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Integer Points on A^4+20021*B^4=C^4+20021*D^4

[2026.06.15]A^4+20021*B^4=C^4+20021*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=20021とする。

Pari/GPで簡単なプログラム("a45s.gp")を作成して、max{A,B,C,D} <= 100000の範囲で、小さい整数解を探すと、2個の整数解
    1427^4+20021*1042^4=9709^4+20021*926^4
    48591^4+20021*617^4=48597^4+20021*293^4
が見つかった。
これらの整数解から、(3a),(3b)を満たす有理数uを求めると、
    u = 404, 89/45
となる。
{gp2cによる計算]
gp > ss(20021,100000)
1:1427^4+20021*1042^4=9709^4+20021*926^4
2:48591^4+20021*617^4=48597^4+20021*293^4
2solutions,
time = 11h, 36min, 1,220 ms.
%1 = 2
{Pari/GPによる計算]
gp > uu(20021,1427,1042,9709,926)
[u - 404, 1; u, -1; 114144*u + 48150505, 1]
[u - 404, 1; u, -1; 114144*u - 48150505, 1]
[u, -1; u + 404, 1; 114144*u + 48150505, 1]
time = 1 ms.
%1 = [404, -48150505/114144]
gp > uu(20021,48591,617,48597,293)
[u, -1; 45*u - 89, 1; 3276*u + 52475041, 1]
[u, -1; 45*u - 89, 1; 3276*u - 52475041, 1]
[u, -1; 45*u + 89, 1; 3276*u + 52475041, 1]
time = 1 ms.
%2 = [89/45, -52475041/3276]

■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200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528個のuが抽出される。
[MAGMAによる計算]
> PP(20021,1,200);
**u= 1/4 , ht= 159 ; ( -159/19 , 32043/19 ), ( 91/795 , -53374/265 )
**u= 1/12 , ht= 3847 ; ( 50875/13787 , -18268870/13787 ), ( -3847/85 , -266733/17 )
**u= 1/24 , ht= 391 ; ( 391/271 , -233183/271 ), ( -2158/2927 , 7129839/11708 )
**u= 1/44 , ht= 4069 ; ( 204373/129543 , -160589299/129543 ), ( 23/4069 , 207733/313 )
**u= 1/80 , ht= 20659 ; ( -20659/1079 , 37025737/2158 ), ( 1061237/255287 , -1953578375/510574 )
**u= 1/140 , ht= 415883 ; ( -694509/108749 , -832203798/108749 ), ( -415883/131487 , -172119521/43829 )
**u= 1/141 , ht= 23675 ; ( -3397/23675 , -1136613/947 ), ( -685075/472817 , 988934675/472817 )
**u= 1/144 , ht= 28375 ; ( 5017/28377 , 103795958/85131 ), ( 5019/28375 , 207580237/170250 )
**u= 1/149 , ht= 246931 ; ( 246931/10243 , 301835249/10243 ), ( 16531/346119 , -141065229/115373 )
**u= 1/165 , ht= 46687 ; ( -46637/46687 , -84810235/46687 ), ( -46687/46637 , 84810235/46637 )
**u= 1/176 , ht= 105657 ; ( 855375/237967 , -1178497098/237967 ), ( 101609/105657 , 64857297/35219 )
**u= 2/17 , ht= 31981 ; ( 28836/31981 , 25122177/63962 ), ( -18163/40824 , 965467/3024 )
**u= 2/37 , ht= 112323 ; ( -269701/6446 , -232193617/12892 ), ( 12182/112323 , 32413837/74882 )
**u= 2/85 , ht= 37666 ; ( -37666/22859 , 57477035/45718 ), ( 22859/37666 , -57477035/75332 )
**u= 4/5 , ht= 799 ; ( 799/81 , -179669/162 ), ( 289/1639 , 372335/3278 )
**u= 4/9 , ht= 323 ; ( -323/161 , -108327/322 ), ( 161/323 , -108327/646 )
**u= 4/25 , ht= 4981 ; ( -369/4981 , 2498637/9962 ), ( -377/4989 , 4171539/16630 )
**u= 4/41 , ht= 2771 ; ( -1657/2771 , 2068421/5542 ), ( -43351/35779 , -36010181/71558 )
**u= 4/65 , ht= 218741 ; ( 218741/181249 , 229149103/362498 ), ( 247939/35649 , -67352159/23766 )
**u= 4/69 , ht= 86957 ; ( 86957/5227 , 72400261/10454 ), ( -1199399/83311 , 999220457/166622 )
**u= 4/81 , ht= 2789 ; ( -1655/2789 , 2920301/5578 ), ( 13929/8275 , -131302049/148950 )
**u= 4/89 , ht= 2075 ; ( 52589/201575 , -39326761/80630 ), ( -2075/1007 , -2177035/2014 )
**u= 4/177 , ht= 3713 ; ( -23113105/2378317 , 30928575685/4756634 ), ( 3713/3595 , -1375893/1438 )
**u= 5/4 , ht= 1133 ; ( 19/1133 , -101410/1133 ), ( -3989/917 , 366298/917 )
**u= 5/24 , ht= 2803 ; ( 149/2803 , 615297/2803 ), ( 14044/12223 , 16324699/48892 )
**u= 5/36 , ht= 7369 ; ( 15927/31909 , -28723283/95727 ), ( -7369/1127 , -2001352/1127 )
**u= 5/49 , ht= 907 ; ( -907/99 , 181855/63 ), ( -74371/4363 , -163338571/30541 )
**u= 5/57 , ht= 192353 ; ( -1620017/208771 , 551793311/208771 ), ( 166589/192353 , 85961791/192353 )
**u= 5/97 , ht= 233869 ; ( -21209421/1416023 , -9367491303/1416023 ), ( 40253/233869 , -104578117/233869 )
**u= 5/124 , ht= 478407 ; ( 550339/422627 , 345736970/422627 ), ( 327469/478407 , -96288111/159469 )
**u= 5/153 , ht= 170419 ; ( 170419/93613 , 107614087/93613 ), ( -2731817/65941 , -1512400099/65941 )
**u= 5/172 , ht= 741541 ; ( 741541/325817 , 475304783/325817 ), ( -3212049/143327 , -1886779756/143327 )
**u= 5/192 , ht= 1087 ; ( 1087/949 , 3578587/3796 ), ( 6300673/6786961 , -11483366355/13573922 )
**u= 9/4 , ht= 443 ; ( -443/117 , 7052/27 ), ( 117/443 , 91676/1329 )
**u= 9/8 , ht= 535 ; ( -535/529 , 212915/1587 ), ( 3001/940 , 711973/2256 )
**u= 9/92 , ht= 138203 ; ( 644405/88789 , -624258410/266367 ), ( -138203/107935 , -33656939/64761 )
**u= 9/100 , ht= 12461 ; ( -11523/12461 , 84905612/186915 ), ( 28853/13509 , 159378064/202635 )
**u= 9/104 , ht= 35173 ; ( -50365/69989 , -87980777/209967 ), ( 35173/14269 , -38729159/42807 )
**u= 9/109 , ht= 175797 ; ( -175797/27161 , 185812583/81483 ), ( 314463/101477 , 345161071/304431 )
**u= 9/133 , ht= 156045 ; ( -156045/109537 , -219986645/328611 ), ( -59581/361155 , 84471115/216693 )
**u= 9/140 , ht= 165301 ; ( 19841/165301 , -197094073/495903 ), ( 743593/22523 , 880696703/67569 )
**u= 9/164 , ht= 618341 ; ( 618341/394399 , 939721187/1183197 ), ( 4973993/2327443 , -7036362391/6982329 )
**u= 9/181 , ht= 5445 ; ( -2881/5445 , -1658413/3267 ), ( 18983585/10300653 , -29072586995/30901959 )
**u= 9/196 , ht= 30323 ; ( 2077935/55087 , -20381608466/1156827 ), ( 30323/24897 , 384699781/522837 )
**u= 10/21 , ht= 1108 ; ( 1108/779 , 392759/1558 ), ( -27256/32363 , 12269429/64726 )
**u= 10/29 , ht= 34412 ; ( -34412/9291 , -12146315/18582 ), ( 23032/38489 , 15284699/76978 )
**u= 10/49 , ht= 1169 ; ( 366/15017 , 6653781/30034 ), ( 1169/522 , 1323209/2436 )
**u= 10/101 , ht= 2373337 ; ( -7392917/421756 , -4709121125/843512 ), ( 829956/2373337 , -1598933719/4746674 )
**u= 10/173 , ht= 179809 ; ( -37652/179809 , 152901271/359618 ), ( -179809/37652 , -152901271/75304 )
**u= 13/17 , ht= 5705 ; ( -3767/5705 , 156439/1141 ), ( 641945/91417 , 74189795/91417 )
**u= 13/44 , ht= 105745 ; ( -105745/21569 , 19865200/21569 ), ( -21569/105745 , -3973040/21149 )
**u= 13/56 , ht= 84261 ; ( 387675/132434 , 340287045/529736 ), ( -84261/7565 , -3513587/1513 )
**u= 13/81 , ht= 44465 ; ( 44465/26339 , -38721035/79017 ), ( -59083/69365 , 13653665/41619 )
**u= 13/137 , ht= 9393 ; ( 1447/9393 , 3086839/9393 ), ( 7263/11095 , 4307129/11095 )
**u= 13/140 , ht= 25171 ; ( 129971/36801 , 44351999/36801 ), ( 25171/41 , -8264591/41 )
**u= 13/144 , ht= 23125 ; ( 23125/9519 , 49964275/57114 ), ( 2337823/78875 , 62314081/6310 )
**u= 13/177 , ht= 2861701 ; ( 3139315/901327 , -1205807459/901327 ), ( 2861701/996947 , -1118769863/996947 )
**u= 13/180 , ht= 83977 ; ( -261789/657721 , -790658837/1973163 ), ( -83977/15693 , 95417449/47079 )
**u= 13/192 , ht= 227 ; ( 227/35 , -17663/7 ), ( 168067/52531 , -135412853/105062 )
**u= 16/125 , ht= 43623 ; ( -25783/43623 , 2163497/6660 ), ( 43623/25783 , -283418107/515660 )
**u= 16/161 , ht= 32965 ; ( 32965/25513 , -52919563/102052 ), ( -3503961833/546099933 , 1500688242349/728133244 )
**u= 16/185 , ht= 7027 ; ( 6953/7027 , -13452779/28108 ), ( 387791/210931 , -600745409/843724 )
**u= 17/22 , ht= 14801 ; ( -14801/7320 , -751753/2928 ), ( -7320/14801 , 3758765/29602 )
**u= 17/78 , ht= 164735 ; ( 164735/105301 , -41901492/105301 ), ( 1034935/295547 , -230668160/295547 )
**u= 17/190 , ht= 3680873 ; ( -3680873/656162 , 2501232197/1312324 ), ( -4057293/179897 , -1358448464/179897 )
**u= 17/198 , ht= 613028 ; ( -1114407/1350011 , -1793214866/4050033 ), ( -613028/34355 , 419302373/68710 )
**u= 18/13 , ht= 2078 ; ( 74146/2255 , -7568543/2706 ), ( 2078/947 , -1165067/5682 )
**u= 18/41 , ht= 36167 ; ( 45200/25197 , -46884925/151182 ), ( -36167/28500 , -1668761/6840 )
**u= 18/85 , ht= 155821 ; ( -155821/87466 , 233106751/524796 ), ( -218579/205826 , -391664179/1234956 )
**u= 18/133 , ht= 17359 ; ( 29630/144941 , 241407301/869646 ), ( 4838/17359 , 29406083/104154 )
**u= 18/193 , ht= 101669 ; ( -101669/6604 , -200273861/39624 ), ( 737936/1343377 , -3012884809/8060262 )
**u= 18/197 , ht= 161139 ; ( 1479591/1211474 , 3797782357/7268844 ), ( -161139/101462 , -378174133/608772 )
**u= 20/21 , ht= 4873 ; ( 70013/15511 , -14703713/31022 ), ( 491/4873 , -1004229/9746 )
**u= 20/33 , ht= 20003 ; ( -28163/69869 , -19363293/139738 ), ( -20003/7739 , -5512993/15478 )
**u= 20/37 , ht= 227939 ; ( 1326693/2791 , 361087101/5582 ), ( 227939/647 , 62039359/1294 )
**u= 20/89 , ht= 7299 ; ( -7299/1067 , 3113811/2134 ), ( 1938133/593739 , -95073143/131942 )
**u= 20/97 , ht= 21997 ; ( -8461/21997 , -10386139/43994 ), ( -1753/35689 , -15746569/71378 )
**u= 20/109 , ht= 33983 ; ( 1664347/473849 , -808395817/947698 ), ( 33983/31149 , -7178357/20766 )
**u= 20/117 , ht= 38407 ; ( -4485373/25679 , 2170906797/51358 ), ( 38407/6141 , -56474123/36846 )
**u= 20/177 , ht= 29801 ; ( -15808943/752239 , -9421570687/1504478 ), ( -25423/29801 , 23318619/59602 )
**u= 20/181 , ht= 553063 ; ( -733323/195391 , -65263857/55826 ), ( 553063/64509 , -111730045/43006 )
**u= 25/12 , ht= 635 ; ( -635/151 , -226201/755 ), ( 7541/13313 , -1060541/13313 )
**u= 25/29 , ht= 11323 ; ( 10337/11323 , 8260771/56615 ), ( 94177/8917 , -50970359/44585 )
**u= 25/48 , ht= 38267 ; ( -102775/9401 , 71538927/47005 ), ( 38267/11519 , 11080779/23038 )
**u= 25/77 , ht= 484903 ; ( -484903/38235 , 427044283/191175 ), ( 2591931/364783 , -459607183/364783 )
**u= 25/84 , ht= 22457 ; ( 564197/106607 , 526520291/533035 ), ( 22457/8777 , -22109964/43885 )
**u= 25/92 , ht= 15175 ; ( 170273/31215 , -166128692/156075 ), ( 14531/15175 , -20162899/75875 )
**u= 25/116 , ht= 92043 ; ( 1089621/164347 , -237490644/164347 ), ( -92043/1273 , -99194977/6365 )
**u= 25/144 , ht= 311 ; ( 380881/43759 , -460304898/218795 ), ( 59/311 , -760111/3110 )
**u= 25/149 , ht= 1388341 ; ( -1388341/71859 , 1697844143/359295 ), ( -5068361/322459 , 6202489627/1612295 )
**u= 25/152 , ht= 19219 ; ( 91179/5930 , -450838251/118600 ), ( -19219/14925 , -10005422/24875 )
**u= 25/168 , ht= 42101 ; ( -58813/36709 , -17981577/36709 ), ( -42101/13675 , -57405742/68375 )
**u= 25/188 , ht= 100065 ; ( -84269/100065 , -179467583/500325 ), ( 104923/164895 , 89374287/274825 )
**u= 25/189 , ht= 2993233 ; ( -2993233/2823431 , -1131964283/2823431 ), ( -4264363/583823 , -17760976823/8757345 )
**u= 26/21 , ht= 1649 ; ( -96073/24152 , 17814541/48304 ), ( -1649/1180 , -364663/2360 )
**u= 26/25 , ht= 1558 ; ( 15503/1322 , -3052943/2644 ), ( 1381/1558 , 2042587/15580 )
**u= 26/49 , ht= 2118 ; ( -2118/83 , 4075899/1162 ), ( -194/3775 , 1038379/7550 )
**u= 26/69 , ht= 456244 ; ( 456244/452081 , 209375841/904162 ), ( -1190528/852769 , -477382563/1705538 )
**u= 26/141 , ht= 24407 ; ( -24407/19660 , 2920885/7864 ), ( 19660/24407 , -14604425/48814 )
**u= 26/181 , ht= 55112 ; ( -8753051/408264 , -4626383447/816528 ), ( 44433/55112 , 37376591/110224 )
**u= 26/185 , ht= 4362658 ; ( -5032509/1980194 , 2886688461/3960388 ), ( 4362658/1275513 , 808720019/850342 )
**u= 29/30 , ht= 2056 ; ( 530837/1254752 , -277292319/2509504 ), ( -2056/1399 , -506139/2798 )
**u= 29/66 , ht= 33631 ; ( 110488/177533 , -63124723/355066 ), ( 33631/3733 , -5107422/3733 )
**u= 29/130 , ht= 351968 ; ( 351968/231393 , 178458305/462786 ), ( -1041257/1966228 , -942637343/3932456 )
**u= 29/150 , ht= 60509 ; ( 459991/195718 , 1137509357/1957180 ), ( 17638/60509 , 143417697/605090 )
**u= 29/166 , ht= 656617 ; ( 656617/199619 , 164282072/199619 ), ( 4696691/1472639 , 1178251898/1472639 )
**u= 29/190 , ht= 70133 ; ( 70133/52303 , -22405606/52303 ), ( 945428/8277 , 161420907/5518 )
**u= 36 , ht= 29 ; ( -29/15 , -641/18 ), ( 237/289 , -37157/1734 )
**u= 36/5 , ht= 37 ; ( 37/13 , 8765/78 ), ( 13/37 , 8765/222 )
**u= 36/49 , ht= 20715 ; ( -42109/7695 , -41971681/64638 ), ( -20715/3049 , 102652285/128058 )
**u= 36/65 , ht= 2633 ; ( -2633/253 , 2133673/1518 ), ( 253/2633 , -2133673/15798 )
**u= 36/121 , ht= 23089 ; ( 7515/23089 , 293957369/1523874 ), ( -127149/60505 , 1704709121/3993330 )
**u= 36/161 , ht= 479363 ; ( 162581/630769 , 826949321/3784614 ), ( 479041/479363 , 860348621/2876178 )
**u= 36/169 , ht= 37429 ; ( 364981/88977 , -907450121/991458 ), ( -37429/34833 , -864549119/2716974 )
**u= 37/8 , ht= 6715 ; ( -4622/6715 , 303463/5372 ), ( 6835/1133 , 322490/1133 )
**u= 37/17 , ht= 5733 ; ( -12241/5475 , 181871/1095 ), ( -5733/4909 , 511877/4909 )
**u= 37/48 , ht= 8833 ; ( -4861/8833 , 208903/1606 ), ( -1580623/280895 , 182950559/280895 )
**u= 37/65 , ht= 42157 ; ( -42157/21803 , -6293933/21803 ), ( -350461/336719 , -64450483/336719 )
**u= 37/72 , ht= 14443 ; ( 39987/5203 , -16884034/15609 ), ( 14443/4362 , 25269059/52344 )
**u= 37/105 , ht= 164201 ; ( -164201/23539 , 27958473/23539 ), ( 19125331/1746991 , -3236952089/1746991 )
**u= 37/113 , ht= 673049 ; ( -3005817/6053393 , -1181742561/6053393 ), ( 673049/227109 , -41400741/75703 )
**u= 37/132 , ht= 2291951 ; ( -507835/2291951 , -443636669/2291951 ), ( -5467613/611053 , 1039699792/611053 )
**u= 37/144 , ht= 38845 ; ( -38845/11847 , -48095765/71082 ), ( -48631/34295 , -2349140/6859 )
**u= 37/156 , ht= 80021 ; ( 467165/94591 , -13988975/13513 ), ( 80021/15295 , -3347461/3059 )
**u= 37/177 , ht= 1248181 ; ( -1145567/1248181 , 370745421/1248181 ), ( 4456841/4312055 , 1357071671/4312055 )
**u= 37/188 , ht= 37638689 ; ( -1060410763/8486361 , -239162762777/8486361 ), ( -37638689/7442113 , 8653035698/7442113 )
**u= 41/6 , ht= 436 ; ( -2417/1253 , -14876/179 ), ( 436/217 , 37307/434 )
**u= 41/30 , ht= 511 ; ( 15737/11827 , -1684784/11827 ), ( -34/511 , -87657/1022 )
**u= 41/34 , ht= 138 ; ( 138/53 , -26937/106 ), ( 20091/25265 , 2941028/25265 )
**u= 41/74 , ht= 105407 ; ( 79371/105407 , 17736054/105407 ), ( 1763689/336579 , -80449506/112193 )
**u= 41/90 , ht= 440039 ; ( 547272/84947 , 492582785/509682 ), ( -440039/67776 , -395998013/406656 )
**u= 41/126 , ht= 30937 ; ( 48691/635921 , -111865164/635921 ), ( 17546/30937 , -12476393/61874 )
**u= 41/150 , ht= 173752 ; ( 78979/173752 , 9871763/46960 ), ( -2119804/313493 , -4100870543/3134930 )
**u= 45/52 , ht= 9329 ; ( 9329/5117 , -490447/2193 ), ( -5117/9329 , 180691/1473 )
**u= 45/73 , ht= 22193 ; ( -8832163/1185481 , -3406784677/3556443 ), ( 22193/13141 , -9860237/39423 )
**u= 45/89 , ht= 104697 ; ( -104697/10291 , 44407601/30873 ), ( 276907/2765169 , -1173069127/8295507 )
**u= 45/112 , ht= 29111 ; ( 113773/59551 , 121618417/357306 ), ( -29111/10393 , 14637287/31179 )
**u= 45/121 , ht= 4363 ; ( -91/4363 , -23627045/143979 ), ( -17431/6617 , 7764995/16797 )
**u= 45/128 , ht= 1538 ; ( 1538/1149 , 15549779/55152 ), ( -64647/106289 , -251909663/1275468 )
**u= 45/136 , ht= 17363 ; ( -189622/20651 , 398125709/247812 ), ( -17363/589 , -9065422/1767 )
**u= 45/161 , ht= 68961 ; ( 29033/68961 , 42480889/206883 ), ( 318103/192201 , -211009199/576603 )
**u= 45/184 , ht= 7974937 ; ( 7974937/1684414 , 19788632189/20212968 ), ( -34663051/6610403 , 21417867703/19831209 )
**u= 45/188 , ht= 632277 ; ( 659889/548123 , -526294495/1644369 ), ( 42521/632277 , -388784051/1896831 )
**u= 49/5 , ht= 727 ; ( 21599/11019 , 5417081/77133 ), ( 727/277 , 174365/1939 )
**u= 49/36 , ht= 2373 ; ( 2119/2373 , -5729551/49833 ), ( 19953/3889 , 36612014/81669 )
**u= 49/60 , ht= 39631 ; ( 39631/18091 , 33762569/126637 ), ( -176953/20873 , -138092168/146111 )
**u= 49/88 , ht= 69315 ; ( 69315/5867 , -65289495/41069 ), ( -53953/154185 , -10221163/71953 )
**u= 