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

[2026.05.15]A^4+B^4=C^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=1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432個のuが抽出される。
[MAGMAによる計算]
> PP(1,1,100);
**u= 1/4 , ht= 1 ; ( 1 , -2 ), ( 1 , 2 )
**u= 1/7 , ht= 5 ; ( -5/2 , 5 ), ( -82/145 , 62/29 )
**u= 1/9 , ht= 3 ; ( 2/3 , 23/9 ), ( -3/2 , 23/6 )
**u= 1/15 , ht= 32 ; ( 23/32 , -27/8 ), ( -92/37 , 272/37 )
**u= 1/16 , ht= 1 ; ( -1 , 4 ), ( -19/15 , -137/30 )
**u= 1/24 , ht= 2 ; ( 1/2 , 31/8 ), ( -25/7 , 90/7 )
**u= 1/25 , ht= 49 ; ( -54/23 , 1037/115 ), ( -49/24 , 193/24 )
**u= 1/36 , ht= 39 ; ( -251/33 , -3221/99 ), ( 39/35 , 667/105 )
**u= 1/39 , ht= 14 ; ( 55/62 , 183/31 ), ( 14 , -62 )
**u= 1/40 , ht= 127 ; ( -283/37 , 1276/37 ), ( 127/7 , -569/7 )
**u= 1/49 , ht= 14 ; ( -14/5 , 103/7 ), ( -20/9 , 760/63 )
**u= 1/64 , ht= 29 ; ( -29/9 , 229/12 ), ( 23/87 , -509/87 )
**u= 1/71 , ht= 114 ; ( 43/114 , -121/19 ), ( -246/251 , -2094/251 )
**u= 1/79 , ht= 75 ; ( -75/4 , 118 ), ( 88/9 , 556/9 )
**u= 1/81 , ht= 1 ; ( -8/27 , -1613/243 ), ( 1 , -9 )
**u= 1/84 , ht= 557 ; ( -557/109 , -3678/109 ), ( -4927/491 , -32091/491 )
**u= 1/89 , ht= 80 ; ( 80/9 , 179/3 ), ( 21/110 , 747/110 )
**u= 1/100 , ht= 1 ; ( -59/53 , 2804/265 ), ( 1 , -10 )
**u= 4 , ht= 33 ; ( -33/31 , 1/62 ), ( -31/33 , -1/66 )
**u= 4/5 , ht= 7 ; ( -11/9 , 7/6 ), ( 3/7 , 9/14 )
**u= 4/9 , ht= 1 ; ( 5/21 , -149/126 ), ( 1 , -3/2 )
**u= 4/21 , ht= 23 ; ( 19/23 , 97/46 ), ( 23/19 , -97/38 )
**u= 4/25 , ht= 1 ; ( -1 , 5/2 ), ( -461/427 , -11119/4270 )
**u= 4/41 , ht= 61 ; ( -307/717 , -1181/478 ), ( -21/61 , 291/122 )
**u= 4/49 , ht= 1 ; ( 1 , 7/2 ), ( 417/319 , 2601/638 )
**u= 4/69 , ht= 359 ; ( 359/83 , 2161/166 ), ( -709/809 , 6317/1618 )
**u= 4/81 , ht= 1 ; ( -1 , 9/2 ), ( -1 , -9/2 )
**u= 5/3 , ht= 2 ; ( -1/2 , 1 ), ( -62/59 , 50/59 )
**u= 5/4 , ht= 49 ; ( -49/33 , 8/11 ), ( -33/49 , 24/49 )
**u= 5/8 , ht= 41 ; ( -79/17 , -58/17 ), ( -41/33 , 49/33 )
**u= 5/11 , ht= 56 ; ( 93/104 , -37/26 ), ( 56/23 , 68/23 )
**u= 5/12 , ht= 13 ; ( 377/89 , -390/89 ), ( 1/13 , -1 )
**u= 5/13 , ht= 17 ; ( -46/33 , 21/11 ), ( -9/17 , -21/17 )
**u= 5/21 , ht= 8 ; ( 1/8 , 3/2 ), ( 67/11 , 101/11 )
**u= 5/24 , ht= 43 ; ( 1/103 , -163/103 ), ( -43/29 , -81/29 )
**u= 5/29 , ht= 23 ; ( 21/37 , -73/37 ), ( -23/6 , 41/6 )
**u= 5/32 , ht= 7 ; ( 46/33 , -269/88 ), ( -1/7 , -25/14 )
**u= 5/36 , ht= 43 ; ( -21/43 , 274/129 ), ( -169/7 , 324/7 )
**u= 5/48 , ht= 143 ; ( 361/1447 , -3283/1447 ), ( -143/109 , 789/218 )
**u= 5/59 , ht= 48 ; ( -41/1752 , 178/73 ), ( 48/11 , 120/11 )
**u= 5/68 , ht= 31 ; ( 31/3 , -27 ), ( 8097/2929 , -22500/2929 )
**u= 5/69 , ht= 112 ; ( -569/443 , -1893/443 ), ( 112/89 , -376/89 )
**u= 5/76 , ht= 1397 ; ( -1397/9 , 427 ), ( 4059/1247 , 11727/1247 )
**u= 5/84 , ht= 29 ; ( -83/419 , -1240/419 ), ( -13/29 , -92/29 )
**u= 5/93 , ht= 287 ; ( -116/287 , -135/41 ), ( 287/194 , 1057/194 )
**u= 5/96 , ht= 11 ; ( -11/8 , 337/64 ), ( -4463/79 , 13849/79 )
**u= 5/99 , ht= 17 ; ( -17/6 , -85/9 ), ( -874/17 , -162 )
**u= 8/5 , ht= 22 ; ( 22/17 , -37/68 ), ( 34/49 , -43/196 )
**u= 8/17 , ht= 31 ; ( 31/20 , -29/16 ), ( 5/56 , 205/224 )
**u= 8/33 , ht= 52 ; ( -56/65 , 99/52 ), ( 52/41 , 383/164 )
**u= 8/37 , ht= 19 ; ( 19/18 , -53/24 ), ( 18/19 , -159/76 )
**u= 8/45 , ht= 3 ; ( 2/3 , 73/36 ), ( -3/2 , -73/24 )
**u= 8/53 , ht= 434 ; ( -169/434 , -3419/1736 ), ( 333/598 , 381/184 )
**u= 8/57 , ht= 644 ; ( -97/644 , 709/368 ), ( 3136/1255 , -25683/5020 )
**u= 8/65 , ht= 9 ; ( -9/4 , 79/16 ), ( 4/9 , -79/36 )
**u= 8/73 , ht= 64 ; ( 64/9 , 61/4 ), ( -8784/10609 , -117549/42436 )
**u= 8/93 , ht= 551 ; ( 1306/97 , -12583/388 ), ( 86/551 , -5359/2204 )
**u= 9 , ht= 11 ; ( -10/11 , 31/33 ), ( 11/10 , 31/30 )
**u= 9/4 , ht= 1 ; ( -1 , -2/3 ), ( -179/193 , 293/579 )
**u= 9/16 , ht= 1 ; ( -1 , 4/3 ), ( 25/19 , -185/114 )
**u= 9/25 , ht= 1 ; ( -61/32 , -293/120 ), ( 1 , -5/3 )
**u= 9/31 , ht= 17 ; ( -17/14 , -43/21 ), ( 38/7 , 158/21 )
**u= 9/40 , ht= 49 ; ( 41/49 , 41/21 ), ( 77/82 , -49/24 )
**u= 9/41 , ht= 33 ; ( -37/45 , -53/27 ), ( 33/8 , 157/24 )
**u= 9/44 , ht= 19 ; ( -3/19 , -92/57 ), ( 247/27 , -1189/81 )
**u= 9/49 , ht= 1 ; ( 1/11 , -389/231 ), ( 1 , -7/3 )
**u= 9/55 , ht= 43 ; ( -43/32 , 35/12 ), ( -64/31 , -376/93 )
**u= 9/56 , ht= 92 ; ( 15/92 , 1997/1104 ), ( 24/199 , 4189/2388 )
**u= 9/64 , ht= 1 ; ( -1355/1601 , -47537/19212 ), ( 1 , 8/3 )
**u= 9/65 , ht= 23 ; ( 23/3 , 131/9 ), ( -587/258 , 3679/774 )
**u= 9/76 , ht= 25 ; ( 1243/2953 , 19847/8859 ), ( -13/25 , -173/75 )
**u= 9/80 , ht= 87 ; ( -87/23 , 566/69 ), ( -261/59 , 1702/177 )
**u= 9/88 , ht= 205 ; ( -205/37 , -1375/111 ), ( 251/955 , -1304/573 )
**u= 9/89 , ht= 129 ; ( 129/49 , 131/21 ), ( 168/257 , -2044/771 )
**u= 9/100 , ht= 127 ; ( -691/73 , -24469/1095 ), ( 127/91 , 5531/1365 )
**u= 13/3 , ht= 8 ; ( 103/148 , 42/37 ), ( -8 , -12 )
**u= 13/5 , ht= 3 ; ( 193/393 , -145/131 ), ( -3/2 , 3/2 )
**u= 13/12 , ht= 395 ; ( -389/395 , 76/79 ), ( -395/389 , 380/389 )
**u= 13/16 , ht= 5 ; ( -635/133 , 320/133 ), ( 1/5 , 1/2 )
**u= 13/27 , ht= 58 ; ( 35/58 , 109/87 ), ( 94/37 , 334/111 )
**u= 13/28 , ht= 165 ; ( -165/59 , -166/59 ), ( -1201/339 , 1409/339 )
**u= 13/32 , ht= 251 ; ( -251/123 , -98/41 ), ( -2199/1550 , -24531/12400 )
**u= 13/37 , ht= 93 ; ( 99/25 , 23/5 ), ( -55/93 , 125/93 )
**u= 13/45 , ht= 3 ; ( -3/2 , 7/3 ), ( 2/3 , 14/9 )
**u= 13/48 , ht= 251 ; ( -241/251 , -947/502 ), ( 263/11 , -741/22 )
**u= 13/59 , ht= 318 ; ( 55/1302 , 335/217 ), ( 318/95 , 102/19 )
**u= 13/60 , ht= 127 ; ( 67/127 , 221/127 ), ( -43/3647 , 5409/3647 )
**u= 13/67 , ht= 368 ; ( -3481/2208 , 547/184 ), ( -368/33 , 604/33 )
**u= 13/68 , ht= 3789 ; ( -355/3789 , -696/421 ), ( 6219/779 , 10314/779 )
**u= 13/77 , ht= 54 ; ( -293/48 , 21/2 ), ( 54/23 , 102/23 )
**u= 13/84 , ht= 85 ; ( -425/2347 , -4335/2347 ), ( -43/85 , -2 )
**u= 13/85 , ht= 1018 ; ( -1018/903 , 117/43 ), ( -399/1606 , -2961/1606 )
**u= 13/88 , ht= 7 ; ( -7/4 , 59/16 ), ( -4/7 , 59/28 )
**u= 13/93 , ht= 721 ; ( 721/256 , 359/64 ), ( -304/1091 , -2124/1091 )
**u= 13/100 , ht= 1655 ; ( 10135/2341 , -101221/11705 ), ( -1655/253 , -16549/1265 )
**u= 16 , ht= 513 ; ( -513/511 , 1/2044 ), ( -511/513 , -1/2052 )
**u= 16/9 , ht= 1 ; ( 1 , 3/4 ), ( 75/67 , 745/804 )
**u= 16/13 , ht= 15 ; ( -15/11 , 35/44 ), ( 17/35 , 5/28 )
**u= 16/21 , ht= 25 ; ( 253 , 531/4 ), ( 11/25 , -69/100 )
**u= 16/25 , ht= 1 ; ( -349/243 , 2321/1620 ), ( -1 , 5/4 )
**u= 16/29 , ht= 79 ; ( 131/159 , 269/212 ), ( 21/79 , 267/316 )
**u= 16/45 , ht= 83 ; ( -131/49 , 631/196 ), ( 83/7 , -419/28 )
**u= 16/49 , ht= 1 ; ( 1 , -7/4 ), ( 1 , 7/4 )
**u= 16/65 , ht= 567 ; ( -1249/441 , 117/28 ), ( -567/47 , -3339/188 )
**u= 16/81 , ht= 129 ; ( -12935/12609 , -1034101/453924 ), ( 129/65 , 8369/2340 )
**u= 16/85 , ht= 27 ; ( -123/47 , -847/188 ), ( -7/27 , 179/108 )
**u= 17/8 , ht= 5 ; ( 28/27 , 23/36 ), ( 5/3 , -5/3 )
**u= 17/12 , ht= 193 ; ( -193/143 , 92/143 ), ( 407/667 , -133/667 )
**u= 17/15 , ht= 4 ; ( -1/4 , 1 ), ( -2044/1019 , 2020/1019 )
**u= 17/23 , ht= 25 ; ( 5039/278 , -1403/139 ), ( -2/25 , -14/25 )
**u= 17/44 , ht= 25 ; ( -25/393 , -160/131 ), ( -19/25 , -7/5 )
**u= 17/47 , ht= 856 ; ( -789/856 , 172/107 ), ( -536/1579 , -1856/1579 )
**u= 17/48 , ht= 13 ; ( -13 , -29/2 ), ( 23/41 , 54/41 )
**u= 17/55 , ht= 467 ; ( 467/138 , 99/23 ), ( 702/5777 , -7050/5777 )
**u= 17/56 , ht= 15 ; ( -15/8 , 85/32 ), ( -37/5 , -10 )
**u= 17/63 , ht= 967 ; ( -967/76 , 318/19 ), ( -2008/127 , -2836/127 )
**u= 17/73 , ht= 106 ; ( 4081/591 , 1961/197 ), ( 33/106 , 3/2 )
**u= 17/80 , ht= 67 ; ( 269/471 , -561/314 ), ( -67/47 , -253/94 )
**u= 17/81 , ht= 14 ; ( 75/448 , 115/72 ), ( -14/5 , 14/3 )
**u= 17/87 , ht= 178 ; ( -241/386 , -367/193 ), ( -178/83 , -318/83 )
**u= 17/92 , ht= 49 ; ( 49/43 , -107/43 ), ( 5999/349 , 10051/349 )
**u= 17/100 , ht= 643 ; ( -643/345 , 2069/575 ), ( 879/785 , 10122/3925 )
**u= 20/21 , ht= 29 ; ( 1247 , 551/2 ), ( 13/29 , -1/2 )
**u= 20/29 , ht= 77 ; ( -19/77 , -23/22 ), ( 77/309 , -413/618 )
**u= 20/33 , ht= 157 ; ( -157/41 , 243/82 ), ( 673/101 , 1437/202 )
**u= 20/41 , ht= 4121 ; ( 6261/767 , 11201/1534 ), ( -4121/963 , 9439/1926 )
**u= 20/53 , ht= 77 ; ( 1459/293 , -3193/586 ), ( -29/77 , -179/154 )
**u= 20/77 , ht= 2789 ; ( -2789/907 , -7913/1814 ), ( -341977/33321 , -984493/66642 )
**u= 20/81 , ht= 3 ; ( -1633/949 , -15881/5694 ), ( 1/3 , 79/54 )
**u= 20/93 , ht= 1253 ; ( 1253/571 , 4135/1142 ), ( 2093/2651 , 10245/5302 )
**u= 20/97 , ht= 123 ; ( 71/123 , 149/82 ), ( 933/1321 , -5001/2642 )
**u= 25 , ht= 79 ; ( -78/79 , 47/79 ), ( 79/78 , 47/78 )
**u= 25/4 , ht= 1 ; ( 1 , 2/5 ), ( -79/21 , -683/105 )
**u= 25/7 , ht= 4 ; ( -47/180 , 101/75 ), ( 4/3 , 4/3 )
**u= 25/9 , ht= 1 ; ( 1 , -3/5 ), ( -52/43 , -224/215 )
**u= 25/12 , ht= 25 ; ( 7/25 , -138/125 ), ( 41/5 , -231/25 )
**u= 25/16 , ht= 1 ; ( -1 , 4/5 ), ( 1 , 4/5 )
**u= 25/24 , ht= 61 ; ( -61/14 , -19/40 ), ( -14/61 , -133/1220 )
**u= 25/28 , ht= 33 ; ( 33/5 , -61/25 ), ( 5/33 , 61/165 )
**u= 25/33 , ht= 10 ; ( 10 , 27/5 ), ( -103/4 , -105/4 )
**u= 25/36 , ht= 61 ; ( -277/111 , 613/333 ), ( -39/61 , -823/915 )
**u= 25/49 , ht= 1 ; ( 1 , 7/5 ), ( -1 , -7/5 )
**u= 25/56 , ht= 121 ; ( 121/72 , 63/32 ), ( 153/167 , -1188/835 )
**u= 25/64 , ht= 1 ; ( -291/521 , -14033/10420 ), ( 1 , 8/5 )
**u= 25/79 , ht= 41 ; ( 41/38 , 35/19 ), ( 38/41 , 70/41 )
**u= 25/81 , ht= 1 ; ( -662/741 , -11443/6669 ), ( -1 , -9/5 )
**u= 25/84 , ht= 127 ; ( 449 , -2778/5 ), ( 127/43 , 899/215 )
**u= 25/88 , ht= 83 ; ( -83/5 , -529/25 ), ( -450/551 , 18723/11020 )
**u= 29/3 , ht= 44 ; ( -25/68 , -35/17 ), ( -44/35 , -12/7 )
**u= 29/5 , ht= 77 ; ( -62/77 , -83/77 ), ( -107/108 , 37/108 )
**u= 29/16 , ht= 193 ; ( -193/141 , -3/94 ), ( -141/193 , 9/386 )
**u= 29/20 , ht= 7 ; ( -3/7 , 1 ), ( -49/79 , -14/79 )
**u= 29/21 , ht= 127 ; ( -127/71 , -3/71 ), ( 2035/1129 , -1985/1129 )
**u= 29/35 , ht= 368 ; ( 543/368 , 55/46 ), ( 368/137 , 376/137 )
**u= 29/36 , ht= 17 ; ( -17 , -8 ), ( -73/93 , -257/279 )
**u= 29/44 , ht= 141 ; ( -1321/1033 , 1382/1033 ), ( 97/141 , 137/141 )
**u= 29/45 , ht= 7 ; ( -7/3 , -17/9 ), ( -1033/1843 , -1647/1843 )
**u= 29/51 , ht= 421 ; ( 421/64 , -83/16 ), ( -284/913 , -768/913 )
**u= 29/53 , ht= 100 ; ( -100/47 , 95/47 ), ( -541/170 , -121/34 )
**u= 29/56 , ht= 5 ; ( 1223/114 , 1381/152 ), ( -2/5 , -19/20 )
**u= 29/60 , ht= 889 ; ( -1381/3029 , -3636/3029 ), ( 889/149 , 1013/149 )
**u= 29/61 , ht= 114 ; ( 53/114 , 23/19 ), ( 114/53 , -138/53 )
**u= 29/69 , ht= 199 ; ( 191/637 , 111/91 ), ( 77/199 , 217/199 )
**u= 29/72 , ht= 125 ; ( -125/37 , -135/37 ), ( 3149/755 , -772/151 )
**u= 29/96 , ht= 101 ; ( -101/35 , 53/14 ), ( 35/101 , 265/202 )
**u= 29/99 , ht= 488 ; ( 205/488 , -89/61 ), ( 488/205 , 712/205 )
**u= 32/5 , ht= 3 ; ( 2/3 , -11/8 ), ( 118/87 , -1187/696 )
**u= 32/13 , ht= 126 ; ( 61/126 , 367/336 ), ( 369/70 , 3489/560 )
**u= 32/33 , ht= 623 ; ( 692/131 , -1429/1048 ), ( -92/623 , 1143/4984 )
**u= 32/49 , ht= 36 ; ( -5/36 , -235/224 ), ( 9/40 , 45/64 )
**u= 32/53 , ht= 27 ; ( -710/191 , -4429/1528 ), ( -26/27 , -271/216 )
**u= 32/57 , ht= 668 ; ( -1336/355 , 1779/568 ), ( 535/668 , -6235/5344 )
**u= 32/77 , ht= 271 ; ( -271/26 , -2177/208 ), ( 481/30 , -4577/240 )
**u= 32/85 , ht= 11 ; ( 11/6 , -37/16 ), ( -618/317 , -6663/2536 )
**u= 32/93 , ht= 41 ; ( 41/10 , 77/16 ), ( 10/41 , -385/328 )
**u= 36 , ht= 2593 ; ( -2593/2591 , 1/15546 ), ( -2591/2593 , -1/15558 )
**u= 36/5 , ht= 7 ; ( 49/69 , -569/414 ), ( 7/3 , -73/18 )
**u= 36/25 , ht= 1 ; ( 1 , 5/6 ), ( -1 , -5/6 )
**u= 36/29 , ht= 39 ; ( 393/277 , 1271/1662 ), ( -19/39 , 37/234 )
**u= 36/49 , ht= 1 ; ( -2015/2209 , 106129/92778 ), ( -1 , -7/6 )
**u= 36/65 , ht= 2239 ; ( 3851/1251 , 20015/7506 ), ( -1401/2239 , -14011/13434 )
**u= 36/77 , ht= 255 ; ( 255/53 , -1445/318 ), ( 411/425 , -43/30 )
**u= 36/85 , ht= 539 ; ( 2351/539 , 187/42 ), ( -539/311 , -4235/1866 )
**u= 37/8 , ht= 39 ; ( 25/39 , 16/13 ), ( 363/365 , 159/365 )
**u= 37/12 , ht= 7 ; ( 709/665 , -7/19 ), ( 7/5 , -7/5 )
**u= 37/13 , ht= 155 ; ( 79/155 , -35/31 ), ( -425/256 , -455/256 )
**u= 37/27 , ht= 11 ; ( -11/8 , -2/3 ), ( -8/11 , -16/33 )
**u= 37/28 , ht= 83 ; ( 57/83 , 79/83 ), ( 307/547 , 116/547 )
**u= 37/35 , ht= 6 ; ( -1/6 , 1 ), ( -15546/5179 , 15510/5179 )
**u= 37/43 , ht= 59 ; ( -59/16 , 7/4 ), ( 16/59 , 28/59 )
**u= 37/45 , ht= 29 ; ( 29/11 , -17/11 ), ( -137/288 , 565/864 )
**u= 37/48 , ht= 173 ; ( 6979/787 , 3669/787 ), ( -109/173 , -142/173 )
**u= 37/52 , ht= 319 ; ( 451/217 , 347/217 ), ( -319/175 , 344/175 )
**u= 37/53 , ht= 1601 ; ( 1966/747 , -473/249 ), ( 837/1601 , 1299/1601 )
**u= 37/60 , ht= 31 ; ( 31 , 22 ), ( 1/31 , -22/31 )
**u= 37/72 , ht= 2191 ; ( 2191/1543 , -2523/1543 ), ( -3707/1115 , 4218/1115 )
**u= 37/80 , ht= 919 ; ( 919/519 , 345/173 ), ( 6561/1589 , 15315/3178 )
**u= 37/84 , ht= 1195 ; ( -2005/143 , 175/13 ), ( -341/1195 , -242/239 )
**u= 37/85 , ht= 1151 ; ( 6369/2731 , -6925/2731 ), ( 124/1151 , -1120/1151 )
**u= 37/91 , ht= 115 ; ( 115/24 , 5 ), ( -24/115 , -24/23 )
**u= 37/93 , ht= 202 ; ( -202/139 , 267/139 ), ( -139/202 , 267/202 )
**u= 37/100 , ht= 99 ; ( -121/171 , 83/57 ), ( -99/35 , -642/175 )
**u= 40 , ht= 2 ; ( -1/2 , 31/8 ), ( 139/98 , -1765/392 )
**u= 40/9 , ht= 42 ; ( 41/42 , -41/72 ), ( -91/82 , -7/8 )
**u= 40/33 , ht= 61 ; ( -719/502 , -1583/2008 ), ( -61/38 , -237/152 )
**u= 40/41 , ht= 59 ; ( 334/117 , -57/52 ), ( 18/59 , -81/236 )
**u= 40/49 , ht= 58 ; ( 58/9 , -37/12 ), ( -198/29 , -5613/812 )
**u= 40/61 , ht= 3084 ; ( -3084/2467 , -13127/9868 ), ( 12736/6697 , -56051/26788 )
**u= 40/69 , ht= 16 ; ( 79/148 , -679/592 ), ( -7/16 , 57/64 )
**u= 40/73 , ht= 377 ; ( 14846/757 , -47567/3028 ), ( 134/377 , -103/116 )
**u= 40/77 , ht= 53 ; ( -2021/432 , 2347/576 ), ( -24/53 , 207/212 )
**u= 40/81 , ht= 337 ; ( -694/297 , -2267/972 ), ( 66/337 , -10945/12132 )
**u= 41/4 , ht= 39 ; ( -23/39 , 24/13 ), ( 699/469 , -1188/469 )
**u= 41/9 , ht= 166 ; ( -105/166 , -307/249 ), ( -262/225 , 698/675 )
**u= 41/12 , ht= 37 ; ( 5/37 , 50/37 ), ( -253/265 , -19/53 )
**u= 41/20 , ht= 923 ; ( 713/923 , -827/923 ), ( 1031/1161 , 544/1161 )
**u= 41/31 , ht= 73 ; ( 73/42 , -3/7 ), ( -42/73 , 18/73 )
**u= 41/39 , ht= 1249 ; ( -22843/5384 , -427/1346 ), ( 376/1249 , 252/1249 )
**u= 