= 73 ; ( 98/101 , 309/404 ), ( -73/54 , 473/216 )
**u= 85/13 , ht= 1194 ; ( -613/1194 , -1961/1194 ), ( -105948/40289 , -185305/40289 )
**u= 85/21 , ht= 304 ; ( -263/304 , 327/304 ), ( -3506/883 , -5305/883 )
**u= 85/23 , ht= 2339 ; ( -6593/6839 , 6537/6839 ), ( 2339/453 , 3469/453 )
**u= 85/24 , ht= 104 ; ( -43/104 , 583/416 ), ( 944/593 , 4869/2372 )
**u= 85/28 , ht= 1999 ; ( 2297/4089 , 5314/4089 ), ( -1473/1999 , 524/1999 )
**u= 85/29 , ht= 644 ; ( -13913/15464 , 16977/15464 ), ( 644/93 , 901/93 )
**u= 85/31 , ht= 43 ; ( -1577/1089 , 475/1089 ), ( -43/19 , 3 )
**u= 85/37 , ht= 38 ; ( 379/1748 , -2331/1748 ), ( 38 , 51 )
**u= 85/44 , ht= 1439 ; ( -1439/673 , 630/673 ), ( -1669/267 , -2200/267 )
**u= 85/47 , ht= 131 ; ( 131/37 , 33/37 ), ( -389/927 , 445/927 )
**u= 85/52 , ht= 29 ; ( -29/3 , -10/3 ), ( 3/29 , -10/29 )
**u= 85/61 , ht= 104 ; ( -1013/5514 , -7369/5514 ), ( 104/43 , -141/43 )
**u= 85/63 , ht= 311 ; ( -127/2701 , 3611/2701 ), ( 3/311 , -617/933 )
**u= 85/69 , ht= 337 ; ( 337/284 , -465/284 ), ( 592/811 , 1019/811 )
**u= 85/72 , ht= 133 ; ( -29/133 , -183/133 ), ( -133/29 , 183/29 )
**u= 85/77 , ht= 1053 ; ( -1053/514 , 1183/514 ), ( -514/1053 , 91/81 )
**u= 85/87 , ht= 7951 ; ( 5573/7951 , 12667/7951 ), ( 8867/8171 , -15133/8171 )
**u= 85/96 , ht= 67 ; ( -25811/88 , -230993/704 ), ( 64/67 , 959/536 )
**u= 87/2 , ht= 62 ; ( 19/170 , -1577/340 ), ( -62/19 , 29/2 )
**u= 87/4 , ht= 117 ; ( -17/117 , -383/117 ), ( -235/123 , -664/123 )
**u= 87/7 , ht= 130 ; ( 81/130 , -259/130 ), ( 998/291 , -2407/291 )
**u= 87/11 , ht= 304 ; ( 139/304 , 557/304 ), ( 452/463 , 199/463 )
**u= 87/13 , ht= 801 ; ( 2241/2137 , -791/2137 ), ( -801/187 , -1477/187 )
**u= 87/14 , ht= 5 ; ( 2/5 , 17/10 ), ( -2689/629 , -4804/629 )
**u= 87/16 , ht= 761 ; ( -1501/1395 , 569/1395 ), ( -761/171 , 2579/342 )
**u= 87/35 , ht= 547 ; ( 547/512 , 541/512 ), ( 11852/16087 , 9689/16087 )
**u= 87/37 , ht= 75 ; ( 1/75 , -101/75 ), ( -15191/13077 , 18071/13077 )
**u= 87/38 , ht= 791 ; ( -368/791 , -2059/1582 ), ( -695/1148 , 947/2296 )
**u= 87/50 , ht= 491 ; ( -491/34 , -19/68 ), ( 61/863 , -116/4315 )
**u= 87/52 , ht= 2013 ; ( 2013/349 , -674/349 ), ( 1113/2101 , 1553/2101 )
**u= 87/59 , ht= 3343 ; ( -14567/2602 , 8443/2602 ), ( -20/3343 , 1769/3343 )
**u= 87/62 , ht= 4282 ; ( -4282/895 , -5711/1790 ), ( -23311/24677 , -34436/24677 )
**u= 87/64 , ht= 129 ; ( -129/11 , -31/4 ), ( -979/519 , -5401/2076 )
**u= 87/76 , ht= 289 ; ( 289/99 , -283/99 ), ( -6427/1173 , -8876/1173 )
**u= 87/79 , ht= 494 ; ( 3807/2938 , 409/226 ), ( 494/99 , 689/99 )
**u= 87/85 , ht= 1831 ; ( -451/5499 , -7745/5499 ), ( -39/1831 , 1789/1831 )
**u= 87/100 , ht= 495 ; ( 495/353 , -3824/1765 ), ( 511/45 , -3763/225 )
**u= 89/5 , ht= 662 ; ( -383/1452 , 4201/1452 ), ( 662/187 , -1905/187 )
**u= 89/10 , ht= 191 ; ( -181/191 , 168/191 ), ( 13943/2088 , 59303/4176 )
**u= 89/18 , ht= 363 ; ( 238/363 , 2957/2178 ), ( -7303/1238 , -24059/2476 )
**u= 89/21 , ht= 1429 ; ( -1429/1238 , 431/1238 ), ( -2788/719 , 4279/719 )
**u= 89/29 , ht= 18 ; ( -11/18 , 23/18 ), ( -73018/8501 , -103497/8501 )
**u= 89/45 , ht= 31 ; ( 31/14 , 11/14 ), ( -108/23 , -427/69 )
**u= 89/50 , ht= 778 ; ( -778/197 , 393/394 ), ( -147/815 , 388/4075 )
**u= 89/53 , ht= 811 ; ( 811/16 , -189/16 ), ( 1170/11929 , -3163/11929 )
**u= 89/54 , ht= 1121 ; ( 1121/245 , 1382/735 ), ( -20720/14107 , -165587/84642 )
**u= 89/58 , ht= 4332 ; ( 4332/185 , 4003/370 ), ( 20297/11133 , 27292/11133 )
**u= 89/70 , ht= 4339 ; ( -4339/1744 , -7941/3488 ), ( -6572/8513 , -21711/17026 )
**u= 89/77 , ht= 799 ; ( 799/246 , 757/246 ), ( 11800/14341 , -20229/14341 )
**u= 89/78 , ht= 487 ; ( -3151/2114 , -7953/4228 ), ( -85/487 , 436/487 )
**u= 89/85 , ht= 1702 ; ( -427/1702 , -105/74 ), ( -54832/4277 , 10971/611 )
**u= 89/86 , ht= 1388 ; ( -101039/12653 , -99144/12653 ), ( -1388/289 , -3933/578 )
**u= 89/93 , ht= 6320 ; ( -311711/15184 , -325999/15184 ), ( -6320/2081 , 9297/2081 )
**u= 89/94 , ht= 191 ; ( 3/191 , 274/191 ), ( -6406/4827 , -21007/9654 )
**u= 96/25 , ht= 16 ; ( -9/16 , -173/128 ), ( -79/72 , -3223/2880 )
**u= 96/37 , ht= 114 ; ( 717/458 , 1231/3664 ), ( -71/114 , -83/912 )
**u= 96/61 , ht= 3315 ; ( -3499/222 , 11651/1776 ), ( 3315/814 , 35083/6512 )
**u= 96/85 , ht= 2706 ; ( -9309/806 , 65671/6448 ), ( 2706/2621 , -34967/20968 )
**u= 97/8 , ht= 337 ; ( 236/337 , -2439/1348 ), ( -1753/1124 , 13569/4496 )
**u= 97/9 , ht= 78 ; ( -67/78 , 301/234 ), ( -78/67 , -301/201 )
**u= 97/11 , ht= 295 ; ( -1007/1105 , -1137/1105 ), ( 295/9 , -631/9 )
**u= 97/16 , ht= 513 ; ( -257/513 , -824/513 ), ( -1081/599 , 1683/599 )
**u= 97/19 , ht= 4205 ; ( 1621/4205 , -6663/4205 ), ( -4413/2123 , -6727/2123 )
**u= 97/20 , ht= 83 ; ( 17/83 , 135/83 ), ( 299/309 , -2/3 )
**u= 97/28 , ht= 3697 ; ( -3697/3249 , 2258/3249 ), ( -581919/141293 , 840977/141293 )
**u= 97/32 , ht= 15407 ; ( -16812/13195 , 62749/105560 ), ( 15407/2001 , 21752/2001 )
**u= 97/36 , ht= 509 ; ( -323/2547 , -10508/7641 ), ( -345/509 , 443/1527 )
**u= 97/43 , ht= 1081 ; ( 1081/565 , -177/565 ), ( 1469/953 , 1857/953 )
**u= 97/44 , ht= 221 ; ( -221/109 , -24/109 ), ( 4145/7071 , 3088/7071 )
**u= 97/49 , ht= 336 ; ( 311/336 , 2927/2352 ), ( 1434/1529 , -12227/10703 )
**u= 97/57 , ht= 8293 ; ( 8293/884 , -143/68 ), ( -19370/82469 , 29289/82469 )
**u= 97/60 , ht= 269 ; ( -269 , 93 ), ( 497/3907 , -1501/3907 )
**u= 97/65 , ht= 10819 ; ( -10819/1726 , 5961/1726 ), ( 634/13309 , -6825/13309 )
**u= 97/68 , ht= 6883 ; ( 6883/185 , -4011/185 ), ( 215/11469 , 6677/11469 )
**u= 97/75 , ht= 443 ; ( -17/859 , -5773/4295 ), ( 143/443 , -371/443 )
**u= 97/76 , ht= 3409 ; ( -3409/1569 , 3272/1569 ), ( -819/4729 , 3637/4729 )
**u= 97/81 , ht= 1050 ; ( -5819/366 , -42583/3294 ), ( -1050/499 , 13361/4491 )
**u= 97/88 , ht= 426 ; ( -4141/2250 , 19417/9000 ), ( -426/157 , -2423/628 )
**u= 97/89 , ht= 199 ; ( 3867/3778 , 6313/3778 ), ( 110/199 , -237/199 )
**u= 97/91 , ht= 1531 ; ( -56999/10341 , -55207/10341 ), ( -1531/1381 , -2493/1381 )
**u= 97/99 , ht= 261 ; ( -1233/1033 , 5801/3099 ), ( 19/261 , 803/783 )
**u= 98/11 , ht= 13 ; ( -302/765 , 21299/10710 ), ( 13/2 , 387/28 )
**u= 98/19 , ht= 35 ; ( -22/35 , 687/490 ), ( 2522/33 , -59809/462 )
**u= 98/23 , ht= 152 ; ( -76/467 , -10203/6538 ), ( -129/152 , -11/112 )
**u= 98/47 , ht= 104 ; ( 104/57 , 661/798 ), ( -14024/217 , 260607/3038 )
**u= 98/51 , ht= 377 ; ( 377/122 , 247/1708 ), ( 571/806 , -723/868 )
**u= 98/55 , ht= 147 ; ( 147/92 , 71/56 ), ( -368/2173 , 759/30422 )
**u= 98/59 , ht= 374 ; ( -374/39 , 1583/546 ), ( -42/1433 , 5459/20062 )
**u= 98/67 , ht= 1562 ; ( 3442/829 , -30327/11606 ), ( 1495/1562 , -30171/21868 )
**u= 98/71 , ht= 239 ; ( -239/116 , -3021/1624 ), ( -116/239 , 3021/3346 )
**u= 98/95 , ht= 743 ; ( -499/1116 , -22949/15624 ), ( 743/268 , 15045/3752 )
**u= 98/99 , ht= 1222 ; ( 1861/1146 , -104359/48132 ), ( 155/1222 , -17553/17108 )
**u= 100/3 , ht= 19949 ; ( 19907/19949 , -15955/39898 ), ( 20093/20051 , 16045/40102 )
**u= 100/27 , ht= 15629 ; ( 15299/15629 , -437447/468870 ), ( 24701/24371 , 682553/731130 )
**u= 100/39 , ht= 241 ; ( -415/469 , -5391/4690 ), ( 163/241 , 191/482 )
**u= 100/51 , ht= 299 ; ( 245/299 , 291/230 ), ( -4303/4775 , -52351/47750 )
**u= 100/59 , ht= 205 ; ( -205/149 , 2001/1490 ), ( -149/205 , -2001/2050 )
**u= 100/91 , ht= 775 ; ( 4001/543 , -7381/1086 ), ( 47/775 , 7029/7750 )
639
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■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。

[2026.05.18追記]u=2/51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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