49/104 , ht= 83422 ; ( -126467/141541 , -193670483/990787 ), ( -18849/83422 , -349056023/2335816 )
**u= 49/124 , ht= 82981 ; ( -428619/39259 , 479537568/274813 ), ( 82981/52737 , -36514461/123053 )
**u= 49/125 , ht= 37651 ; ( -37651/2521 , -211056683/88235 ), ( 140197/29257 , -801031601/1023995 )
**u= 49/144 , ht= 72889 ; ( 72889/17579 , 180043219/246106 ), ( 42519/88103 , -352359239/1850163 )
**u= 49/176 , ht= 153061 ; ( -153061/18649 , -409333997/261086 ), ( -1181573/697579 , 1821288673/4883053 )
**u= 50 , ht= 4 ; ( -4/3 , -139/6 ), ( 576/935 , -150961/9350 )
**u= 50/21 , ht= 9754 ; ( -9754/6647 , 1530635/13294 ), ( -437243/223270 , -318364747/2232700 )
**u= 50/57 , ht= 11624 ; ( -11624/5419 , 13700467/54190 ), ( 48932/27823 , -60132421/278230 )
**u= 50/61 , ht= 32067 ; ( 673767/307706 , -163711323/615412 ), ( 7850/32067 , 4053733/35630 )
**u= 50/73 , ht= 131488 ; ( 131488/2497 , 158987743/24970 ), ( -2848724/113689 , -3446714699/1136890 )
**u= 50/129 , ht= 100115 ; ( 100115/48988 , 179121261/489880 ), ( -432760/69437 , 704378411/694370 )
**u= 50/193 , ht= 2953 ; ( -328/2953 , 5840477/29530 ), ( -2953/328 , -5840477/3280 )
**u= 52 , ht= 61 ; ( 45/61 , -2145/122 ), ( 263/15 , -519/2 )
**u= 52/17 , ht= 28079 ; ( -628989/163789 , 74350137/327578 ), ( 28079/8913 , -1123751/5942 )
**u= 52/33 , ht= 49277 ; ( 628427/74291 , 100868491/148582 ), ( 49277/3077 , 7870987/6154 )
**u= 52/37 , ht= 79101 ; ( 79101/27449 , 14132277/54898 ), ( -169997/59625 , -3378829/13250 )
**u= 52/45 , ht= 1679 ; ( -112341/65519 , 72625643/393114 ), ( -1679/1381 , 404689/2762 )
**u= 52/57 , ht= 36329 ; ( 23599/36329 , -9076027/72658 ), ( 473645/33187 , 99476155/66374 )
**u= 52/73 , ht= 173553 ; ( 173553/30409 , 41774451/60818 ), ( -1226453/202011 , -98234761/134674 )
**u= 52/77 , ht= 4257041 ; ( -5087455/921317 , -1258939129/1842634 ), ( -1012069/4257041 , -1065476593/8514082 )
**u= 52/125 , ht= 29311 ; ( 1749/29311 , 45549699/293110 ), ( 29311/1749 , -15183233/5830 )
**u= 52/149 , ht= 191305 ; ( -32811455/1243091 , 11122053295/2486182 ), ( 30981/191305 , 13128829/76522 )
**u= 52/153 , ht= 103829 ; ( -103829/32477 , -37341263/64954 ), ( -660599/93385 , -229001159/186770 )
**u= 52/185 , ht= 456301 ; ( 6113198869/336794019 , 2310829159615/673588038 ), ( 456301/125299 , -178599395/250598 )
**u= 52/197 , ht= 618961 ; ( -2826747/1014359 , -1169709753/2028718 ), ( 618961/108757 , 244768787/217514 )
**u= 53/25 , ht= 4579 ; ( 6847/1095 , -2382143/5475 ), ( 2285/4579 , -1758149/22895 )
**u= 53/40 , ht= 29348 ; ( 11937/71137 , -6269940/71137 ), ( 4713/29348 , 10334023/117392 )
**u= 53/97 , ht= 122099 ; ( 10357073/2213341 , -1433530837/2213341 ), ( -40473/122099 , 17410961/122099 )
**u= 53/132 , ht= 1373969 ; ( 86639/1373969 , -217378858/1373969 ), ( 4340641/544607 , 690755218/544607 )
**u= 53/180 , ht= 71707 ; ( 71707/46527 , -47283149/139581 ), ( -46527/71707 , 47283149/215121 )
**u= 61 , ht= 21 ; ( 21/13 , 303/13 ), ( 46175/10941 , 218815/3647 )
**u= 61/9 , ht= 21 ; ( -13/21 , -407/9 ), ( -21/13 , 2849/39 )
**u= 61/20 , ht= 25987 ; ( 3735043/429907 , 215344220/429907 ), ( 19897/25987 , 144227/1999 )
**u= 61/64 , ht= 114901 ; ( 153663/20113 , 31763727/40226 ), ( 114901/3285 , 3926836/1095 )
**u= 61/80 , ht= 3469 ; ( 6273/19837 , 4767771/39674 ), ( -461/3469 , -400966/3469 )
**u= 61/104 , ht= 2849 ; ( 808/2849 , 140683/1036 ), ( 4381113/219431 , 573075514/219431 )
**u= 61/105 , ht= 364843 ; ( -6017/364843 , 47898989/364843 ), ( 8666401/4942121 , 1309600431/4942121 )
**u= 61/120 , ht= 63259 ; ( 63259/7786 , -35776519/31144 ), ( 12029387/59797 , 1688127542/59797 )
**u= 61/129 , ht= 1289 ; ( -1732747/322481 , -256439357/322481 ), ( 1289/515 , -201963/515 )
**u= 61/145 , ht= 704619 ; ( -704619/37549 , -108846681/37549 ), ( -13968679/814221 , 719481265/271407 )
**u= 