41/40 , ht= 9 ; ( 1/9 , 1 ), ( 9 , -9 )
**u= 41/44 , ht= 35 ; ( -35/13 , -16/13 ), ( -59/4143 , 1103/4143 )
**u= 41/56 , ht= 321 ; ( 2231/481 , -1349/481 ), ( 104/321 , 839/1284 )
**u= 41/60 , ht= 1879 ; ( 7151/1301 , 4665/1301 ), ( 1879/839 , 2016/839 )
**u= 41/63 , ht= 86 ; ( 605/74 , -205/37 ), ( 86/5 , -18 )
**u= 41/65 , ht= 227 ; ( 227/7 , -157/7 ), ( 331/396 , 443/396 )
**u= 41/76 , ht= 993 ; ( 6679/579 , 1817/193 ), ( -993/575 , 1182/575 )
**u= 41/80 , ht= 181 ; ( 181/11 , -14 ), ( 6083/1423 , 13717/2846 )
**u= 41/84 , ht= 2711 ; ( 10793/295 , -1908/59 ), ( 1111/2711 , -2702/2711 )
**u= 41/89 , ht= 149 ; ( 624 , -577 ), ( -149/29 , -173/29 )
**u= 41/96 , ht= 23 ; ( 23/2 , -181/16 ), ( 349/517 , -1303/1034 )
**u= 45/8 , ht= 91 ; ( 122/119 , 281/1428 ), ( 91/82 , 907/984 )
**u= 45/13 , ht= 17 ; ( 9/17 , 61/51 ), ( -654/173 , -2606/519 )
**u= 45/16 , ht= 37 ; ( 21/37 , -121/111 ), ( -117/101 , -287/303 )
**u= 45/19 , ht= 16 ; ( 3/16 , -7/6 ), ( -168/149 , -400/447 )
**u= 45/28 , ht= 53 ; ( 53/39 , 53/117 ), ( -39/53 , 1/3 )
**u= 45/29 , ht= 28 ; ( -19/77 , -239/231 ), ( 28 , -88/3 )
**u= 45/37 , ht= 24 ; ( 3/71 , 215/213 ), ( -13/24 , -23/72 )
**u= 45/44 , ht= 347 ; ( -347/71 , -145/213 ), ( 533/401 , -1589/1203 )
**u= 45/52 , ht= 27 ; ( 1751/87 , -2018/261 ), ( 1/27 , -31/81 )
**u= 45/53 , ht= 91 ; ( -863/91 , 155/39 ), ( 91/68 , -287/204 )
**u= 45/59 , ht= 103 ; ( 103/74 , -139/111 ), ( -94/743 , -1202/2229 )
**u= 45/61 , ht= 349 ; ( 403/24 , -169/18 ), ( 349/78 , 83/18 )
**u= 45/64 , ht= 413 ; ( -1363/279 , 10397/3348 ), ( 413/111 , -5171/1332 )
**u= 45/77 , ht= 703 ; ( -703/221 , 1735/663 ), ( 9734/1957 , 31594/5871 )
**u= 45/83 , ht= 667 ; ( -667/584 , 313/219 ), ( -1096/17 , 3592/51 )
**u= 45/91 , ht= 201 ; ( -201/58 , -281/87 ), ( -526/657 , 2470/1971 )
**u= 45/92 , ht= 839 ; ( -2573/49 , -6809/147 ), ( 839/103 , 2846/309 )
**u= 49 , ht= 40 ; ( 21/40 , 59/14 ), ( -40/21 , -1180/147 )
**u= 49/4 , ht= 819 ; ( -1571/1593 , 1853/3717 ), ( 819/803 , -3228/5621 )
**u= 49/9 , ht= 27 ; ( -14/27 , -823/567 ), ( -153/155 , 1087/3255 )
**u= 49/16 , ht= 163 ; ( -4225/4291 , -36457/60074 ), ( -163/161 , -1369/2254 )
**u= 49/25 , ht= 23 ; ( -96/77 , 887/2695 ), ( 23/16 , 757/560 )
**u= 49/31 , ht= 72 ; ( -41/72 , 41/42 ), ( -72/41 , -12/7 )
**u= 49/32 , ht= 27 ; ( 20/27 , -155/168 ), ( -27/20 , 279/224 )
**u= 49/36 , ht= 1 ; ( -1 , 6/7 ), ( 231/67 , 4903/1407 )
**u= 49/40 , ht= 3 ; ( 2/3 , -27/28 ), ( 3/2 , 81/56 )
**u= 49/55 , ht= 13 ; ( -13/2 , -17/7 ), ( -178/937 , 2558/6559 )
**u= 49/57 , ht= 89 ; ( -1975/113 , 5445/791 ), ( -89/70 , 131/98 )
**u= 49/60 , ht= 617 ; ( 617/167 , 2280/1169 ), ( -641/2419 , 8889/16933 )
**u= 49/64 , ht= 1 ; ( 1389/1127 , 37951/31556 ), ( -1 , 8/7 )
**u= 49/68 , ht= 183 ; ( 115/183 , -465/427 ), ( -33/475 , -387/665 )
**u= 49/81 , ht= 1 ; ( -181/258 , -9571/8127 ), ( -1 , -9/7 )
**u= 49/96 , ht= 262 ; ( -437/13 , 5211/182 ), ( -262/239 , -19889/13384 )
**u= 49/100 , ht= 1 ; ( 1 , -10/7 ), ( -47/149 , -4951/5215 )
**u= 52/33 , ht= 113 ; ( -1649/1453 , 2053/2906 ), ( -113/85 , 207/170 )
**u= 52/37 , ht= 799 ; ( -799/607 , -821/1214 ), ( 9157/4939 , -17921/9878 )
**u= 52/45 , ht= 29 ; ( -29/11 , -1/22 ), ( -121/279 , -355/1674 )
**u= 52/57 , ht= 329 ; ( 253/329 , 677/658 ), ( 217/331 , 479/662 )
**u= 52/61 , ht= 145 ; ( 145/23 , 125/46 ), ( -259/3645 , -593/1458 )
**u= 52/85 , ht= 343 ; ( -713/647 , 1709/1294 ), ( 343/207 , -785/414 )
**u= 52/93 , ht= 193 ; ( -55/193 , 427/386 ), ( 389/203 , -901/406 )
**u= 52/97 , ht= 18371 ; ( -93341/1191 , -50739/794 ), ( 8089/18371 , -34807/36742 )
**u= 53/8 , ht= 19 ; ( 13/19 , -26/19 ), ( 19/13 , 2 )
**u= 53/12 , ht= 101 ; ( 97/101 , -63/101 ), ( 19319/11993 , 23772/11993 )
**u= 53/20 , ht= 119 ; ( -19/119 , 145/119 ), ( 127/47 , -148/47 )
**u= 53/28 , ht= 9 ; ( -5/9 , -1 ), ( 3627/1483 , -3798/1483 )
**u= 53/29 , ht= 113 ; ( 205/263 , 235/263 ), ( 113/55 , 23/11 )
**u= 53/32 , ht= 71 ; ( 203/949 , 998/949 ), ( 71/10 , -601/80 )
**u= 53/37 , ht= 291 ; ( -227/291 , -89/97 ), ( 339/524 , -147/524 )
**u= 53/45 , ht= 7 ; ( 1853/3717 , 1571/1593 ), ( -7/2 , -7/2 )
**u= 53/48 , ht= 911 ; ( -1069/419 , 251/419 ), ( -353/911 , -207/911 )
**u= 53/68 , ht= 7441 ; ( 7441/1331 , -361/121 ), ( 51931/72297 , -64009/72297 )
**u= 53/72 , ht= 7 ; ( 797/1795 , -378/359 ), ( -7/2 , -29/8 )
**u= 53/75 , ht= 344 ; ( 1811/472 , -739/295 ), ( -344/227 , -1896/1135 )
**u= 53/77 , ht= 163 ; ( 145/163 , -191/163 ), ( -7393/8889 , 9419/8889 )
**u= 53/92 , ht= 229 ; ( -139/229 , -268/229 ), ( 2870677/212981 , -3090694/212981 )
**u= 53/93 , ht= 4067 ; ( 5702/6601 , 8361/6601 ), ( -4067/428 , -4397/428 )
**u= 53/100 , ht= 849 ; ( -401/849 , -330/283 ), ( 3447/1795 , -20337/8975 )
**u= 61/11 , ht= 1410 ; ( -167/1410 , 79/47 ), ( -3570/1051 , -5790/1051 )
**u= 61/21 , ht= 109 ; ( 200/179 , 29/179 ), ( 109/101 , 79/101 )
**u= 61/24 , ht= 404 ; ( -253/404 , -1653/1616 ), ( 1312/269 , 6259/1076 )
**u= 61/29 , ht= 351 ; ( -287/351 , 101/117 ), ( -1467/887 , 1461/887 )
**u= 61/40 , ht= 353 ; ( -1724/1091 , 311/4364 ), ( 223/353 , -8/353 )
**u= 61/45 , ht= 59 ; ( 117/67 , 65/201 ), ( 59/39 , -13/9 )
**u= 61/48 , ht= 685 ; ( 3175/1619 , -1015/3238 ), ( -337/685 , -23/274 )
**u= 61/52 , ht= 485 ; ( 485/277 , 200/277 ), ( -717/1055 , 117/211 )
**u= 61/56 , ht= 23 ; ( -8543/3993 , -1041/1331 ), ( -9/23 , 6/23 )
**u= 61/60 , ht= 20141 ; ( 5491/20141 , -1830/1831 ), ( 20141/5491 , 20130/5491 )
**u= 61/64 , ht= 95 ; ( -107/43 , -49/43 ), ( 95/69 , 385/276 )
**u= 61/69 , ht= 16 ; ( 16/7 , -9/7 ), ( 571/3170 , -1253/3170 )
**u= 61/75 , ht= 125 ; ( -125/94 , -277/235 ), ( -5590/1279 , -28398/6395 )
**u= 61/76 , ht= 653 ; ( 1611/91 , 764/91 ), ( -259/653 , -404/653 )
**u= 61/96 , ht= 97 ; ( 97/31 , -74/31 ), ( -31/97 , 74/97 )
**u= 61/99 , ht= 147 ; ( 147/92 , -107/69 ), ( -84916/27955 , -92012/27955 )
**u= 64 , ht= 1 ; ( -1 , 1/8 ), ( 1 , 1/8 )
**u= 64/9 , ht= 1 ; ( -755/741 , -1559/17784 ), ( -1 , 3/8 )
**u= 64/25 , ht= 1 ; ( 1 , -5/8 ), ( -1 , 5/8 )
**u= 64/45 , ht= 557 ; ( -1009/611 , -1401/4888 ), ( -433/557 , 2369/4456 )
**u= 64/49 , ht= 1 ; ( 1 , 7/8 ), ( -1 , -7/8 )
**u= 64/61 , ht= 179 ; ( 179/65 , -83/104 ), ( 65/179 , -415/1432 )
**u= 64/81 , ht= 1 ; ( 17/273 , -19937/19656 ), ( -1 , -9/8 )
**u= 64/85 , ht= 521 ; ( -7083/737 , 30977/5896 ), ( 11/521 , 2237/4168 )
**u= 65/7 , ht= 114 ; ( -37/114 , -39/19 ), ( 162/139 , -186/139 )
**u= 65/8 , ht= 79 ; ( -396/2713 , -21809/10852 ), ( -79/47 , 130/47 )
**u= 65/9 , ht= 83 ; ( 79/83 , 57/83 ), ( -397/269 , -569/269 )
**u= 65/16 , ht= 9 ; ( -7/9 , -1 ), ( -9/7 , -9/7 )
**u= 65/24 , ht= 166 ; ( 67/166 , 771/664 ), ( 661/757 , -46/757 )
**u= 65/33 , ht= 7 ; ( -11873/21014 , 1505/1501 ), ( 7/4 , 7/4 )
**u= 65/36 , ht= 1589 ; ( -873/1589 , 4745/4767 ), ( 9811/2307 , 31496/6921 )
**u= 65/41 , ht= 239 ; ( 157/239 , -227/239 ), ( -517/271 , -511/271 )
**u= 65/48 , ht= 703 ; ( -12377/6781 , 1951/13562 ), ( -421/703 , -365/1406 )
**u= 65/56 , ht= 2207 ; ( -2207/1584 , 163/192 ), ( -1584/2207 , 5379/8828 )
**u= 65/63 , ht= 7288 ; ( 1821/7288 , -2731/2733 ), ( 7288/1821 , 21848/5463 )
**u= 65/71 , ht= 109 ; ( 109/38 , 25/19 ), ( 12626/12413 , 13178/12413 )
**u= 65/72 , ht= 191 ; ( 97/191 , -194/191 ), ( 179/388 , -9/16 )
**u= 65/79 , ht= 4236 ; ( -4157/4236 , 388/353 ), ( -18996/18917 , -20928/18917 )
**u= 65/81 , ht= 73 ; ( -711/32 , -757/72 ), ( -64/73 , 220/219 )
**u= 65/88 , ht= 17 ; ( 19897/3441 , 3861/1147 ), ( 6/17 , -45/68 )
**u= 65/92 , ht= 5323 ; ( -5323/3831 , -1687/1277 ), ( 11487/79 , 11832/79 )
**u= 65/97 , ht= 2123 ; ( 2123/981 , -189/109 ), ( 2133/2426 , -2709/2426 )
**u= 68/5 , ht= 67 ; ( 47/67 , 251/134 ), ( 373/327 , -955/654 )
**u= 68/13 , ht= 937 ; ( -937/909 , -37/202 ), ( -45621/28583 , -119727/57166 )
**u= 68/49 , ht= 42325 ; ( -47765/28461 , 44965/132818 ), ( -25197/42325 , 3387/16930 )
**u= 68/53 , ht= 1505 ; ( 1861/1001 , 73/182 ), ( -869/1505 , -913/3010 )
**u= 68/57 , ht= 79 ; ( 539/235 , 27/94 ), ( 41/79 , -49/158 )
**u= 72/29 , ht= 19 ; ( -10/19 , -245/228 ), ( -31109/17450 , 78857/41880 )
**u= 72/37 , ht= 1257 ; ( -334/1257 , 16381/15084 ), ( -8238/5341 , 95279/64092 )
**u= 72/53 , ht= 283 ; ( -698/415 , -407/996 ), ( -230/283 , 2095/3396 )
**u= 72/65 , ht= 97 ; ( -2716/989 , -5723/11868 ), ( -32/97 , 1/12 )
**u= 72/73 , ht= 332 ; ( -332/33 , 613/396 ), ( 52/855 , 1357/10260 )
**u= 72/77 , ht= 142 ; ( -142/23 , -521/276 ), ( -1522/4099 , 22291/49188 )
**u= 72/85 , ht= 87 ; ( -4122/553 , 21275/6636 ), ( 2/87 , -427/1044 )
**u= 72/97 , ht= 585 ; ( 488/585 , -121/108 ), ( -585/488 , 7865/5856 )
**u= 73/7 , ht= 92 ; ( 71/92 , 34/23 ), ( -15916/7611 , -32152/7611 )
**u= 73/8 , ht= 81 ; ( -280/333 , 177/148 ), ( 81/76 , 261/304 )
**u= 73/17 , ht= 117 ; ( 53/117 , 53/39 ), ( -117/53 , -3 )
**u= 73/28 , ht= 27 ; ( -155/827 , -998/827 ), ( 27 , 33 )
**u= 73/33 , ht= 1357 ; ( 1427/2467 , -2513/2467 ), ( -1357/664 , -1437/664 )
**u= 73/40 , ht= 1063 ; ( -3809/4439 , -3757/4439 ), ( -1063/377 , 86/29 )
**u= 73/48 , ht= 715 ; ( 113/715 , -27/26 ), ( -2365/2867 , -1595/2867 )
**u= 73/55 , ht= 8 ; ( -3/8 , -1 ), ( -9288/4943 , 9096/4943 )
**u= 73/57 , ht= 47 ; ( 47/28 , -4/7 ), ( 1844/3725 , -192/3725 )
**u= 73/63 , ht= 25 ; ( 25/18 , 23/27 ), ( -134/251 , 94/251 )
**u= 73/72 , ht= 59 ; ( -59/13 , -11/13 ), ( 13/59 , -11/59 )
**u= 73/87 , ht= 43556 ; ( 51593/6668 , 5671/1667 ), ( 43556/785 , 43892/785 )
**u= 73/97 , ht= 3397 ; ( -10470/1061 , 5723/1061 ), ( -3397/1356 , 3541/1356 )
**u= 73/100 , ht= 175 ; ( 175/93 , -229/155 ), ( -395/1161 , 3857/5805 )
**u= 80/9 , ht= 221 ; ( -103/221 , 1665/884 ), ( 503/341 , -3171/1364 )
**u= 80/17 , ht= 1069 ; ( -287/1069 , 6481/4276 ), ( -9459/3767 , -54867/15068 )
**u= 80/21 , ht= 79 ; ( -43/979 , -5583/3916 ), ( -79/47 , -375/188 )
**u= 80/33 , ht= 269 ; ( 1663/1531 , -2947/6124 ), ( 247/269 , -469/1076 )
**u= 80/37 , ht= 41 ; ( 41/33 , 1/44 ), ( -1767/1841 , 4413/7364 )
**u= 80/41 , ht= 121 ; ( -43/121 , 47/44 ), ( -29843/14919 , -122441/59676 )
**u= 80/57 , ht= 1603 ; ( 7801/4837 , -7723/19348 ), ( 919/1603 , -303/6412 )
**u= 80/69 , ht= 223 ; ( 2399/1433 , -4421/5732 ), ( -191/223 , -687/892 )
**u= 80/77 , ht= 963 ; ( -963/269 , 769/1076 ), ( -269/963 , 769/3852 )
**u= 80/89 , ht= 247 ; ( 2377/1131 , -141/116 ), ( -247/109 , -1001/436 )
**u= 81 , ht= 13 ; ( 11/13 , -397/117 ), ( -44/5 , -2504/45 )
**u= 81/4 , ht= 1 ; ( 2629/2623 , 1319/23607 ), ( -1 , 2/9 )
**u= 81/16 , ht= 1 ; ( -1 , 4/9 ), ( -1 , -4/9 )
**u= 81/17 , ht= 49 ; ( 375/392 , -40/63 ), ( 49/15 , 133/27 )
**u= 81/20 , ht= 117 ; ( 1689/1589 , 95/14301 ), ( 117/7 , 1541/63 )
**u= 81/25 , ht= 1 ; ( -1 , -5/9 ), ( -169/32 , -9979/1440 )
**u= 81/40 , ht= 14 ; ( 14/11 , -5/36 ), ( 11/14 , 55/504 )
**u= 81/49 , ht= 1 ; ( 1 , -7/9 ), ( -1 , 7/9 )
**u= 81/55 , ht= 161 ; ( -161/106 , 191/477 ), ( -542/377 , 4586/3393 )
**u= 81/56 , ht= 11 ; ( 3/11 , -101/99 ), ( 75/107 , 370/963 )
**u= 81/64 , ht= 1 ; ( -1 , -8/9 ), ( 1 , 8/9 )
**u= 81/65 , ht= 36 ; ( 177/148 , -280/333 ), ( 36/29 , 304/261 )
**u= 81/76 , ht= 65 ; ( 91/23 , -13/207 ), ( 49/65 , 32/45 )
**u= 81/92 , ht= 45 ; ( 475/261 , 2810/2349 ), ( 1/45 , 29/81 )
**u= 81/100 , ht= 1 ; ( -1 , -10/9 ), ( 143/181 , 7511/8145 )
**u= 85/12 , ht= 217 ; ( -101/217 , -367/217 ), ( 2033/2051 , 573/2051 )
**u= 85/13 , ht= 3843 ; ( -1721/3843 , -301/183 ), ( 5481/1048 , -9849/1048 )
**u= 85/16 , ht= 13 ; ( -1/13 , -43/26 ), ( 503/521 , 20/521 )
**u= 85/19 , ht= 314 ; ( 1613/2966 , 1945/1483 ), ( 314/257 , -302/257 )
**u= 85/21 , ht= 13 ; ( -1/13 , -19/13 ), ( 214/193 , 166/193 )
**u= 85/28 , ht= 923 ; ( -491/1063 , 1255/1063 ), ( 923/881 , -619/881 )
**u= 85/32 , ht= 9 ; ( 1123/2019 , 726/673 ), ( -9/2 , -87/16 )
**u= 85/36 , ht= 11 ; ( -7/11 , 1 ), ( -12243/5741 , -39886/17223 )
**u= 85/37 , ht= 683 ; ( 2437/2336 , -347/584 ), ( 683/464 , 661/464 )
**u= 85/52 , ht= 269 ; ( 7119/6833 , -5150/6833 ), ( 269/243 , 226/243 )
**u= 85/64 , ht= 409 ; ( -183/409 , 811/818 ), ( -1277/2371 , -283/2371 )
**u= 85/72 , ht= 281 ; ( -281/223 , 193/223 ), ( 757/1494 , -5483/17928 )
**u= 85/77 , ht= 9 ; ( 2/9 , 1 ), ( -124623/31519 , 124533/31519 )
**u= 85/84 , ht= 3571 ; ( -1373/3571 , 3568/3571 ), ( -3571/1373 , 3568/1373 )
**u= 85/88 , ht= 723 ; ( 5289/383 , -1057/383 ), ( 19/723 , 136/723 )
**u= 85/93 , ht= 1889 ; ( 11581/3707 , 5087/3707 ), ( 1889/587 , 1901/587 )
**u= 85/96 , ht= 736 ; ( 5677/4 , 15863/32 ), ( -736/343 , 5987/2744 )
**u= 89 , ht= 213 ; ( -56/213 , -457/71 ), ( -471/83 , 3093/83 )
**u= 89/9 , ht= 28 ; ( -27/28 , -2/3 ), ( 28/27 , -56/81 )
**u= 89/15 , ht= 1334 ; ( 1303/1622 , 885/811 ), ( -1334/1301 , -742/1301 )
**u= 89/39 , ht= 8 ; ( 50833/92680 , 2411/2317 ), ( -8/5 , -8/5 )
**u= 89/41 , ht= 1928 ; ( -3281/2707 , 661/2707 ), ( -1928/493 , 2164/493 )
**u= 89/48 , ht= 1613 ; ( -605/1613 , -1695/1613 ), ( 6817/4205 , 2653/1682 )
**u= 89/56 , ht= 988 ; ( 4673/4967 , 4109/4967 ), ( 988/673 , 3721/2692 )
**u= 89/60 , ht= 1519 ; ( -27817/17063 , 1004/17063 ), ( 1519/419 , -1555/419 )
**u= 89/71 , ht= 239 ; ( -239/116 , -7/29 ), ( 5028/1975 , -5004/1975 )
**u= 89/80 , ht= 13 ; ( 15121/34749 , -1769/1782 ), ( 13/3 , 13/3 )
**u= 89/84 , ht= 1163 ; ( -1163/293 , -88/293 ), ( 30559/68065 , 25834/68065 )
**u= 89/96 , ht= 2779 ; ( -2779/146 , -6231/1168 ), ( -3967/1289 , 7977/2578 )
**u= 97/7 , ht= 654 ; ( -647/654 , -51/109 ), ( -1426/929 , -2866/929 )
**u= 97/12 , ht= 59 ; ( -14401/21263 , 32092/21263 ), ( -59/37 , -94/37 )
**u= 97/20 , ht= 3231 ; ( 7897/7947 , -1285/2649 ), ( -3231/2891 , -2643/2891 )
**u= 97/28 , ht= 281 ; ( -281/265 , -16/53 ), ( 265/281 , -80/281 )
**u= 97/52 , ht= 2279 ; ( 745/2331 , -827/777 ), ( 2279/1369 , 2234/1369 )
**u= 97/60 , ht= 5411 ; ( -3499/6071 , -5923/6071 ), ( 5261/5411 , 4040/5411 )
**u= 97/63 , ht= 148 ; ( -139/148 , 31/37 ), ( 284/445 , 12/445 )
**u= 97/65 , ht= 9 ; ( 4/9 , -1 ), ( 9/4 , -9/4 )
**u= 97/72 , ht= 13 ; ( -5/13 , 1 ), ( -3185/36 , 39065/432 )
**u= 97/73 , ht= 142 ; ( -212/299 , -283/299 ), ( -77/142 , 19/142 )
**u= 97/88 , ht= 1493 ; ( -619/1493 , -1484/1493 ), ( -2217/6385 , -981/6385 )
**u= 100 , ht= 31 ; ( -1/31 , -2191/310 ), ( 31 , -2191/10 )
**u= 100/9 , ht= 1 ; ( -19801/19839 , -198371/595170 ), ( -1 , -3/10 )
**u= 100/13 , ht= 47 ; ( -61/95 , 1457/950 ), ( 47/21 , 167/42 )
**u= 100/17 , ht= 739 ; ( 739/723 , -99/482 ), ( 2127/665 , -35253/6650 )
**u= 100/37 , ht= 795 ; ( -721/795 , -2013/2650 ), ( -1257/1405 , -3537/14050 )
**u= 100/49 , ht= 1 ; ( 681/911 , -58171/63770 ), ( -1 , -7/10 )
**u= 100/53 , ht= 195 ; ( 17/195 , 713/650 ), ( -393/349 , 645/698 )
**u= 100/57 , ht= 797 ; ( 2095/1639 , 7231/16390 ), ( -571/797 , 105/1594 )
**u= 100/73 , ht= 509 ; ( 301/885 , 2969/2950 ), ( -363/509 , 471/1018 )
**u= 100/81 , ht= 1 ; ( -1 , 9/10 ), ( -1 , -9/10 )
**u= 100/93 , ht= 95 ; ( -95/29 , 137/290 ), ( -305/811 , 2133/8110 )
432
