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

[2026.05.18]A^4+4*B^4=C^4+4*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=4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675個のuが抽出される。
[MAGMAによる計算]
> PP(4,1,100);
**u= 1 , ht= 1 ; ( -1 , -2 ), ( -1 , 2 )
**u= 1/4 , ht= 1 ; ( -1/3 , -3 ), ( 1 , 4 )
**u= 1/5 , ht= 7 ; ( -7/3 , 8 ), ( -111/31 , -366/31 )
**u= 1/9 , ht= 1 ; ( 5/23 , -100/23 ), ( 1 , -6 )
**u= 1/12 , ht= 215 ; ( -71/215 , 222/43 ), ( -16715/3461 , 83690/3461 )
**u= 1/13 , ht= 21 ; ( -31/5 , -32 ), ( -5/21 , 110/21 )
**u= 1/16 , ht= 15 ; ( -15 , -85 ), ( 1/15 , -17/3 )
**u= 1/21 , ht= 71 ; ( -71/55 , -582/55 ), ( 23/121 , 798/121 )
**u= 1/24 , ht= 43 ; ( -79/49 , -92/7 ), ( -14/43 , 1253/172 )
**u= 1/25 , ht= 1 ; ( -1 , -10 ), ( -49/3 , 1736/15 )
**u= 1/36 , ht= 99 ; ( 13/99 , -2542/297 ), ( 15/101 , 2599/303 )
**u= 1/37 , ht= 1055 ; ( 219/1255 , 2192/251 ), ( -1055/241 , 9310/241 )
**u= 1/40 , ht= 9 ; ( 467/333 , 570/37 ), ( 9 , 81 )
**u= 1/41 , ht= 73 ; ( -73/9 , 74 ), ( -1197/115 , -2178/23 )
**u= 1/49 , ht= 45 ; ( -83/109 , -9494/763 ), ( -45/13 , -3246/91 )
**u= 1/60 , ht= 2123 ; ( 1093/2123 , -2378/193 ), ( 30137/2617 , 331388/2617 )
**u= 1/61 , ht= 1373 ; ( 1373/275 , -1406/25 ), ( -27313/717 , 301796/717 )
**u= 1/64 , ht= 143 ; ( 143/109 , -8137/436 ), ( 145/111 , 2066/111 )
**u= 1/69 , ht= 53 ; ( 53/7 , -628/7 ), ( 1993/1211 , 27396/1211 )
**u= 1/76 , ht= 211 ; ( 131/211 , -3062/211 ), ( -963/13 , -11874/13 )
**u= 1/81 , ht= 165 ; ( -1133/159 , -131056/1431 ), ( 165/161 , 26408/1449 )
**u= 1/84 , ht= 13 ; ( -1415/377 , -1460/29 ), ( -13 , -169 )
**u= 1/85 , ht= 4927 ; ( 4967/1833 , -1770/47 ), ( -4927/207 , 64298/207 )
**u= 1/88 , ht= 485 ; ( -307/485 , -1523/97 ), ( 5510/6907 , -468865/27628 )
**u= 1/96 , ht= 86 ; ( 11/86 , -9611/688 ), ( 262/121 , 31991/968 )
**u= 1/100 , ht= 219 ; ( -219/181 , 4018/181 ), ( -221/179 , -4022/179 )
**u= 2/3 , ht= 7 ; ( -29/38 , 42/19 ), ( 2/7 , -12/7 )
**u= 2/5 , ht= 38 ; ( -79/14 , -88/7 ), ( -38/7 , -88/7 )
**u= 2/15 , ht= 4 ; ( 1/4 , -4 ), ( -14044/3389 , -56064/3389 )
**u= 2/17 , ht= 36 ; ( 19/36 , -14/3 ), ( 44/7 , 184/7 )
**u= 2/33 , ht= 160 ; ( -61/160 , 123/20 ), ( -160/61 , 984/61 )
**u= 2/35 , ht= 6 ; ( -257/2378 , 7078/1189 ), ( -6 , -36 )
**u= 2/37 , ht= 31 ; ( -31/6 , 32 ), ( 1146/223 , -7104/223 )
**u= 2/47 , ht= 161 ; ( 161/20 , -278/5 ), ( 2236/1813 , -19736/1813 )
**u= 2/63 , ht= 8 ; ( -1/8 , -8 ), ( -8 , 64 )
**u= 2/65 , ht= 57 ; ( -57/8 , -58 ), ( -4616/521 , -37456/521 )
**u= 2/95 , ht= 341 ; ( -451/584 , -899/73 ), ( -56/341 , 3368/341 )
**u= 2/99 , ht= 169 ; ( 247/490 , 78/7 ), ( -70/169 , 140/13 )
**u= 4 , ht= 1 ; ( 1 , 1 ), ( -7/9 , -1/9 )
**u= 4/9 , ht= 1 ; ( 7/9 , -73/27 ), ( -1 , -3 )
**u= 4/25 , ht= 1 ; ( -43/93 , -121/31 ), ( 1 , 5 )
**u= 4/49 , ht= 1 ; ( -11/19 , -761/133 ), ( -1 , -7 )
**u= 4/65 , ht= 53 ; ( 911/2601 , 5239/867 ), ( 27/53 , 339/53 )
**u= 4/73 , ht= 295 ; ( -441/295 , -641/59 ), ( -295/289 , 2495/289 )
**u= 4/81 , ht= 13439 ; ( -13439/13113 , -1075429/118017 ), ( -13455/13129 , -1076741/118161 )
**u= 5/2 , ht= 2 ; ( 2 , 1/2 ), ( -27/26 , 69/52 )
**u= 5/3 , ht= 23 ; ( -137/31 , -94/31 ), ( -1/23 , -14/23 )
**u= 5/6 , ht= 29 ; ( -13/29 , 52/29 ), ( -29/13 , 4 )
**u= 5/7 , ht= 37 ; ( -139/153 , 116/51 ), ( -9/37 , 60/37 )
**u= 5/11 , ht= 57 ; ( -31/57 , 46/19 ), ( 79/13 , 172/13 )
**u= 5/14 , ht= 59 ; ( -143/74 , 755/148 ), ( 59/3 , 142/3 )
**u= 5/18 , ht= 57 ; ( 143 , -380 ), ( 57/14 , 953/84 )
**u= 5/23 , ht= 103 ; ( -81/103 , 398/103 ), ( -139/2207 , 6668/2207 )
**u= 5/26 , ht= 36 ; ( -7219/3077 , -25226/3077 ), ( 23/36 , 275/72 )
**u= 5/27 , ht= 721 ; ( -467/721 , 404/103 ), ( 1201/13 , -3964/13 )
**u= 5/34 , ht= 178 ; ( 178/9 , 437/6 ), ( -1509/922 , -13059/1844 )
**u= 5/38 , ht= 136 ; ( 59/192 , -523/128 ), ( 136/3 , 1063/6 )
**u= 5/39 , ht= 5761 ; ( -5761/713 , -22882/713 ), ( 9637/6959 , -46980/6959 )
**u= 5/42 , ht= 52 ; ( -52/11 , 435/22 ), ( 67/109 , 524/109 )
**u= 5/47 , ht= 227 ; ( 1481/2703 , -4458/901 ), ( 227/39 , 1000/39 )
**u= 5/51 , ht= 1129 ; ( -199/2317 , -10516/2317 ), ( 1129/143 , -5146/143 )
**u= 5/54 , ht= 757 ; ( -9613/10904 , -405409/65424 ), ( 379/757 , 3932/757 )
**u= 5/59 , ht= 77 ; ( 409/173 , -2156/173 ), ( -77/41 , -424/41 )
**u= 5/62 , ht= 1443 ; ( -2627/3046 , -40067/6092 ), ( -386/1443 , -14867/2886 )
**u= 5/63 , ht= 963 ; ( 1119/4213 , 5972/1149 ), ( 319/963 , 15268/2889 )
**u= 5/66 , ht= 2407 ; ( -3434/3727 , 52081/7454 ), ( 2407/394 , 25081/788 )
**u= 5/71 , ht= 779 ; ( -779/637 , -766/91 ), ( 7/4821 , -25676/4821 )
**u= 5/79 , ht= 399 ; ( 233/399 , 866/133 ), ( -579/263 , 3576/263 )
**u= 5/83 , ht= 413 ; ( -413/81 , 808/27 ), ( 24597/10819 , 154878/10819 )
**u= 5/86 , ht= 101 ; ( -101/28 , -1229/56 ), ( -19216/29503 , 412943/59006 )
**u= 5/87 , ht= 1937 ; ( -3527/1729 , -1782/133 ), ( 1937/151 , -11466/151 )
**u= 5/98 , ht= 91 ; ( 218/61 , 19837/854 ), ( -91/43 , -4412/301 )
**u= 8/5 , ht= 3 ; ( -2/3 , -3/2 ), ( 3/2 , -9/4 )
**u= 8/13 , ht= 7 ; ( -7/6 , 11/4 ), ( 6/19 , 69/38 )
**u= 8/21 , ht= 290 ; ( -17/290 , 271/116 ), ( -2410/793 , -11795/1586 )
**u= 8/29 , ht= 19 ; ( 26/235 , 257/94 ), ( 10/19 , -115/38 )
**u= 8/45 , ht= 134 ; ( 134/91 , 155/26 ), ( -441/94 , 9107/564 )
**u= 8/53 , ht= 35 ; ( -18/35 , -41/10 ), ( 14/39 , -301/78 )
**u= 8/77 , ht= 494 ; ( -494/45 , -483/10 ), ( -84501/20758 , 764451/41516 )
**u= 8/85 , ht= 3022 ; ( 3022/1143 , 3307/254 ), ( -3411/1366 , -33903/2732 )
**u= 8/93 , ht= 4390 ; ( 21629/3494 , -211101/6988 ), ( -4390/1097 , -43673/2194 )
**u= 9 , ht= 5 ; ( -1/5 , -32/15 ), ( 19/13 , 94/39 )
**u= 9/4 , ht= 1 ; ( -1 , 4/3 ), ( -1 , 4/3 )
**u= 9/5 , ht= 7 ; ( -7/3 , -16/9 ), ( 7/117 , -164/351 )
**u= 9/8 , ht= 5 ; ( -3/5 , -5/3 ), ( -5/3 , -25/9 )
**u= 9/16 , ht= 1 ; ( -31/29 , -479/174 ), ( -1 , -8/3 )
**u= 9/17 , ht= 19 ; ( -49/15 , -58/9 ), ( -15/19 , -140/57 )
**u= 9/20 , ht= 41 ; ( 41/21 , -41/9 ), ( -511/369 , 98/27 )
**u= 9/25 , ht= 1 ; ( -143/79 , 5726/1185 ), ( 1 , -10/3 )
**u= 9/29 , ht= 207 ; ( 207/91 , 244/39 ), ( -315/149 , 2674/447 )
**u= 9/40 , ht= 26 ; ( 26 , 925/12 ), ( -263/167 , -2794/501 )
**u= 9/44 , ht= 227 ; ( -227/15 , 2123/45 ), ( 279/235 , -685/141 )
**u= 9/49 , ht= 131 ; ( 371/115 , 26822/2415 ), ( 65/131 , 1444/393 )
**u= 9/56 , ht= 65 ; ( 345/89 , 3760/267 ), ( 33/65 , -154/39 )
**u= 9/61 , ht= 79 ; ( -79/43 , 992/129 ), ( -413/563 , -7706/1689 )
**u= 9/64 , ht= 1 ; ( -31/113 , 2657/678 ), ( -1 , 16/3 )
**u= 9/65 , ht= 2761 ; ( 100461/33121 , 109432/9033 ), ( 2761/411 , 31900/1233 )
**u= 9/80 , ht= 63 ; ( -119/3861 , 7529/1782 ), ( -13/63 , -1625/378 )
**u= 9/85 , ht= 1501 ; ( 311/1501 , 20012/4503 ), ( -1429/1631 , -28268/4893 )
**u= 9/89 , ht= 377 ; ( -3213/2483 , -4166/573 ), ( -377/21 , 5044/63 )
**u= 9/92 , ht= 1125 ; ( 623/1125 , -3491/675 ), ( 2085/1871 , -38005/5613 )
**u= 9/100 , ht= 1 ; ( -1 , 20/3 ), ( -1 , 20/3 )
**u= 10 , ht= 62 ; ( -63/94 , 82/47 ), ( -62/49 , 92/49 )
**u= 10/3 , ht= 11 ; ( 11/8 , -1/2 ), ( 16/13 , -20/13 )
**u= 10/7 , ht= 19 ; ( -19/2 , 8 ), ( 22 , 32 )
**u= 10/13 , ht= 23 ; ( 67/24 , -9/2 ), ( 16/23 , -44/23 )
**u= 10/19 , ht= 23 ; ( 23/4 , 11 ), ( -44/13 , -92/13 )
**u= 10/33 , ht= 139 ; ( -139/62 , 194/31 ), ( 274/67 , -732/67 )
**u= 10/39 , ht= 53 ; ( -53/14 , -76/7 ), ( -74/43 , -240/43 )
**u= 10/61 , ht= 92 ; ( 91/92 , 113/23 ), ( -92/31 , 340/31 )
**u= 10/67 , ht= 43 ; ( 4933/3576 , 1857/298 ), ( 24/43 , -180/43 )
**u= 10/77 , ht= 53 ; ( -53/24 , -19/2 ), ( -11208 , 44076 )
**u= 10/97 , ht= 453 ; ( 453/226 , 1114/113 ), ( -30502/2769 , 135076/2769 )
**u= 13/2 , ht= 3 ; ( -62/171 , 203/114 ), ( -3/2 , -9/4 )
**u= 13/3 , ht= 5 ; ( 181/307 , -438/307 ), ( -5 , 8 )
**u= 13/6 , ht= 43 ; ( 43/20 , -9/8 ), ( -25/73 , 20/73 )
**u= 13/7 , ht= 11 ; ( 11/3 , 2 ), ( -3/11 , -6/11 )
**u= 13/10 , ht= 29 ; ( 24/29 , -97/58 ), ( -61/59 , 106/59 )
**u= 13/15 , ht= 181 ; ( -5953/1097 , 8350/1097 ), ( -181/161 , 372/161 )
**u= 13/18 , ht= 9 ; ( -9/7 , 8/3 ), ( 679/366 , -7973/2196 )
**u= 13/19 , ht= 385 ; ( -4269/1925 , -1538/385 ), ( 385/233 , 790/233 )
**u= 13/22 , ht= 7 ; ( 21/131 , -254/131 ), ( -4/7 , 29/14 )
**u= 13/27 , ht= 2477 ; ( -2477/901 , -15754/2703 ), ( 5779/971 , 12266/971 )
**u= 13/30 , ht= 907 ; ( 907/37 , 1904/37 ), ( -401/959 , -2196/959 )
**u= 13/31 , ht= 193 ; ( 327/1505 , 98/43 ), ( -55/193 , -430/193 )
**u= 13/38 , ht= 1347 ; ( -2387/12 , 3791/8 ), ( 1347/781 , 3792/781 )
**u= 13/42 , ht= 68 ; ( 136/745 , -3893/1490 ), ( -37/68 , 23/8 )
**u= 13/43 , ht= 527 ; ( -269/527 , -1532/527 ), ( 1233/1433 , 4854/1433 )
**u= 13/47 , ht= 711 ; ( 241/711 , 226/79 ), ( 261/1201 , -3276/1201 )
**u= 13/55 , ht= 719 ; ( 719/59 , 2084/59 ), ( 17579/639 , 51524/639 )
**u= 13/58 , ht= 53 ; ( 5/53 , -160/53 ), ( -5/92 , -547/184 )
**u= 13/63 , ht= 10429 ; ( -10429/5445 , -109546/16335 ), ( 37477/132733 , -427434/132733 )
**u= 13/67 , ht= 77 ; ( -815/1463 , 70/19 ), ( -77/57 , 308/57 )
**u= 13/70 , ht= 659 ; ( -2699/584 , 18053/1168 ), ( -344/659 , 4867/1318 )
**u= 13/75 , ht= 1511 ; ( -1511/67 , -25592/335 ), ( 3083/829 , -54398/4145 )
**u= 13/82 , ht= 35 ; ( -6/35 , 253/70 ), ( 35/6 , 253/12 )
**u= 13/83 , ht= 3695 ; ( -14785/1671 , -17670/557 ), ( -873/3695 , -2706/739 )
**u= 13/87 , ht= 5245 ; ( 58807/3823 , 215002/3823 ), ( 5245/199 , -19256/199 )
**u= 13/90 , ht= 2096 ; ( 2096/951 , -51299/5706 ), ( 8829/1024 , 198943/6144 )
**u= 13/94 , ht= 2271 ; ( 2271/1865 , 2234/373 ), ( 5155/1707 , 20690/1707 )
**u= 16 , ht= 1 ; ( 1 , 1/2 ), ( -9 , -51/2 )
**u= 16/9 , ht= 17 ; ( 1/33 , 281/198 ), ( -15/17 , 137/102 )
**u= 16/17 , ht= 49 ; ( -49/15 , 9/2 ), ( -135 , 435/2 )
**u= 16/25 , ht= 1 ; ( -1 , 5/2 ), ( 1/33 , 553/330 )
**u= 16/33 , ht= 35 ; ( 13/35 , -31/14 ), ( -8875/2917 , -38835/5834 )
**u= 16/49 , ht= 1 ; ( -7/17 , 643/238 ), ( -1 , -7/2 )
**u= 16/81 , ht= 1 ; ( 1 , 9/2 ), ( -1 , 9/2 )
**u= 16/89 , ht= 3807 ; ( 5959/1161 , 4483/258 ), ( 3807/247 , -25551/494 )
**u= 17/4 , ht= 75 ; ( -131/105 , -1/7 ), ( 75/11 , 120/11 )
**u= 17/5 , ht= 27 ; ( -47/33 , 2/11 ), ( 23/27 , -20/27 )
**u= 17/8 , ht= 49 ; ( 49/25 , 31/25 ), ( 19/77 , 2/77 )
**u= 17/12 , ht= 121 ; ( 197/11 , 166/11 ), ( -121/59 , -183/59 )
**u= 17/13 , ht= 339 ; ( 1133/483 , 426/161 ), ( 339/157 , 522/157 )
**u= 17/20 , ht= 229 ; ( -727/953 , 1882/953 ), ( -229/219 , -488/219 )
**u= 17/21 , ht= 155 ; ( -127/155 , -64/31 ), ( -1135/1121 , -2510/1121 )
**u= 17/29 , ht= 2087 ; ( 1939/3455 , -1496/691 ), ( -1565/2087 , -4760/2087 )
**u= 17/32 , ht= 28 ; ( -1144/513 , 2111/456 ), ( 27/28 , -603/224 )
**u= 17/33 , ht= 749 ; ( 659/749 , -282/107 ), ( -1211 , 2464 )
**u= 17/40 , ht= 99 ; ( 29/99 , 76/33 ), ( 99/29 , -228/29 )
**u= 17/45 , ht= 47 ; ( -18107/40213 , -102666/40213 ), ( -33/47 , 394/141 )
**u= 17/48 , ht= 13 ; ( 404591/30347 , 1898173/60694 ), ( 1/13 , 61/26 )
**u= 17/53 , ht= 61 ; ( -8005/7369 , 27140/7369 ), ( -61/45 , 38/9 )
**u= 17/57 , ht= 2851 ; ( -4679/2617 , 13802/2617 ), ( 2851/343 , 7516/343 )
**u= 17/60 , ht= 12899 ; ( -12899 , 33925 ), ( 16873/763 , -45321/763 )
**u= 17/65 , ht= 1233 ; ( 1233/427 , -512/61 ), ( 5579/3599 , -18424/3599 )
**u= 17/77 , ht= 3089 ; ( 15739/1593 , -15776/531 ), ( 1153/3089 , -9878/3089 )
**u= 17/92 , ht= 5383 ; ( 11583/6385 , 43414/6385 ), ( 4141/5383 , 22319/5383 )
**u= 17/96 , ht= 290 ; ( -24001/1270 , 128731/2032 ), ( 290/157 , -8885/1256 )
**u= 18/7 , ht= 4 ; ( -3/4 , -4/3 ), ( 156/149 , 592/447 )
**u= 18/13 , ht= 15 ; ( 15/2 , -20/3 ), ( 2/15 , -8/9 )
**u= 18/23 , ht= 16 ; ( -7/16 , 11/6 ), ( 64/41 , -376/123 )
**u= 18/25 , ht= 20 ; ( -3/20 , -134/75 ), ( 116/45 , -3256/675 )
**u= 18/41 , ht= 361 ; ( -361/320 , 77/24 ), ( 800/553 , 6280/1659 )
**u= 18/55 , ht= 8 ; ( 3/8 , -8/3 ), ( 8/3 , 64/9 )
**u= 18/59 , ht= 490 ; ( -727/550 , -698/165 ), ( 490/41 , 3820/123 )
**u= 18/73 , ht= 1893 ; ( 1893/1528 , -2594/573 ), ( -9128/873 , 78928/2619 )
**u= 18/77 , ht= 559 ; ( 559/398 , -3004/597 ), ( 6250/2099 , 58168/6297 )
**u= 18/91 , ht= 5610 ; ( -59/5610 , 5378/1683 ), ( 28590/9721 , 289180/29163 )
**u= 20 , ht= 333 ; ( -49/333 , -349/111 ), ( 801/197 , 2469/197 )
**u= 20/9 , ht= 3 ; ( -299/97 , 7/97 ), ( 1/3 , 1/9 )
**u= 20/17 , ht= 37 ; ( 37/31 , 61/31 ), ( 537/469 , 951/469 )
**u= 20/33 , ht= 47 ; ( 989/593 , 2049/593 ), ( -19/47 , -89/47 )
**u= 20/41 , ht= 83 ; ( -81/83 , -235/83 ), ( 1009/1173 , -3119/1173 )
**u= 20/57 , ht= 481 ; ( -481/203 , 1233/203 ), ( 8861/3047 , 22641/3047 )
**u= 20/73 , ht= 339 ; ( 47/339 , -311/113 ), ( 633741/84287 , -1742925/84287 )
**u= 20/89 , ht= 249 ; ( 249/107 , 805/107 ), ( 467/289 , -1643/289 )
**u= 25 , ht= 1 ; ( -1 , 2/5 ), ( -87/59 , -228/59 )
**u= 25/4 , ht= 1 ; ( 1 , -4/5 ), ( -1 , 4/5 )
**u= 25/8 , ht= 4 ; ( -95/63 , 34/105 ), ( 3/4 , 9/16 )
**u= 25/9 , ht= 1 ; ( -19/33 , -136/99 ), ( -1 , -6/5 )
**u= 25/12 , ht= 7 ; ( 193/49 , 6/7 ), ( -7/5 , -49/25 )
**u= 25/13 , ht= 31 ; ( 31/5 , 56/25 ), ( 479/1597 , -812/1597 )
**u= 25/16 , ht= 1 ; ( 249/107 , 467/214 ), ( -1 , -8/5 )
**u= 25/17 , ht= 19 ; ( 47/21 , 16/7 ), ( -15/19 , 132/95 )
**u= 25/21 , ht= 433 ; ( 307/433 , 3636/2165 ), ( 463/127 , 3554/635 )
**u= 25/24 , ht= 7 ; ( -7/2 , -177/40 ), ( 49/209 , 1294/1045 )
**u= 25/28 , ht= 9 ; ( -5/9 , 9/5 ), ( -9/5 , -81/25 )
**u= 25/36 , ht= 1 ; ( -1 , 12/5 ), ( -3/97 , -2317/1455 )
**u= 25/37 , ht= 39 ; ( -1915/1323 , 934/315 ), ( 35/39 , 448/195 )
**u= 25/44 , ht= 163 ; ( 163/141 , 134/47 ), ( -453/97 , 4503/485 )
**u= 25/48 , ht= 451 ; ( -295/451 , -973/410 ), ( 1529/197 , 6237/394 )
**u= 25/49 , ht= 1 ; ( 1/31 , 2218/1085 ), ( 1 , 14/5 )
**u= 25/52 , ht= 3973 ; ( -3973/161 , -7873/161 ), ( 4853/329 , -10201/329 )
**u= 25/53 , ht= 61 ; ( -151/45 , 176/25 ), ( 45/61 , -774/305 )
**u= 25/57 , ht= 2795 ; ( 2795/2567 , -40436/12835 ), ( -41213/9859 , 92410/9859 )
**u= 25/61 , ht= 217 ; ( -1179/203 , 370/29 ), ( -217/27 , 2464/135 )
**u= 25/64 , ht= 297 ; ( -397/477 , -4697/1590 ), ( -297/29 , -2751/116 )
**u= 25/69 , ht= 1223 ; ( -2569/2951 , -46066/14755 ), ( -1223/203 , 14788/1015 )
**u= 25/72 , ht= 13 ; ( 5/13 , -13/5 ), ( 65/191 , 2392/955 )
**u= 25/73 , ht= 215 ; ( 2567/231 , -186/7 ), ( 77/215 , 2728/1075 )
**u= 25/76 , ht= 3789 ; ( 3789/179 , 46132/895 ), ( 6217/137 , 77696/685 )
**u= 25/81 , ht= 269 ; ( 1049/243 , -24404/2187 ), ( 111/269 , -33052/12105 )
**u= 25/88 , ht= 69 ; ( 69/58 , 955/232 ), ( 1907/499 , -5276/499 )
**u= 25/93 , ht= 1727 ; ( 1727/239 , 4714/239 ), ( -18959/1315 , 261496/6575 )
**u= 25/97 , ht= 129 ; ( -129/65 , 154/25 ), ( 22399/1075 , 314912/5375 )
**u= 26/5 , ht= 41 ; ( -41/36 , -1/3 ), ( -324/281 , 372/281 )
**u= 26/9 , ht= 3 ; ( -135/118 , -190/177 ), ( -2/3 , -4/9 )
**u= 26/11 , ht= 7 ; ( 9/20 , -7/5 ), ( -4/7 , 4/7 )
**u= 26/21 , ht= 1016 ; ( -811/1520 , 603/380 ), ( -1016/517 , 1604/517 )
**u= 26/31 , ht= 42 ; ( 11/42 , 12/7 ), ( -54/23 , 96/23 )
**u= 26/41 , ht= 35 ; ( -659/202 , -586/101 ), ( 6/35 , 12/7 )
**u= 26/59 , ht= 64 ; ( -183/112 , -113/28 ), ( 64/5 , -28 )
**u= 26/63 , ht= 1207 ; ( 647/3070 , -704/307 ), ( -10/1207 , 2600/1207 )
**u= 26/69 , ht= 43 ; ( -1301/148 , -741/37 ), ( -20/43 , -108/43 )
**u= 26/75 , ht= 20 ; ( 17/20 , -79/25 ), ( -148/155 , -2572/775 )
**u= 26/85 , ht= 1848 ; ( -3989/504 , -121/6 ), ( 1848/107 , -4788/107 )
**u= 26/95 , ht= 454 ; ( -21/454 , -620/227 ), ( -454/21 , -1240/21 )
**u= 29/2 , ht= 5 ; ( -2/5 , 5/2 ), ( -5485/2069 , -13850/2069 )
**u= 29/6 , ht= 55 ; ( -151/217 , -296/217 ), ( 55/7 , -92/7 )
**u= 29/10 , ht= 147 ; ( -208/147 , 11/14 ), ( 147/53 , -210/53 )
**u= 29/11 , ht= 25 ; ( -25/19 , 20/19 ), ( 251/141 , 346/141 )
**u= 29/14 , ht= 5 ; ( 5/2 , 5/4 ), ( 563/1565 , -136/313 )
**u= 29/15 , ht= 121 ; ( 3337/817 , 1446/817 ), ( 31/121 , 54/121 )
**u= 29/19 , ht= 4383 ; ( -20297/1791 , 5078/597 ), ( -4383/1837 , 6456/1837 )
**u= 29/23 , ht= 875 ; ( 781/875 , 302/175 ), ( 1525/13747 , 13630/13747 )
**u= 29/26 , ht= 183 ; ( 11/219 , -112/73 ), ( 183/79 , -294/79 )
**u= 29/27 , ht= 901 ; ( 901/295 , 680/177 ), ( 905/2363 , -180/139 )
**u= 29/30 , ht= 46 ; ( 59/46 , 209/92 ), ( 46 , 147/2 )
**u= 29/35 , ht= 643 ; ( 47281/2391 , -22326/797 ), ( -643/27 , -1082/27 )
**u= 29/39 , ht= 50911 ; ( 28025/50911 , -14110/7273 ), ( -84679/23995 , -30520/4799 )
**u= 29/46 , ht= 66 ; ( 49/66 , -9/4 ), ( 498/31 , 1863/62 )
**u= 29/50 , ht= 505 ; ( -505/463 , 6338/2315 ), ( 505 , 4882/5 )
**u= 29/51 , ht= 1279 ; ( 1279/1211 , -3296/1211 ), ( 1285/211 , -2534/211 )
**u= 29/54 , ht= 271 ; ( -271/4 , 3023/24 ), ( 1423/991 , 3388/991 )
**u= 29/55 , ht= 2653 ; ( -134413/58923 , -31002/6547 ), ( -603/2653 , 5130/2653 )
**u= 29/59 , ht= 507 ; ( 507/35 , 142/5 ), ( 3521/1161 , -7658/1161 )
**u= 29/71 , ht= 149 ; ( -149/135 , -148/45 ), ( -135/149 , -444/149 )
**u= 29/74 , ht= 1932 ; ( -646277/67453 , 1440086/67453 ), ( -103/1932 , -8573/3864 )
**u= 29/75 , ht= 781 ; ( -169/781 , -168/71 ), ( 3113/1135 , 38214/5675 )
**u= 29/90 , ht= 1019 ; ( 309/1019 , 8044/3057 ), ( -1019/309 , -8044/927 )
**u= 29/91 , ht= 185217 ; ( -1404107/217929 , -1172104/72643 ), ( 185217/8599 , -470334/8599 )
**u= 29/95 , ht= 1469 ; ( -1469/431 , -3880/431 ), ( 7331/681 , -19060/681 )
**u= 29/98 , ht= 155 ; ( 54/155 , 1205/434 ), ( -135/257 , -750/257 )
**u= 29/99 , ht= 1053 ; ( 443/1053 , -694/243 ), ( 1053/311 , 8684/933 )
**u= 32/9 , ht= 5 ; ( -4/5 , 5/4 ), ( 55/48 , -785/576 )
**u= 32/17 , ht= 21 ; ( 21/4 , 37/16 ), ( 56/141 , -373/564 )
**u= 32/25 , ht= 37 ; ( -37/12 , -53/16 ), ( -93/100 , -3363/2000 )
**u= 32/33 , ht= 7 ; ( 4/7 , 7/4 ), ( -1099/736 , 7931/2944 )
**u= 32/41 , ht= 1103 ; ( -1103/160 , 1319/128 ), ( 1760/1227 , -14095/4908 )
**u= 32/57 , ht= 20 ; ( 1/20 , 157/80 ), ( -1421/4 , -11143/16 )
**u= 32/65 , ht= 9 ; ( -37/28 , -53/16 ), ( -9/4 , 81/16 )
**u= 32/97 , ht= 12844 ; ( 12844/2385 , 14113/1060 ), ( 15921/1468 , 