>

■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。

例えば、n=1,u=1/4のときの計算は以下のようになる。
(i)２次曲線(3a)の有理点(1,-2)より、(3a)の有理点(xk(k),yk(k))を求めると、
    xk(k)=(-8*k^2-32*k-15)/(-8*k^2+15)
となる。 [Pari/GPによる計算]
gp > uxy(1,1/4,1,-2)
(3a+) YY2(x)=15/8*x^2 + 17/8
%15 = (-8*k^2 - 32*k - 15)/(-8*k^2 + 15)
gp > xk(k)=(-8*k^2 - 32*k - 15)/(-8*k^2 + 15)
%16 = (k)->(-8*k^2-32*k-15)/(-8*k^2+15)
(ii)２次曲線(3b)上に、有理点(xk(k),tk(k))が存在するので、楕円曲線
    E: Y^2=256*X^4 + 1088*X^3 + 2236*X^2 + 2040*X + 900
を得る。
[Pari/GPによる計算]
gp > PP4(TT2(1, u0, xk(x)))
%17 = 256*x^4 + 1088*x^3 + 2236*x^2 + 2040*x + 900
(iii)楕円曲線Eを2-descentの結果と見なして、MAGMAで-descentを実行すると、成功する。
4-descentの結果から、Eのminimal standard model
    E2: y^2 = x^3 + x^2 - 23040*x - 1089612
の有理点を計算すると、E2の有理点
P(-7433076/66049, 5110591170/16974593); height 11.1518978623326526359851761363
が求まる。
[MAGMAによる計算]
> fd:=C0(256*x^4 + 1088*x^3 + 2236*x^2 + 2040*x + 900);
Time: 1.656
2
> fd;
[
    Curve over Rational Field defined by
    $.1*$.4 + 8*$.2*$.3 - $.2*$.4 + $.4^2,
    2*$.1*$.2 + 2*$.1*$.3 + 2*$.2*$.4 + $.3^2 - 14*$.3*$.4 - 2*$.4^2,
    Curve over Rational Field defined by
    2*$.1*$.2 + 3*$.1*$.3 + 3*$.1*$.4 + 2*$.2*$.3 - 2*$.2*$.4 + $.3^2 - $.4^2,
    -5*$.1*$.3 + 3*$.1*$.4 + $.2^2 + 2*$.2*$.3 + 6*$.2*$.4 + 2*$.3*$.4 - $.4^2
]
> RP4(fd,10^2);
J=1
Elliptic Curve defined by y^2 = x^3 + x^2 - 23040*x - 1089612 over Rational Field
rootno=-1
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-2549211301295770854036/22286547681928698529 : -26265584217822723268141244995230/105211738861943558208070593167 : 1)
height 44.6075914493306105439407045454
true (-2549211301295770854036/22286547681928698529 : 26265584217822723268141244995230/105211738861943558208070593167 :
1)
(-1417987018543371498516/24942003744727758049 : -24828060474509765738948895541470/124565280450502161278135959343 : 1)
height 44.6075914493306105439407045454
true (-1417987018543371498516/24942003744727758049 : 24828060474509765738948895541470/124565280450502161278135959343 :
1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-2549211301295770854036/22286547681928698529 : -26265584217822723268141244995230/105211738861943558208070593167 : 1)
height 44.6075914493306105439407045454
true (-2549211301295770854036/22286547681928698529 : 26265584217822723268141244995230/105211738861943558208070593167 :
1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
(-84 : 510 : 1)
height 0.000000000000000000000000000000
true (-84 : 510 : 1)
J=2
Elliptic Curve defined by y^2 = x^3 + x^2 - 23040*x - 1089612 over Rational Field
rootno=-1
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-4284084/73441 : 4846575390/19902511 : 1)
height 11.1518978623326526359851761363
true (-4284084/73441 : 4846575390/19902511 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
(-7433076/66049 : 5110591170/16974593 : 1)
height 11.1518978623326526359851761363
true (-7433076/66049 : 5110591170/16974593 : 1)
realtime=35.668
2
>
(iv)有理点Pから、Eのsyzygyによる楕円曲線
    E0: y-2=x^3 - 29860272*x - 50478615936
の有理点
    Q(-266798148/66049, 1103887692720/16974593)
を求める。
gp > v=V(256*x^4 + 1088*x^3 + 2236*x^2 + 2040*x + 900)
%4 = [256, 1088, 2236, 2040, 900]
gp > Q=chpi(v,-7433076/66049)
v=[256, 1088, 2236, 2040, 900]
I=1105936
J=1869578368
rr=[6, 12, 0, 0]
v=[256, 1088, 2236, 2040, 900]
I=1105936
J=1869578368
%5 = [-266798148/66049, 1103887692720/16974593]
(v)有理点Qのm倍点から、Eの有理点のX座標(つまりk)をいくつか求める。
    k=-11/16, -8/21, -30/11, -315/64, -108065151403/55574463248, -104202118590/108065151403, 111928184216/3083775093, 46256626395/895425473728, 1206897228315835392955786994453/572360750632430207104549770512, 1073176407435806638321030819710/1206897228315835392955786994453
[Pari/GPによる計算]
gp > L=ss1x(v,Q,20);
v=[256, 1088, 2236, 2040, 900]
I=1105936
J=1869578368
time = 39 ms.
gp > L[1..10]
%7 = [-11/16, -8/21, -30/11, -315/64, -108065151403/55574463248, -104202118590/108065151403, 111928184216/3083775093, 46256626395/895425473728, 1206897228315835392955786994453/572360750632430207104549770512, 1073176407435806638321030819710/1206897228315835392955786994453]
(vi)Eの有理点のX座標から、(1)の整数点をいくつか求める。
[Pari/GPによる計算]
(14:47) gp > sss4(1, u0, L)
[u0,x0,y0]=[1/4, 1, -2]