159561/5872 )
**u= 34 , ht= 4 ; ( -49/60 , 12/5 ), ( 4 , 16 )
**u= 34/15 , ht= 317 ; ( 317/112 , 1/14 ), ( -448/503 , 584/503 )
**u= 34/19 , ht= 337 ; ( 337/90 , 34/15 ), ( 378/815 , 132/163 )
**u= 34/21 , ht= 314 ; ( -8801/2462 , 3368/1231 ), ( 314/223 , 472/223 )
**u= 34/31 , ht= 35 ; ( 35/4 , -10 ), ( 4/35 , -8/7 )
**u= 34/33 , ht= 1061 ; ( -3683/860 , -230/43 ), ( -380/1061 , -1400/1061 )
**u= 34/49 , ht= 881 ; ( 881/736 , 241/92 ), ( -2576/1749 , 37816/12243 )
**u= 34/79 , ht= 232 ; ( -163/232 , 77/29 ), ( 2008/1929 , -6008/1929 )
**u= 34/81 , ht= 65 ; ( -65/56 , -10/3 ), ( 56/65 , 112/39 )
**u= 36 , ht= 1 ; ( 1 , 1/3 ), ( -41/25 , 83/15 )
**u= 36/25 , ht= 1 ; ( 47 , -577/15 ), ( -1 , -5/3 )
**u= 36/41 , ht= 49 ; ( 141/469 , -2389/1407 ), ( -49/33 , -277/99 )
**u= 36/49 , ht= 37 ; ( -11/37 , -1411/777 ), ( 1/65 , 419/273 )
**u= 36/65 , ht= 279 ; ( 279/71 , 1585/213 ), ( 1589/549 , 9869/1647 )
**u= 36/89 , ht= 249 ; ( 12583/9735 , 105581/29205 ), ( 249/71 , -1757/213 )
**u= 37/3 , ht= 61 ; ( -41/61 , 116/61 ), ( 61/41 , 116/41 )
**u= 37/6 , ht= 311 ; ( -311/280 , 1/16 ), ( -805/131 , 1470/131 )
**u= 37/7 , ht= 209 ; ( -209/181 , -8/181 ), ( 2165/737 , 3596/737 )
**u= 37/10 , ht= 301 ; ( -461/509 , -584/509 ), ( -229/301 , 82/301 )
**u= 37/15 , ht= 653 ; ( 1577/787 , -464/787 ), ( -337/653 , -228/653 )
**u= 37/18 , ht= 6 ; ( 6 , 1/6 ), ( 23/121 , 16/121 )
**u= 37/19 , ht= 1209 ; ( 1209/335 , 110/67 ), ( 1595/1937 , 2300/1937 )
**u= 37/22 , ht= 19 ; ( -19/3 , 4 ), ( -180/7 , 513/14 )
**u= 37/23 , ht= 379 ; ( 131/453 , -218/151 ), ( 57/379 , 264/379 )
**u= 37/30 , ht= 97 ; ( 127/22 , 263/44 ), ( -47/97 , 120/97 )
**u= 37/31 , ht= 249 ; ( 1981/65 , -412/13 ), ( -125/249 , -320/249 )
**u= 37/42 , ht= 47 ; ( 1835/2981 , 500/271 ), ( 23/47 , 74/47 )
**u= 37/43 , ht= 1379 ; ( -347/1379 , -334/197 ), ( -1379/1207 , 2828/1207 )
**u= 37/50 , ht= 123 ; ( -331/123 , 916/205 ), ( 123/109 , -1362/545 )
**u= 37/55 , ht= 4511 ; ( -77129/4529 , 17930/647 ), ( -3829/4511 , -10108/4511 )
**u= 37/58 , ht= 20 ; ( 20/9 , -25/6 ), ( -479/247 , 982/247 )
**u= 37/63 , ht= 4889 ; ( 2809/7233 , -44302/21699 ), ( 4889/1487 , 9762/1487 )
**u= 37/67 , ht= 63929 ; ( 63929/2289 , 39006/763 ), ( 125349/13325 , 49734/2665 )
**u= 37/70 , ht= 719 ; ( -681/719 , -1930/719 ), ( -568/2113 , -8251/4226 )
**u= 37/75 , ht= 79 ; ( 67/79 , 1048/395 ), ( -1799/101 , -3736/101 )
**u= 37/78 , ht= 50 ; ( -881/3214 , -14013/6428 ), ( -50/41 , -267/82 )
**u= 37/82 , ht= 24337 ; ( -24337/21665 , -68396/21665 ), ( 21221/27371 , -72452/27371 )
**u= 37/87 , ht= 737 ; ( -1699/389 , -3702/389 ), ( 389/737 , 1784/737 )
**u= 37/94 , ht= 15285 ; ( 1541/15285 , 2352/1019 ), ( -17310/4291 , 81765/8582 )
**u= 37/95 , ht= 3671 ; ( 6407/813 , 4788/271 ), ( 1839/3671 , -9198/3671 )
**u= 40/13 , ht= 23 ; ( 4/43 , 127/86 ), ( -23/16 , 61/32 )
**u= 40/21 , ht= 29 ; ( -29/8 , 29/16 ), ( -47/464 , 11/32 )
**u= 40/29 , ht= 113 ; ( 293/84 , 27/8 ), ( 84/113 , 315/226 )
**u= 40/37 , ht= 48 ; ( 48/11 , 115/22 ), ( -137/196 , -617/392 )
**u= 40/61 , ht= 556 ; ( -556/237 , 677/158 ), ( 897/836 , -4299/1672 )
**u= 40/69 , ht= 896 ; ( -367/896 , -529/256 ), ( -1708/3709 , -14749/7418 )
**u= 40/93 , ht= 256 ; ( 931/652 , 4863/1304 ), ( -23/256 , 1083/512 )
**u= 41/5 , ht= 121 ; ( 1013/1573 , -2606/1573 ), ( -121/59 , -224/59 )
**u= 41/8 , ht= 5 ; ( -4/5 , 5/4 ), ( 155/163 , -115/163 )
**u= 41/9 , ht= 33 ; ( -33/31 , -74/93 ), ( -841/549 , -3508/1647 )
**u= 41/16 , ht= 33 ; ( 31/33 , -14/11 ), ( 51/49 , -129/98 )
**u= 41/17 , ht= 539 ; ( -5873/2575 , 142/515 ), ( 505/539 , -640/539 )
**u= 41/20 , ht= 167 ; ( -383/243 , 37/27 ), ( 27/167 , -9/167 )
**u= 41/21 , ht= 17 ; ( -17/11 , 16/11 ), ( 1/337 , -74/337 )
**u= 41/24 , ht= 23 ; ( 23/2 , 53/8 ), ( 269/371 , -436/371 )
**u= 41/29 , ht= 37373 ; ( 38441/1405 , -32404/1405 ), ( 24033/37373 , 47052/37373 )
**u= 41/33 , ht= 1465 ; ( -4537/1315 , -984/263 ), ( -1465/229 , -2200/229 )
**u= 41/36 , ht= 361 ; ( -4617/677 , 15403/2031 ), ( -215/361 , -512/361 )
**u= 41/44 , ht= 371 ; ( -14785/3013 , -19775/3013 ), ( -223/371 , -601/371 )
**u= 41/45 , ht= 893 ; ( -1261/541 , -1882/541 ), ( 893/83 , -1458/83 )
**u= 41/56 , ht= 705 ; ( -2565/613 , 4090/613 ), ( -359/705 , -251/141 )
**u= 41/61 , ht= 729 ; ( -38671/9153 , -7222/1017 ), ( -729/247 , 1386/247 )
**u= 41/65 , ht= 85549 ; ( 85549/58509 , 60398/19503 ), ( 30433311/943201 , -56844822/943201 )
**u= 41/69 , ht= 11605 ; ( 90385/16703 , -161530/16703 ), ( 9371/11605 , -5422/2321 )
**u= 41/80 , ht= 929 ; ( -929/659 , -4451/1318 ), ( -3629/3091 , 9466/3091 )
**u= 41/84 , ht= 709 ; ( 709/149 , -1426/149 ), ( 373/5161 , -10161/5161 )
**u= 41/89 , ht= 6691 ; ( 44367/7343 , -91318/7343 ), ( -6691/2921 , -15484/2921 )
**u= 41/96 , ht= 818 ; ( -433/818 , 16227/6544 ), ( 679/863 , 4727/1726 )
**u= 45/2 , ht= 7 ; ( -1/7 , 10/3 ), ( -7 , 70/3 )
**u= 45/7 , ht= 27 ; ( -1/27 , -152/81 ), ( -199/207 , -370/621 )
**u= 45/11 , ht= 47 ; ( -47/41 , -86/123 ), ( 1583/1561 , 4798/4683 )
**u= 45/14 , ht= 53 ; ( 53/46 , -265/276 ), ( -74/53 , 11/6 )
**u= 45/19 , ht= 479 ; ( 479/213 , 356/639 ), ( -491/3 , 2098/9 )
**u= 45/22 , ht= 293 ; ( 293/51 , -110/153 ), ( 1857/11449 , -3010/34347 )
**u= 45/26 , ht= 43 ; ( 4/43 , 367/258 ), ( 1801/1372 , 15989/8232 )
**u= 45/31 , ht= 531 ; ( 933/499 , 3134/1497 ), ( 531/283 , -2410/849 )
**u= 45/34 , ht= 129 ; ( -129/7 , -358/21 ), ( -567/1711 , 5354/5133 )
**u= 45/38 , ht= 53 ; ( -53/4 , 335/24 ), ( -947/421 , -4490/1263 )
**u= 45/43 , ht= 281 ; ( 1253/4521 , -21620/13563 ), ( -281/63 , -1334/189 )
**u= 45/46 , ht= 2142 ; ( 6453/4291 , -31666/12873 ), ( 2142/739 , -21191/4434 )
**u= 45/47 , ht= 863 ; ( 863/581 , -616/249 ), ( -3199/5543 , 26110/16629 )
**u= 45/59 , ht= 1931 ; ( -12403/8129 , -69920/24387 ), ( -1931/1373 , 11782/4119 )
**u= 45/62 , ht= 1181 ; ( 1181/338 , 887/156 ), ( -949/2437 , -12376/7311 )
**u= 45/67 , ht= 7779 ; ( 1883/7779 , 43462/23337 ), ( 15837/1499 , 86794/4497 )
**u= 45/71 , ht= 947 ; ( 2303/1451 , -14188/4353 ), ( 947/379 , 5630/1137 )
**u= 45/74 , ht= 104 ; ( -104/7 , -1081/42 ), ( 7693/256 , 87533/1536 )
**u= 45/82 , ht= 691 ; ( 271/927 , 5704/2781 ), ( 691/117 , 4154/351 )
**u= 45/83 , ht= 1667 ; ( 10937/1309 , -5560/357 ), ( 671/1667 , 10076/5001 )
**u= 45/91 , ht= 1189 ; ( -813/1189 , -8786/3567 ), ( 23553/2441 , 147074/7323 )
**u= 45/98 , ht= 121 ; ( 121/23 , 752/69 ), ( -23/121 , -752/363 )
**u= 49 , ht= 1 ; ( 1 , 2/7 ), ( 1 , -2/7 )
**u= 49/4 , ht= 155 ; ( -941/935 , 3271/6545 ), ( 155/41 , 2629/287 )
**u= 49/9 , ht= 1 ; ( -385/417 , -9110/8757 ), ( -1 , -6/7 )
**u= 49/12 , ht= 107 ; ( 107/85 , -32/119 ), ( 28465/5951 , -312750/41657 )
**u= 49/13 , ht= 335 ; ( -179/335 , -668/469 ), ( 335/179 , -3340/1253 )
**u= 49/16 , ht= 9 ; ( 85/63 , 75/98 ), ( 9/7 , -81/49 )
**u= 49/20 , ht= 143 ; ( 143/93 , 225/217 ), ( 99/221 , 48/1547 )
**u= 49/24 , ht= 19 ; ( 137/23 , 118/161 ), ( -10/19 , -381/532 )
**u= 49/25 , ht= 13 ; ( 13 , -104/35 ), ( -27/91 , -114/245 )
**u= 49/29 , ht= 37 ; ( 37/21 , -86/49 ), ( -21/37 , 258/259 )
**u= 49/36 , ht= 101 ; ( -101/11 , 58/7 ), ( 11/101 , 638/707 )
**u= 49/37 , ht= 1271 ; ( -12621/3215 , 17572/4501 ), ( -605/1271 , 10300/8897 )
**u= 49/40 , ht= 19 ; ( 34/19 , 1249/532 ), ( -19/6 , 815/168 )
**u= 49/45 , ht= 191 ; ( 29/191 , 2076/1337 ), ( 293/1003 , 8586/7021 )
**u= 49/60 , ht= 937 ; ( 937/637 , 925/343 ), ( 13949/4541 , -171184/31787 )
**u= 49/61 , ht= 717 ; ( 251/1335 , -5364/3115 ), ( 717/503 , 9936/3521 )
**u= 49/64 , ht= 1 ; ( 1 , -16/7 ), ( 271/187 , 899/308 )
**u= 49/65 , ht= 67 ; ( -67/63 , -116/49 ), ( 8217/1483 , 101340/10381 )
**u= 49/69 , ht= 143 ; ( 143/97 , -1982/679 ), ( -2701/1853 , -39342/12971 )
**u= 49/76 , ht= 43 ; ( -8719/4885 , -119827/34195 ), ( -5/43 , -506/301 )
**u= 49/81 , ht= 1 ; ( -239/111 , 29282/6993 ), ( -1 , -18/7 )
**u= 49/85 , ht= 253 ; ( 4353/1903 , -5470/1211 ), ( 253/83 , -3586/581 )
**u= 49/88 , ht= 15 ; ( -7/15 , 15/7 ), ( -1030/763 , -68785/21364 )
**u= 49/100 , ht= 1 ; ( 151/347 , 27301/12145 ), ( 1 , 20/7 )
**u= 50/7 , ht= 48 ; ( -103/96 , -1/4 ), ( -48/13 , 456/65 )
**u= 50/9 , ht= 4 ; ( -59/140 , 288/175 ), ( 4/3 , 16/9 )
**u= 50/11 , ht= 6 ; ( -5/6 , -6/5 ), ( 6/5 , -36/25 )
**u= 50/23 , ht= 87 ; ( -13/148 , -262/185 ), ( -28/87 , 88/435 )
**u= 50/29 , ht= 19 ; ( 19/10 , -44/25 ), ( -66/37 , -96/37 )
**u= 50/39 , ht= 103 ; ( -103/40 , -72/25 ), ( 680/139 , -5088/695 )
**u= 50/41 , ht= 21 ; ( 21/20 , 46/25 ), ( -20/61 , 344/305 )
**u= 50/43 , ht= 479 ; ( 479/6 , -426/5 ), ( -558/517 , 5052/2585 )
**u= 50/57 , ht= 64 ; ( 913/512 , 937/320 ), ( -64/31 , 568/155 )
**u= 50/61 , ht= 150 ; ( 89/150 , -236/125 ), ( 7314/2873 , -13008/2873 )
**u= 50/71 , ht= 200 ; ( 129/200 , 257/125 ), ( 1816/325 , -16424/1625 )
**u= 50/89 , ht= 49 ; ( 129/40 , 154/25 ), ( -40/49 , 592/245 )
**u= 50/93 , ht= 3734 ; ( 4553/2662 , 24976/6655 ), ( 3734/913 , 7648/913 )
**u= 52 , ht= 5 ; ( 1/5 , -5 ), ( 5 , -25 )
**u= 52/17 , ht= 321 ; ( 337/411 , -173/137 ), ( 321/253 , -411/253 )
**u= 52/25 , ht= 77 ; ( 6573/1327 , -367/1327 ), ( 77/73 , -107/73 )
**u= 52/33 , ht= 53 ; ( -31/53 , 79/53 ), ( 625/317 , -923/317 )
**u= 52/49 , ht= 401 ; ( -401/205 , 113/41 ), ( -132385/26667 , 1456465/186669 )
**u= 52/57 , ht= 383 ; ( 13955/2479 , -18825/2479 ), ( -383/35 , 125/7 )
**u= 52/73 , ht= 521 ; ( 521/83 , -829/83 ), ( -43931/1007 , -78157/1007 )
**u= 52/97 , ht= 2237 ; ( 879/2237 , 4763/2237 ), ( -4021/4127 , -11119/4127 )
**u= 53/2 , ht= 7 ; ( -2/7 , 7/2 ), ( 329/330 , 35/132 )
**u= 53/3 , ht= 259 ; ( -863/875 , -596/875 ), ( -259/233 , -356/233 )
**u= 53/11 , ht= 145 ; ( 57/145 , -46/29 ), ( -48245/22287 , -74720/22287 )
**u= 53/14 , ht= 405 ; ( -323/645 , -62/43 ), ( -405/13 , 630/13 )
**u= 53/15 , ht= 641 ; ( 641/521 , -374/521 ), ( -817/983 , -630/983 )
**u= 53/18 , ht= 55 ; ( 46/55 , -139/110 ), ( -25983/6394 , -226103/38364 )
**u= 53/22 , ht= 31 ; ( -31/24 , -19/16 ), ( -987/619 , 1356/619 )
**u= 53/23 , ht= 75 ; ( -695/1293 , 600/431 ), ( -29/75 , -2/15 )
**u= 53/27 , ht= 65 ; ( -27895/1133 , 16880/3399 ), ( -11/65 , -4/13 )
**u= 53/30 , ht= 2141 ; ( 1019/3434 , -9803/6868 ), ( 2126/2141 , 6403/4282 )
**u= 53/35 , ht= 473 ; ( -4241/5111 , -8026/5111 ), ( 67/473 , -368/473 )
**u= 53/38 , ht= 16 ; ( -6279/257 , -5402/257 ), ( -16/3 , -47/6 )
**u= 53/39 , ht= 395 ; ( 31/395 , 116/79 ), ( -7650595/592939 , -11233470/592939 )
**u= 53/42 , ht= 856 ; ( -856/131 , -1717/262 ), ( 401/4769 , 4696/4769 )
**u= 53/47 , ht= 737 ; ( 737/203 , 124/29 ), ( 18529/9693 , 30268/9693 )
**u= 53/50 , ht= 258 ; ( 514/739 , -2591/1478 ), ( 161/258 , 3911/2580 )
**u= 53/58 , ht= 5003 ; ( 23587/11349 , 926/291 ), ( -1755/5003 , 7176/5003 )
**u= 53/62 , ht= 7138 ; ( -20913/6119 , -30658/6119 ), ( 7138/2829 , -25003/5658 )
**u= 53/63 , ht= 73 ; ( 73/17 , 106/17 ), ( 5527/3259 , 10308/3259 )
**u= 53/67 , ht= 729 ; ( 193/729 , -142/81 ), ( -729/1147 , -2088/1147 )
**u= 53/78 , ht= 821 ; ( 821/197 , -1372/197 ), ( 890/253 , -3327/506 )
**u= 53/83 , ht= 137 ; ( 1521/4343 , 8462/4343 ), ( -29/137 , -236/137 )
**u= 53/86 , ht= 3085 ; ( -73159/18775 , -26056/3755 ), ( -3085/677 , 5930/677 )
**u= 53/87 , ht= 9031 ; ( -7391/54197 , 9404/4927 ), ( -9031/3439 , 18106/3439 )
**u= 53/90 , ht= 8557 ; ( 34121/3751 , -60524/3751 ), ( -8557/128 , -32881/256 )
**u= 53/95 , ht= 2399 ; ( -2953/2253 , -2320/751 ), ( 261/2399 , 4392/2399 )
**u= 58/15 , ht= 94 ; ( -61/94 , -64/47 ), ( 2458/293 , -3832/293 )
**u= 58/21 , ht= 163 ; ( -163/100 , -3/5 ), ( -1460/787 , -2020/787 )
**u= 58/25 , ht= 5 ; ( -941/366 , 14/61 ), ( -2/5 , -4/25 )
**u= 58/37 , ht= 128 ; ( -3249/104 , -571/26 ), ( 128/57 , -188/57 )
**u= 58/43 , ht= 136 ; ( 1093/240 , -87/20 ), ( 136/79 , -212/79 )
**u= 58/47 , ht= 475 ; ( 475/42 , -80/7 ), ( -7746/4879 , -12576/4879 )
**u= 58/57 , ht= 1994 ; ( 989/1994 , 1678/997 ), ( 1994/989 , 3356/989 )
**u= 58/79 , ht= 4723 ; ( 4723/570 , -244/19 ), ( 21990/1789 , 38760/1789 )
**u= 58/85 , ht= 3212 ; ( 26703/13732 , 12409/3433 ), ( -3212/577 , 5884/577 )
**u= 61/6 , ht= 140 ; ( 137/140 , -31/40 ), ( 2863/383 , 6524/383 )
**u= 61/7 , ht= 145 ; ( -9/145 , 62/29 ), ( 2035/1157 , 3670/1157 )
**u= 61/10 , ht= 261 ; ( 841/1119 , -506/373 ), ( 261/179 , -378/179 )
**u= 61/18 , ht= 6 ; ( -1754/1385 , -397/554 ), ( 5/6 , 25/36 )
**u= 61/19 , ht= 1633 ; ( -1633/1557 , 556/519 ), ( 3321/4811 , -1356/4811 )
**u= 61/26 , ht= 53 ; ( 53/27 , 8/9 ), ( 219/275 , -54/55 )
**u= 61/30 , ht= 91 ; ( 91/86 , -241/172 ), ( -121/599 , -132/599 )
**u= 61/31 , ht= 8623 ; ( -8623/181 , -1576/181 ), ( -6041/71087 , 15406/71087 )
**u= 61/34 , ht= 479 ; ( 3373/1571 , 2728/1571 ), ( 479/343 , -698/343 )
**u= 61/35 , ht= 267 ; ( -253/267 , 134/89 ), ( -633/1333 , 1140/1333 )
**u= 61/39 , ht= 70997 ; ( -304043/111229 , -267218/111229 ), ( -70997/15661 , -102514/15661 )
**u= 61/43 , ht= 11647 ; ( 11647/4165 , -328/119 ), ( 7945/13621 , 16240/13621 )
**u= 61/46 , ht= 3181 ; ( 3181/99 , -976/33 ), ( -405670/49983 , 1199047/99966 )
**u= 61/51 , ht= 139 ; ( -65/139 , 220/139 ), ( 144817/9305 , -43682/1861 )
**u= 61/54 , ht= 101 ; ( -101/13 , -26/3 ), ( 2132/1619 , -7423/3238 )
**u= 61/55 , ht= 109 ; ( -109/21 , -6 ), ( -21/109 , -126/109 )
**u= 61/66 , ht= 35842 ; ( 35842/8521 , 97507/17042 ), ( -121751/199537 , 326626/199537 )
**u= 61/70 , ht= 991 ; ( 187/1237 , -2060/1237 ), ( -991/409 , -1730/409 )
**u= 61/75 , ht= 755 ; ( -1075/2309 , 21006/11545 ), ( -113/755 , -5492/3775 )
**u= 61/79 , ht= 16105 ; ( 368759/268429 , -717056/268429 ), ( -16105/11 , -27784/11 )
**u= 61/82 , ht= 2615 ; ( -4846/10545 , 2649/1406 ), ( -2615/583 , 4660/583 )
**u= 61/90 , ht= 121 ; ( -121/111 , 842/333 ), ( -44847/74308 , 870283/445848 )
**u= 61/91 , ht= 967 ; ( -967/99 , 16 ), ( -9999/67 , 18216/67 )
**u= 61/94 , ht= 9453 ; ( 9453/1181 , 15848/1181 ), ( 8219/10851 , 23554/10851 )
**u= 64 , ht= 1 ; ( 1 , 1/4 ), ( 1 , 1/4 )
**u= 64/9 , ht= 1 ; ( 19/21 , -271/252 ), ( -1 , -3/4 )
**u= 64/25 , ht= 13 ; ( 13/11 , -257/220 ), ( 37/59 , 659/1180 )
**u= 64/41 , ht= 73 ; ( 73/9 , 71/12 ), ( 501/91 , 2889/364 )
**u= 64/49 , ht= 27 ; ( 125/27 , -55/12 ), ( -27/5 , 225/28 )
**u= 64/57 , ht= 97 ; ( -97/55 , -109/44 ), ( -55/97 , -545/388 )
**u= 64/81 , ht= 1 ; ( -1 , -9/4 ), ( 21/13 , -1463/468 )
**u= 65 , ht= 213 ; ( 157/291 , -466/97 ), ( -213/209 , 240/209 )
**u= 65/8 , ht= 7 ; ( -4/7 , -7/4 ), ( -7/4 , 49/16 )
**u= 65/9 , ht= 3307 ; ( -2859/3307 , -11722/9921 ), ( 18901/12527 , 29424/12527 )
**u= 65/12 , ht= 4333 ; ( -2711/4333 , -6371/4333 ), ( -29119/3827 , 50747/3827 )
**u= 65/17 , ht= 2591 ; ( -2591/2047 , 938/2047 ), ( -5601/6367 , 4482/6367 )
**u= 65/21 , ht= 157 ; ( -2911/1877 , -300/1877 ), ( -109/157 , 62/157 )
**u= 65/24 , ht= 103 ; ( -367/271 , 266/271 ), ( 103/31 , 147/31 )
**u= 65/28 , ht= 1427 ; ( 1427/561 , -5/17 ), ( 561/1427 , 165/1427 )
**u= 65/29 , ht= 341 ; ( 341/167 , -172/167 ), ( 1903/4861 , -1376/4861 )
**u= 65/32 , ht= 33799 ; ( -33799/4217 , 382/4217 ), ( 4217/33799 , 382/33799 )
**u= 65/33 , ht= 263 ; ( -263/89 , 134/89 ), ( 89/263 , 134/263 )
**u= 65/36 , ht= 679 ; ( -3201/977 , -6014/2931 ), ( 167/679 , 4/7 )
**u= 65/37 , ht= 81 ; ( -46129/2913 , -7962/971 ), ( 81/67 , 120/67 )
**u= 65/41 , ht= 413 ; ( -249/413 , -88/59 ), ( -14567/11509 , -22316/11509 )
**u= 65/44 , ht= 239 ; ( -157/239 , -367/239 ), ( -7617/941 , -11043/941 )
**u= 65/53 , ht= 2213 ; ( 3673/1309 , -4192/1309 ), ( 2213/1471 , -3632/1471 )
**u= 65/56 , ht= 11 ; ( 2189/22853 , 34763/22853 ), ( 11/3 , -17/3 )
**u= 65/57 , ht= 797 ; ( -7177/1981 , 1196/283 ), ( 721/797 , 1400/797 )
**u= 65/64 , ht= 9 ; ( 7/9 , 11/6 ), ( -9/7 , 33/14 )
**u= 65/68 , ht= 281 ; ( 77/281 , 461/281 ), ( -1433/1389 , 2897/1389 )
**u= 65/72 , ht= 929 ; ( 412/929 , 6451/3716 ), ( 4013/604 , 26409/2416 )
**u= 65/73 , ht= 729 ; ( -437/729 , 148/81 ), ( -3069/2777 , -6264/2777 )
**u= 65/77 , ht= 2053 ; ( 41477/37319 , -85072/37319 ), ( 1591/2053 , -3908/2053 )
**u= 65/81 , ht= 501 ; ( -501/277 , 7778/2493 ), ( -3389/3127 , 21980/9381 )
**u= 65/84 , ht= 2897 ; ( 2897/29 , 4295/29 ), ( 26297/10169 , 47772/10169 )
**u= 65/89 , ht= 3681 ; ( -2461/8157 , 4954/2719 ), ( -3681/3617 , 8550/3617 )
**u= 65/92 , ht= 977 ; ( 13055583/296011 , -20556409/296011 ), ( -977/81 , 1748/81 )
**u= 65/93 , ht= 1331 ; ( 1273/1331 , -284/121 ), ( 1331/1273 , -3124/1273 )
**u= 65/96 , ht= 973 ; ( -973/374 , 13709/2992 ), ( 1582/149 , 23045/1192 )
**u= 68/9 , ht= 7315 ; ( 4723/7315 , 2349/1463 ), ( -11495/7009 , 19015/7009 )
**u= 68/25 , ht= 5 ; ( -5/3 , 3/5 ), ( 3/5 , 9/25 )
**u= 68/33 , ht= 113 ; ( -827/163 , -111/163 ), ( -41/113 , -51/113 )
**u= 68/41 , ht= 137 ; ( -137/55 , 23/11 ), ( -775/1439 , 1415/1439 )
**u= 68/65 , ht= 11927 ; ( -1983/11927 , -18757/11927 ), ( 12411/2869 , -19659/2869 )
**u= 68/97 , ht= 281 ; ( 56677/18789 , 31919/6263 ), ( 87/281 , 471/281 )
**u= 72/5 , ht= 94 ; ( -58/97 , -1277/582 ), ( -94/91 , -479/546 )
**u= 72/13 , ht= 105 ; ( -171/154 , 53/132 ), ( 105/38 , -1057/228 )
**u= 72/37 , ht= 31 ; ( -31/6 , -67/36 ), ( -27/158 , -319/948 )
**u= 72/53 , ht= 79 ; ( -277/518 , 4793/3108 ), ( 79/26 , 709/156 )
**u= 72/61 , ht= 31 ; ( 31/30 , 67/36 ), ( 210/461 , 3475/2766 )
**u= 72/77 , ht= 170 ; ( -6290/2033 , -52615/12198 ), ( 149/170 , -23/12 )
**u= 72/85 , ht= 11 ; ( 43/42 , 79/36 ), ( 11/6 , 121/36 )
**u= 73 , ht= 95 ; ( -83/95 , -56/19 ), ( 1465/281 , -8690/281 )
**u= 73/5 , ht= 1017 ; ( 887/1017 , 478/339 ), ( -1299/259 , -3474/259 )
**u= 73/12 , ht= 133 ; ( 59/133 , 32/19 ), ( -133/59 , 224/59 )
**u= 73/13 , ht= 27 ; ( 1/27 , 16/9 ), ( -122281/4729 , -217378/4729 )
**u= 73/24 , ht= 605 ; ( 163/605 , 16/11 ), ( 550/761 , 1595/3044 )
**u= 73/25 , ht= 95 ; ( 331/469 , -622/469 ), ( 95/83 , -592/415 )
**u= 73/28 , ht= 37 ; ( -323/769 , -1081/769 ), ( 23/37 , 19/37 )
**u= 73/32 , ht= 8 ; ( 8/3 , 3/8 ), ( 4071/7175 , -891/1435 )
**u= 73/33 , ht= 719 ; ( 719/433 , -522/433 ), ( -445/1271 , -268/1271 )
**u= 73/37 , ht= 3261 ; ( 3261/301 , -98/43 ), ( -34895/33407 , -49658/33407 )
**u= 73/40 , ht= 274 ; ( -274/171 , 359/228 ), ( 19/294 , 515/1176 )
**u= 73/45 , ht= 3559 ; ( -3559/851 , -2612/851 ), ( 1711/7361 , 5370/7361 )
**u= 73/48 , ht= 227 ; ( -227/143 , -41/22 ), ( -8333/151 , -24011/302 )
**u= 73/49 , ht= 1471 ; ( 1471/5 , -1592/7 ), ( -5295/2159 , -7830/2159 )
**u= 73/52 , ht= 521 ; ( -51/521 , -761/521 ), ( 521/51 , -761/51 )
**u= 73/57 , ht= 1225 ; ( -1225/67 , 1180/67 ), ( -6439/325 , -1912/65 )
**u= 73/60 , ht= 11497 ; ( -11497/3043 , 12559/3043 ), ( 22151/13931 , -36139/13931 )
**u= 73/68 , ht= 65 ; ( -65/3 , 25 ), ( -14073/5335 , 4530/1067 )
**u= 73/76 , ht= 1015 ; ( 2469/1255 , -743/251 ), ( -1015/103 , -1630/103 )
**u= 73/77 , ht= 153 ; ( 64483/6579 , 9238/731 ), ( -153/43 , 252/43 )
**u= 73/85 , ht= 849 ; ( 22409/711 , -3434/79 ), ( -849 , -1410 )
**u= 73/88 , ht= 249 ; ( 249/7 , -352/7 ), ( -1618/1209 , -12847/4836 )
**u= 73/97 , ht= 5639 ; ( -5639/3091 , 916/281 ), ( -10703/23313 , -39842/23313 )
**u= 74 , ht= 6 ; ( 89/90 , -14/15 ), ( 6 , -36 )
**u= 74/29 , ht= 1493 ; ( -1493/884 , 179/221 ), ( 1076/2201 , -28/2201 )
**u= 74/35 , ht= 173 ; ( 173/72 , -7/6 ), ( -72/247 , -60/247 )
**u= 74/39 , ht= 391 ; ( 391/274 , 204/137 ), ( 230/1343 , -32/79 )
**u= 74/45 , ht= 299 ; ( -329/24 , 157/18 ), ( 56/299 , 204/299 )
**u= 74/51 , ht= 236 ; ( -121/236 , 89/59 ), ( 604/331 , -916/331 )
**u= 74/65 , ht= 242 ; ( 83/242 , -190/121 ), ( 6206/421 , -9476/421 )
**u= 74/71 , ht= 241 ; ( -241/114 , 56/19 ), ( 114/611 , -744/611 )
**u= 74/99 , ht= 103 ; ( 7099/7000 , 4073/1750 ), ( 48/103 , -532/309 )
**u= 80/9 , ht= 161 ; ( 123/161 , -205/138 ), ( 161/123 , -35/18 )
**u= 80/17 , ht= 37 ; ( -37/31 , -11/62 ), ( 311/243 , -787/486 )
**u= 80/33 , ht= 119 ; ( -29/367 , -1047/734 ), ( 53/119 , -31/238 )
**u= 80/41 , ht= 83 ; ( -83 , 37/2 ), ( 1/83 , 37/166 )
**u= 80/49 , ht= 87 ; ( -11/87 , -83/58 ), ( 87/109 , -285/218 )
**u= 80/57 , ht= 197 ; ( -31/197 , -577/394 ), ( 1129/5803 , 10405/11606 )
**u= 80/81 , ht= 77 ; ( -39/77 , 215/126 ), ( 77/159 , -4169/2862 )
**u= 80/89 , ht= 1603 ; ( -1/1603 , -749/458 ), ( 1603 , -5243/2 )
**u= 81 , ht= 1 ; ( -83/241 , 12964/2169 ), ( 1 , -2/9 )
**u= 81/4 , ht= 157 ; ( 115/167 , 3515/1503 ), ( -157/115 , 622/207 )
**u= 81/16 , ht= 35 ; ( -47/77 , 289/198 ), ( -35/13 , -511/117 )
**u= 81/20 , ht= 11 ; ( -9/11 , -11/9 ), ( -11/9 , -121/81 )
**u= 81/25 , ht= 59 ; ( 59/41 , -148/369 ), ( -271/79 , -3578/711 )
**u= 81/28 , ht= 95 ; ( 493/305 , 223/549 ), ( -95/67 , -1135/603 )
**u= 81/29 , ht= 299 ; ( -299/183 , -896/1647 ), ( -509/45 , 1330/81 )
**u= 81/41 , ht= 91 ; ( 91/9 , 172/81 ), ( 431/2973 , -6904/26757 )
**u= 81/44 , ht= 13 ; ( 9/13 , -13/9 ), ( -117/337 , -1937/3033 )
**u= 81/49 , ht= 133 ; ( 133/89 , 9536/5607 ), ( -305/109 , -3964/981 )
**u= 81/53 , ht= 885 ; ( 8473/855 , 11470/1539 ), ( 885/37 , -310/9 )
**u= 81/56 , ht= 153 ; ( -381/80 , -11897/2880 ), ( 76/153 , -5981/5508 )
**u= 81/64 , ht= 1 ; ( 1 , -16/9 ), ( -115/209 , -9569/7524 )
**u= 81/65 , ht= 693 ; ( -811/21 , 1034/27 ), ( -277/693 , 7214/6237 )
**u= 81/76 , ht= 37 ; ( -89/25 , -992/225 ), ( -37 , -517/9 )
**u= 81/80 , ht= 393 ; ( 393/203 , -10327/3654 ), ( 3429/1829 , -52550/16461 )
**u= 81/85 , ht= 463 ; ( -20851/2609 , 241972/23481 ), ( 293/463 , 6788/4167 )
**u= 81/97 , ht= 5755 ; ( 2587/5755 , -18566/10359 ), ( 9445/15653 , 244070/140877 )
**u= 81/100 , ht= 4751 ; ( 4751/3917 , -428539/176265 ), ( -93923/93121 , -9358861/4190445 )
**u= 82/7 , ht= 740 ; ( -23/740 , 454/185 ), ( -1004/221 , 2408/221 )
**u= 82/9 , ht= 200 ; ( 119/200 , -9/5 ), ( -360 , -2360/3 )
**u= 82/13 , ht= 235 ; ( 235/222 , 20/37 ), ( -898/415 , 304/83 )
**u= 82/23 , ht= 811 ; ( 811/696 , 24/29 ), ( 4536/1201 , -6816/1201 )
**u= 82/25 , ht= 5 ; ( 203/428 , 152/107 ), ( 4/5 , 16/25 )
**u= 82/27 , ht= 74 ; ( 91/206 , -146/103 ), ( -74/31 , -316/93 )
**u= 82/45 , ht= 118 ; ( 249/14 , 164/21 ), ( 118/37 , 168/37 )
**u= 82/55 , ht= 113 ; ( -113/8 , 11 ), ( -4856/8321 , -9512/8321 )
**u= 82/57 , ht= 199 ; ( -199/28 , -6 ), ( 28/199 , -168/199 )
**u= 82/59 , ht= 2719 ; ( 2719/1470 , -106/49 ), ( -20370/6623 , -30300/6623 )
**u= 82/73 , ht= 793 ; ( 5447/848 , 769/106 ), ( -448/793 , 1112/793 )
**u= 82/77 , ht= 317 ; ( -109/1110 , -288/185 ), ( 222/317 , 504/317 )
**u= 82/87 , ht= 295 ; ( -485/224 , -45/14 ), ( 256/295 , -112/59 )
**u= 82/91 , ht= 603 ; ( -603/190 , 86/19 ), ( -51650/18253 , -87820/18253 )
**u= 85/2 , ht= 9 ; ( 199/207 , 30/23 ), ( -9/2 , -81/4 )
**u= 85/6 , ht= 743 ; ( 439/743 , -1628/743 ), ( 743/751 , 270/751 )
**u= 85/7 , ht= 3321 ; ( 2917/3411 , 1574/1137 ), ( -3321/143 , 8286/143 )
**u= 85/11 , ht= 69 ; ( -47/69 , 36/23 ), ( 69/47 , 108/47 )
**u= 85/18 , ht= 7 ; ( -6/7 , 7/6 ), ( -1639/1558 , -3333/3116 )
**u= 85/19 , ht= 351 ; ( 351/313 , -206/313 ), ( 5981/4347 , 7886/4347 )
**u= 85/22 , ht= 739 ; ( 739/563 , -64/563 ), ( 2036/67 , -6371/134 )
**u= 85/27 , ht= 281 ; ( 4307/2849 , -720/2849 ), ( -281/257 , -1006/771 )
**u= 85/31 , ht= 349 ; ( 179/427 , 86/61 ), ( 287/349 , -308/349 )
**u= 85/39 , ht= 319 ; ( 9949/3163 , 1746/3163 ), ( 137/319 , 142/319 )
**u= 85/42 , ht= 1808 ; ( 1808/299 , -49/46 ), ( -1313/4064 , -3497/8128 )
**u= 85/43 , ht= 859 ; ( 859/87 , 60/29 ), ( 87/859 , 180/859 )
**u= 85/46 , ht= 6899 ; ( -6899/277 , 2774/277 ), ( -11902/13421 , 35369/26842 )
**u= 85/47 , ht= 9317 ; ( -47441/20427 , 11984/6809 ), ( -6099/9317 , 9606/9317 )
**u= 85/54 , ht= 19 ; ( -101/13 , -218/39 ), ( 8/19 , 35/38 )
**u= 85/58 , ht= 913 ; ( -4443/6229 , -9688/6229 ), ( -913/711 , 1438/711 )
**u= 85/63 , ht= 1061 ; ( -16459/7747 , 18662/7747 ), ( 487/1061 , -1192/1061 )
**u= 85/67 , ht= 973 ; ( -2069/343 , -296/49 ), ( 973/169 , 112/13 )
**u= 85/74 , ht= 4529 ; ( -8692/1329 , -6403/886 ), ( 1983/4529 , 5748/4529 )
**u= 85/78 , ht= 1093 ; ( 30307/1954 , 69119/3908 ), ( -779/1093 , 1728/1093 )
**u= 85/79 , ht= 5229 ; ( -149647/18939 , -58156/6313 ), ( -5229/1723 , -8334/1723 )
**u= 85/82 , ht= 3114 ; ( 3114/1433 , 8651/2866 ), ( 4439/3783 , 8272/3783 )
**u= 85/83 , ht= 13887 ; ( -38071/13887 , 5626/1543 ), ( 13887/12061 , -26172/12061 )
**u= 85/91 , ht= 213 ; ( -10483/6079 , 16744/6079 ), ( -151/213 , -368/213 )
**u= 85/99 , ht= 113 ; ( -2103/31 , -8698/93 ), ( -41/113 , -170/113 )
**u= 89/4 , ht= 171 ; ( 389/747 , -238/83 ), ( -171/149 , 288/149 )
**u= 89/5 , ht= 617 ; ( -293/617 , -1636/617 ), ( -1167/607 , 3006/607 )
**u= 89/9 , ht= 293 ; ( 3697/4687 , -6946/4687 ), ( 293/273 , 892/819 )
**u= 89/20 , ht= 1337 ; ( -1229/2359 , 499/337 ), ( 1337/377 , -2128/377 )
**u= 89/21 , ht= 415 ; ( 5099/5435 , -5806/5435 ), ( 415/397 , -432/397 )
**u= 89/24 , ht= 137 ; ( -137/119 , -96/119 ), ( 1358/515 , -1613/412 )
**u= 89/29 , ht= 133 ; ( -969/1351 , -254/193 ), ( 133/17 , 196/17 )
**u= 89/32 , ht= 8 ; ( -8/5 , -5/8 ), ( -795/419 , -2205/838 )
**u= 89/40 , ht= 794 ; ( -3893/1443 , -25/37 ), ( 741/794 , 3939/3176 )
**u= 89/41 , ht= 2101 ; ( 29579/8469 , 218/2823 ), ( 1035/2101 , 1194/2101 )
**u= 89/44 , ht= 263 ; ( -441/131 , -173/131 ), ( 131/263 , 181/263 )
**u= 89/45 , ht= 1609 ; ( 1609/29 , 244/29 ), ( 21/12191 , 5468/36573 )
**u= 89/49 , ht= 1261 ; ( 1261/243 , 1522/567 ), ( 1323/1105 , -2712/1547 )
**u= 89/56 , ht= 57 ; ( 57/20 , -193/80 ), ( -263/487 , -502/487 )
**u= 89/60 , ht= 151 ; ( -197/23 , -157/23 ), ( 61/151 , 147/151 )
**u= 89/65 , ht= 3449 ; ( 382219/23029 , 338398/23029 ), ( -659/3449 , -3188/3449 )
**u= 89/68 , ht= 1395 ; ( -641/3905 , -1160/781 ), ( -1395/883 , 2220/883 )
**u= 89/69 , ht= 2621 ; ( 2621/1933 , -3798/1933 ), ( -869/4181 , 4182/4181 )
**u= 89/76 , ht= 3677 ; ( -3677/561 , 1329/187 ), ( 5247/1555 , -1623/311 )
**u= 89/80 , ht= 169 ; ( -169/139 , -569/278 ), ( -11399/11719 , 21826/11719 )
**u= 89/81 , ht= 29 ; ( 7/29 , 136/87 ), ( -29/7 , -136/21 )
**u= 89/84 , ht= 1699 ; ( 1699/173 , -1998/173 ), ( -30091/4489 , 47077/4489 )
**u= 89/93 , ht= 581 ; ( -4231/3085 , -1460/617 ), ( 85/581 , -750/581 )
**u= 89/96 , ht= 143 ; ( 143/122 , 2171/976 ), ( -8339/2197 , 2123/338 )
**u= 90/31 , ht= 334 ; ( -317/334 , -604/501 ), ( 1654/1127 , -6632/3381 )
**u= 90/37 , ht= 44 ; ( -7/44 , -47/33 ), ( -44/7 , -188/21 )
**u= 90/41 , ht= 59 ; ( -6357/1966 , -718/2949 ), ( -18/59 , -4/177 )
**u= 90/43 , ht= 71 ; ( -101/28 , 19/21 ), ( 28/71 , -100/213 )
**u= 90/53 , ht= 783 ; ( -783/224 , -59/24 ), ( 224/1443 , 2660/4329 )
**u= 90/73 , ht= 1243 ; ( 2017/1214 , -4078/1821 ), ( -874/1243 , 5420/3729 )
**u= 90/91 , ht= 1376 ; ( -491/2792 , 3353/2094 ), ( 1376/613 , -6932/1839 )
**u= 97/8 , ht= 189 ; ( 2432/2385 , 327/1060 ), ( 189/181 , 171/181 )
**u= 97/12 , ht= 473 ; ( 167/473 , -84/43 ), ( -7315/7753 , 1199/7753 )
**u= 97/13 , ht= 1103 ; ( 1103/1051 , -452/1051 ), ( -1123/359 , 2144/359 )
**u= 97/17 , ht= 12807 ; ( -8761/12807 , 6086/4269 ), ( 116307/53747 , -190014/53747 )
**u= 97/25 , ht= 781 ; ( -781/595 , -14/425 ), ( -19943/985 , 156142/4925 )
**u= 97/28 , ht= 2713 ; ( -2713/1971 , -229/657 ), ( -13423/2657 , 20198/2657 )
**u= 97/33 , ht= 1433 ; ( 971/1433 , 1916/1433 ), ( -5693/8167 , -4128/8167 )
**u= 97/36 , ht= 845 ; ( 1421/845 , -8/13 ), ( -845/451 , -1170/451 )
**u= 97/37 , ht= 2999 ; ( -2407/2999 , 3934/2999 ), ( 12647/4521 , -17902/4521 )
**u= 97/40 , ht= 694 ; ( -2039/1111 , -950/1111 ), ( -349/694 , -991/2776 )
**u= 97/45 , ht= 1073 ; ( 1073/353 , 278/353 ), ( -4259/5331 , -17002/15993 )
**u= 97/48 , ht= 215 ; ( -1555/163 , 55/163 ), ( -23/215 , -2/43 )
**u= 97/52 , ht= 2595 ; ( 169805/2573 , -63520/2573 ), ( 733/2595 , 284/519 )
**u= 97/57 , ht= 49 ; ( -439/55 , 262/55 ), ( 11/49 , -32/49 )
**u= 97/61 , ht= 2065 ; ( 131/2065 , -592/413 ), ( -9415/53579 , -38860/53579 )
**u= 97/64 , ht= 125 ; ( 1687/425 , -281/85 ), ( 125/63 , 745/252 )
**u= 97/72 , ht= 433 ; ( 1193/41 , -1076/41 ), ( 433/368 , -2869/1472 )
**u= 97/73 , ht= 515 ; ( -6565/2331 , 110/37 ), ( -369/515 , -144/103 )
**u= 97/80 , ht= 47 ; ( 47/33 , 23/11 ), ( -429/29 , -645/29 )
**u= 97/85 , ht= 16171 ; ( 12091/16171 , 27934/16171 ), ( 45563/22613 , -73670/22613 )
**u= 97/89 , ht= 311 ; ( -52771/52605 , 2242/1169 ), ( 45/311 , -360/311 )
**u= 97/93 , ht= 30707 ; ( -30707/14305 , 42608/14305 ), ( 91795/39529 , -150776/39529 )
**u= 97/100 , ht= 125 ; ( -63/125 , 1073/625 ), ( 125/63 , 1073/315 )
**u= 98/15 , ht= 8 ; ( 31/56 , -80/49 ), ( 8/7 , -64/49 )
**u= 98/17 , ht= 9 ; ( -9/8 , 1/7 ), ( -8/9 , -8/63 )
**u= 98/33 , ht= 116 ; ( 83/116 , -268/203 ), ( -724/1157 , 1488/8099 )
**u= 98/47 , ht= 39 ; ( 2683/568 , -187/497 ), ( 8/39 , -8/273 )
**u= 98/51 , ht= 10 ; ( 7/10 , 10/7 ), ( 310/1279 , -3980/8953 )
**u= 98/53 , ht= 39 ; ( 277/94 , 604/329 ), ( -14/39 , 176/273 )
**u= 98/65 , ht= 2149 ; ( -2149/4 , 2846/7 ), ( -3644/5789 , -47896/40523 )
**u= 98/83 , ht= 93 ; ( -289/206 , -1522/721 ), ( -10/93 , 692/651 )
**u= 98/85 , ht= 1914 ; ( 3749/4854 , -9812/5663 ), ( -1914/1081 , 21936/7567 )
**u= 98/95 , ht= 12 ; ( 7/12 , 12/7 ), ( -2348/847 , -26704/5929 )
**u= 100 , ht= 1 ; ( -1 , 1/5 ), ( -721/711 , -4291/3555 )
**u= 100/9 , ht= 1 ; ( -11/29 , 323/145 ), ( -1 , -3/5 )
**u= 100/33 , ht= 353 ; ( 767/1031 , 1345/1031 ), ( -329/353 , -359/353 )
**u= 100/49 , ht= 13 ; ( -39/23 , -221/161 ), ( -3/13 , -9/35 )
**u= 100/57 , ht= 823 ; ( 1081/967 , -1481/967 ), ( 25/823 , -2117/4115 )
**u= 100/73 , ht= 311 ; ( -273/311 , -515/311 ), ( 25/609 , -2687/3045 )
**u= 100/81 , ht= 107 ; ( -107/99 , -1645/891 ), ( 1083/1381 , 19169/12429 )
**u= 100/97 , ht= 379 ; ( -2015/119 , -1723/85 ), ( 203/379 , 553/379 )
675
>

■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。

[2026.05.20追記]u=4/81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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