103^4+542^4=359^4+514^4
6553597382445311099047^4+12416872071510589133918^4=5887100137809620288089^4+12500065387625324926786^4
4833967387977622932389399227844307124630957930685120267040473^4+6732117210931278917381556685913800620634033429405725224458718^4=842356865595472662646264276427110306934014002763539306377689^4+7140435634222255566117756314157630141431697165300873682821826^4
59698737512122790297963709810886750778921255332552835146120211940301408913131090158260118586246294842199924294187709977^4+89081013148941668894102352104773954246602473482616250007545303509390166547550588492970414780572491253833594863260012702^4=24283044582122941762196014074848286291872408768227495399332723977338641045880703955312375025999852876370068449102986521^4+93161124699269867282795493645819438408583017261969555477763501141900017175679240598593290469963801485128508446179910274^4
11683673854706312810689922652427322476000750588061567959202237680744299764473585158307521479532752
... 省略 ...
18453547021038601484030664443230917377332299768981228151980715173776558490260235100806153417563361979725380860509454729269427385763523776256747941941805930923634743153421776362146865549638162105425909606565724148642691050700973122935183800298802543743518431837126880497269337413625653312290967689258105354396410152443150003775021676591518871738510160691870754887477866534454586386804962153076341362275410226975782928687771176144276513574377392204955215271748089286703428022739499075440108347786960642441664131700055028696928171442329654777698233670032502045958134535682853336347465709381568834332623612987394056031752637874618375945198950677677961271048119844676245237837386880378482732194626540496430428528570420276816059535515103022692023217218746106865866222538063186511505718483784422205535564354326034519532240618781043970517582943226595452304196941807966917380124222553^4+31266652649554881791050427206222325115352440565326582141937793069658134649693911224047261072938029746076451975496731585095372292195879676550582775158875981595273318961864506243956499180061231990302529043866916121479588220927177389657463550197410767062307519864036933061766501030144604030006682212279138601322674940441224481193465426687223289763178892059396626876992018938984865160661580679331814904772673369730157049251026603183274568349906549215365486357717690747141822578530983682067331770831098230104652167712101532885975181189280938672675202177862246276816185293742629995418403777442479249743299229130265583578486346809907890224619573293391287255928928590513213326379632450375387651225515726582942315836412055609657124073334160600478601583436056149727447317231435831580634032375849272379808233246040342586332584965166705196837518470242354318532703247683530978822695108958^4=12632422287389975869113811573342141195105606328070679770140467629566736427060346627057679441614143876160581211463951552757882919600644864083565024412066124841469278564647605570733562238547834618728876536167537832395104708457345167040598980763092799934148952602598671742579696228330318902273899794269555403843115963482592253983896606378831946842496306776150887391332256127738970682037698243254019597548220848644033480535365671914235426354618929668838591659458060581230841735723316499023630627970370486418319242211597591285190751296089536731127111764771814209071227582788263765708372154299162243792324807270742622605518106429238475589244675936095642464423567176423378775854137024271474196333563678655824827833094957033718786832134672107545692972423772875809712665506065241690752678782830896434187424623146231336086570870393309901362763506218948393521905937731714312495809099609^4+31981907293196277146941207901163303286349736752312900323039823099265592331090641646990431461571039138364231808776157164498908282015438374448292630230377662093628986620986054121770783007186959322012301388030492279099593513935454183102337700466138232316113798857229857723220574321939571204171725401518782336856910638562420059711094269913928215274851937697117191348163100579627997324648322660486749084831389624876128314247573094278813317797400878088640831917438343753539148231283813416790515724760404775664540413111405105534148353056400254415760661470609162015941652171717262900468868861975227814288040834863719505399728785184226980240244550919659258989361506525738541873145595661786494568042378671821682101549748832145570376052649300290336236919972157870628081321788419929245989564187708745774404797493855753989496101302482922081453014192067598873594478599898861869034761144386^4
time = 5 ms.
%8 = 10

他のuについても、同様に計算できるので、ここからは、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。

[2026.05.17追記]u=1/4のときの整数解を求める計算手墳を追加した。

[参考文献]

Last Update: 2026.06.16
H.Nakao

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