61/180 , ht= 102073 ; ( -3054223/1508867 , 585491029/1508867 ), ( 102073/12163 , 17667411/12163 )
**u= 61/185 , ht= 158261 ; ( -1360871/3109839 , 591471337/3109839 ), ( -44721/158261 , 28655201/158261 )
**u= 61/200 , ht= 19871 ; ( 19871/5655 , 18714467/28275 ), ( -32399/160945 , -148713536/804725 )
**u= 64/9 , ht= 4611 ; ( 32183/11229 , -30662881/269496 ), ( -4611/4489 , -5794961/107736 )
**u= 64/49 , ht= 1625 ; ( -33231/2245 , 4665195/3592 ), ( 1625/1059 , 3169755/19768 )
**u= 64/65 , ht= 10471 ; ( -10471/7989 , -10624021/63912 ), ( -29369/1571 , -23724913/12568 )
**u= 64/105 , ht= 1843 ; ( -118927/45587 , -130577737/364696 ), ( -383/1843 , 1929857/14744 )
**u= 64/121 , ht= 1227 ; ( 1227/289 , -15260871/25432 ), ( -289/1227 , 5086957/35992 )
**u= 64/149 , ht= 60391 ; ( -513533/163085 , -131608069/260936 ), ( -7255/60391 , 74285285/483128 )
**u= 65/16 , ht= 584777 ; ( 937087/92899 , -93452131/185798 ), ( -584777/264521 , 31868672/264521 )
**u= 65/37 , ht= 196893 ; ( 196893/25921 , 2141427/3703 ), ( -340583/9211 , 25720963/9211 )
**u= 65/44 , ht= 35943 ; ( -44481/42353 , 5055936/42353 ), ( 35591/35943 , 1387966/11981 )
**u= 65/48 , ht= 198997 ; ( -198997/82091 , -18507588/82091 ), ( -996029/107587 , -86139456/107587 )
**u= 65/61 , ht= 226997 ; ( -1418237/666351 , -21696671/95193 ), ( 117929/226997 , 24793265/226997 )
**u= 65/64 , ht= 11189 ; ( -323199/192227 , 149332413/768908 ), ( 9093/11189 , -2862803/22378 )
**u= 65/72 , ht= 88709 ; ( -88709/42213 , 31034470/126639 ), ( -2018243/414124 , 867829415/1656496 )
**u= 65/93 , ht= 70237 ; ( 27661/70237 , -9034243/70237 ), ( 102061/575617 , -69962101/575617 )
**u= 65/132 , ht= 202603 ; ( -437557/582671 , 103894001/582671 ), ( 202603/60791 , -30159591/60791 )
**u= 65/148 , ht= 4354079 ; ( 1139297/4354079 , -679486370/4354079 ), ( -123516777/28402781 , -19134573757/28402781 )
**u= 65/157 , ht= 102043 ; ( -102043/61551 , 18530375/61551 ), ( 88356393/2532971 , 13744826209/2532971 )
**u= 65/196 , ht= 104467 ; ( -102019/104467 , 25369120/104467 ), ( -124401/2177 , 151317388/15239 )
**u= 73/5 , ht= 9783 ; ( -9783/703 , 255471/703 ), ( 290041/52601 , 7756967/52601 )
**u= 73/20 , ht= 3137 ; ( -64591/56369 , -4489414/56369 ), ( -2063/3137 , -196600/3137 )
**u= 73/45 , ht= 21471 ; ( 238893/10343 , 56347625/31029 ), ( -8419/21471 , -5434771/64413 )
**u= 73/68 , ht= 67717 ; ( -29637/67717 , -7138098/67717 ), ( 489715/43839 , -5275545/4871 )
**u= 73/108 , ht= 120479 ; ( -17281/120479 , 44436302/361437 ), ( -64559461/50560763 , 9979379474/50560763 )
**u= 73/112 , ht= 225163 ; ( 3282003/585133 , -413147244/585133 ), ( 225163/66177 , -19389931/44118 )
**u= 73/128 , ht= 21263 ; ( -21263/19097 , 15145783/76388 ), ( -92809/11489 , -49559551/45956 )
**u= 73/149 , ht= 105735 ; ( 105735/7991 , 248475/131 ), ( 9535459/1311255 , 30574481/29139 )
**u= 73/152 , ht= 23158 ; ( -11555/23158 , 14946031/92632 ), ( -7085998/1848391 , 4229079869/7393564 )
**u= 73/165 , ht= 12520643 ; ( -12520643/573433 , -269333209/81919 ), ( 50086799/65902679 , 12451216911/65902679 )
**u= 73/176 , ht= 462329 ; ( -285605/462329 , 168849125/924658 ), ( -4925453/982515 , 52017860/65501 )
**u= 73/180 , ht= 11561 ; ( -1751/11561 , -167006/1051 ), ( 7030111/9201 , 3313510112/27603 )
**u= 73/188 , ht= 13813 ; ( -437453/2797 , -70239922/2797 ), ( -6491/13813 , -2450524/13813 )
**u= 73/192 , ht= 473761 ; ( -884977/169633 , 292423877/339266 ), ( 280417/473761 , 89329979/473761 )
**u= 74 , ht= 38 ; ( -331/234 , -8989/468 ), ( 31/38 , -1109/76 )
**u= 74/9 , ht= 835 ; ( 614/835 , 14473/334 ), ( -20629/19434 , 5934031/116604 )
**u= 74/49 , ht= 5654 ; ( -19498/10539 , 3608897/21078 ), ( 3991/5654 , 7888211/79156 )
**u= 74/81 , ht= 369407 ; ( -439246/11843 , -91990103/23686 ), ( -369407/318638 , 306399481/1911828 )
**u= 74/109 , ht= 3935 ; ( -1188/3935 , 199653/1574 ), ( 298880/752211 , 65524085/501474 )
**u= 74/141 , ht= 180128 ; ( -154325/180128 , -65518023/360256 ), ( -180128/154325 , -65518023/308650 )
**u= 74/161 , ht= 1826 ; ( 699/1826 , -577101/3652 ), ( -1036249/1239486 , -158951633/826324 )
**u= 74/181 , ht= 125913 ; ( 43286103/1077808 , -13550726313/2155616 ), ( -11828/125913 , 13192789/83942 )
**u= 80/13 , ht= 3481 ; ( 2883/3481 , 729315/13924 ), ( 6709/10167 , 654813/13556 )
**u= 80/57 , ht= 1445657 ; ( -1445657/1055161 , -604605163/4220644 ), ( 14899019/2496883 , 5103554805/9987532 )
**u= 80/61 , ht= 35569 ; ( -41194883/9801239 , 14797600579/39204956 ), ( -3227/35569 , 12480763/142276 )
**u= 80/73 , ht= 162819 ; ( -2964173/45179 , 1133301019/180716 ), ( 162819/117413 , -76742957/469652 )
**u= 80/93 , ht= 23149 ; ( -23149/12377 , 11326883/49508 ), ( -12377/23149 , -11326883/92596 )
**u= 80/157 , ht= 24929 ; ( -123/24929 , 13976733/99716 ), ( 645959/10974867 , 2054560241/14633156 )
**u= 80/169 , ht= 2037 ; ( 229/2037 , 15500659/105924 ), ( 278993/12479 , -2111831767/648908 )
**u= 80/189 , ht= 1619 ; ( 3069197/809849 , -5857788281/9718188 ), ( 1577/1619 , -4170847/19428 )
**u= 81/4 , ht= 2539 ; ( 13721/3579 , 2812037/32211 ), ( 1605/2539 , 598420/22851 )
**u= 81/5 , ht= 4777 ; ( 28643/7303 , 6574979/65727 ), ( 4777/2137 , -1175899/19233 )
**u= 81/16 , ht= 501 ; ( 2343/2347 , 2654461/42246 ), ( 501/437 , 266086/3933 )
**u= 81/20 , ht= 2423 ; ( 19033/11723 , 10000241/105507 ), ( -2167/2423 , 1454438/21807 )
**u= 81/29 , ht= 5555 ; ( 3757/5555 , 65701/909 ), ( 590095/52487 , -319259105/472383 )
**u= 81/44 , ht= 19933 ; ( 19933/8449 , 14367487/76041 ), ( -15791/34061 , 24915187/306549 )
**u= 81/53 , ht= 69301 ; ( 9245/69301 , -50928425/623709 ), ( -96547/26095 , -14570251/46971 )
**u= 81/61 , ht= 48793 ; ( 136771985/7329009 , 107027052161/65961081 ), ( -48793/5619 , 38382193/50571 )
**u= 81/100 , ht= 15339 ; ( -15339/13013 , 100628746/585585 ), ( 15501/12851 , 20145902/115659 )
**u= 81/116 , ht= 4765 ; ( 327389/152867 , -389354362/1375803 ), ( -2659/4765 , -5880086/42885 )
**u= 81/125 , ht= 29597 ; ( 29597/8893 , -172848737/400185 ), ( 143423/175463 , -1267516421/7895835 )
**u= 81/140 , ht= 10373 ; ( 10373/2843 , -12732722/25587 ), ( -1847/20453 , 24311345/184077 )
**u= 81/149 , ht= 98461 ; ( 602573/156169 , -760229173/1405521 ), ( -98461/19249 , -122527303/173241 )
**u= 81/184 , ht= 11764 ; ( -4067271/71464 , -22083343373/2572704 ), ( -11764/8775 , 6128711/24300 )
**u= 81/196 , ht= 6165 ; ( 40587/20369 , -63609251/183321 ), ( 6165/4891 , 11023100/44019 )
**u= 85/2 , ht= 8948 ; ( -56832/18551 , 1766925/37102 ), ( -1489/8948 , -266285/17896 )
**u= 85/18 , ht= 103517 ; ( 3235909/311927 , -149596652/311927 ), ( 103517/3401 , 251124/179 )
**u= 85/26 , ht= 52271 ; ( -32333/52271 , 3401492/52271 ), ( -267309/62462 , -30387625/124924 )
**u= 85/54 , ht= 120167 ; ( 120167/117494 , 26804955/234988 ), ( 2129759/66083 , -509803484/198249 )
**u= 85/78 , ht= 134132 ; ( 634957/867856 , 206130959/1735712 ), ( -134132/12469 , 697923/674 )
**u= 85/86 , ht= 10762 ; ( -4331/10762 , 2335031/21524 ), ( -10210282/39241 , -2055176771/78482 )
**u= 85/114 , ht= 2879021 ; ( 1230353/2879021 , -362780650/2879021 ), ( -39668851/887048 , -9195281757/1774096 )
**u= 85/162 , ht= 889384 ; ( -889384/194477 , -251498087/388954 ), ( 3174233/44636 , -876981933/89272 )
**u= 85/182 , ht= 430743 ; ( -403791/770213 , 127319640/770213 ), ( -163994/430743 , -44985511/287162 )
**u= 89/4 , ht= 17139 ; ( 10131/18221 , -445086/18221 ), ( -10103/17139 , 139821/5713 )
**u= 89/40 , ht= 315869 ; ( -3347389/463229 , -226639984/463229 ), ( 315869/229259 , -26180407/229259 )
**u= 89/45 , ht= 7517 ; ( -54011/26559 , -12845257/79677 ), ( -2077/7517 , -554783/7517 )
**u= 89/64 , ht= 25269 ; ( 19787/25269 , -2723056/25269 ), ( 1107833/1579207 , 163665287/1579207 )
**u= 89/124 , ht= 4201 ; ( 2837/4201 , 598669/4201 ), ( -469951/41549 , 55717619/41549 )
**u= 89/180 , ht= 3090701 ; ( -114427277/3197433 , -48863749469/9592299 ), ( 1654531/3090701 , -498817792/3090701 )
**u= 90/13 , ht= 543 ; ( 244/543 , -135929/3258 ), ( -17421/4432 , -4105603/26592 )
**u= 90/17 , ht= 38027 ; ( -71047/3566 , 18546919/21396 ), ( -38027/6394 , 10067359/38364 )
**u= 90/41 , ht= 3757 ; ( -2434/3757 , 1813901/22542 ), ( -3757/2434 , -1813901/14604 )
**u= 90/49 , ht= 11538 ; ( 26562/7399 , -85489123/310758 ), ( 11538/10051 , 47446507/422142 )
**u= 90/101 , ht= 377492 ; ( 6813389/751268 , -4359106813/4507608 ), ( 124891/377492 , 252856897/2264952 )
**u= 90/133 , ht= 277 ; ( 277/144 , -227827/864 ), ( -475259/257172 , 394352143/1543032 )
**u= 90/157 , ht= 79361 ; ( -5956432/1841601 , 4943280505/11049606 ), ( 79361/18288 , 64573369/109728 )
**u= 90/173 , ht= 224968 ; ( 309997/430924 , -441822119/2585544 ), ( 224968/83281 , 199660411/499686 )
**u= 97/10 , ht= 4152 ; ( 324516/69979 , -21283653/139958 ), ( 1037/4152 , 91463/2768 )
**u= 97/38 , ht= 1751492 ; ( -3105691/1529644 , -433552861/3059288 ), ( -1751492/13057 , -219409127/26114 )
**u= 97/42 , ht= 231937 ; ( 71224/231937 , -31950879/463874 ), ( 742849/2450296 , -337101407/4900592 )
**u= 97/114 , ht= 103745 ; ( 366650170/5895707 , 79547149295/11791414 ), ( -10787/103745 , 2262658/20749 )
**u= 97/122 , ht= 35091 ; ( 27527/35091 , 5004416/35091 ), ( 1300117/141727 , -146749388/141727 )
**u= 97/138 , ht= 802453 ; ( -394145/802453 , 106692676/802453 ), ( 9920707/15296681 , 2175783516/15296681 )
**u= 97/150 , ht= 72235 ; ( -72235/47161 , -53666376/235805 ), ( 6097184/3155 , -7586152037/31550 )
**u= 97/158 , ht= 1058633 ; ( 1058633/46722 , -270622853/93444 ), ( -7291154/295927 , 1863624101/591854 )
**u= 97/170 , ht= 125011 ; ( -125011/38801 , -17337388/38801 ), ( -4210501/1836791 , 608459846/1836791 )
**u= 97/198 , ht= 25133 ; ( 25133/9480 , 4607693/11376 ), ( 389260/19907 , 111433085/39814 )
**u= 98/5 , ht= 169 ; ( -9306/14341 , 5430099/200774 ), ( 169/66 , 19269/308 )
**u= 98/57 , ht= 2216 ; ( -823312/78865 , -176695979/220822 ), ( 2216/791 , 2513717/11074 )
**u= 98/93 , ht= 53842 ; ( -55174/50815 , -20470391/142282 ), ( 13697/53842 , 75807427/753788 )
**u= 98/117 , ht= 19137 ; ( -67110/9187 , -311005943/385854 ), ( -14858/19137 , -111241967/803754 )
**u= 98/157 , ht= 16087 ; ( -132582/230245 , 94210371/644686 ), ( -7138/16087 , 31202569/225218 )
**u= 100/49 , ht= 8211 ; ( 8211/5161 , -47544321/361270 ), ( 3509/9231 , 2305301/30770 )
**u= 100/81 , ht= 12461 ; ( 12661/9921 , 130355681/892890 ), ( 12461/10121 , 43367227/303630 )
**u= 100/93 , ht= 2933 ; ( 2933/19 , 565989/38 ), ( -129421/146603 , -37737179/293206 )
**u= 100/129 , ht= 89369 ; ( -89369/16411 , 103253587/164110 ), ( -533113/1291 , 121165729/2582 )
**u= 100/177 , ht= 30083 ; ( 30083/5185 , -40633963/51850 ), ( -62347/351479 , 95031583/702958 )
**u= 100/193 , ht= 378405 ; ( -378405/232247 , 617135451/2322470 ), ( 1793023/413739 , -170517329/275826 )
**u= 101/12 , ht= 40207 ; ( 33745/40207 , -1810830/40207 ), ( 59839/17135 , -429968/3427 )
**u= 101/16 , ht= 3457 ; ( 1285/3457 , 293959/6914 ), ( 3275/4571 , -223855/4571 )
**u= 101/36 , ht= 81361 ; ( 1999099/2610001 , 196391841/2610001 ), ( 81361/24699 , 15239453/74097 )
**u= 101/44 , ht= 2141519 ; ( -5668527/1043057 , -380575674/1043057 ), ( -2141519/7617 , -47147004/2539 )
**u= 101/105 , ht= 2100323 ; ( 14361511/3373681 , -1504932267/3373681 ), ( 2100323/611633 , -223167927/611633 )
**u= 101/120 , ht= 63973 ; ( 86579/217334 , 7850417/66872 ), ( 63973/58483 , -9452781/58483 )
**u= 101/124 , ht= 507575 ; ( 507575/38173 , 56428159/38173 ), ( -8442291/19655 , 935937247/19655 )
**u= 101/129 , ht= 27257 ; ( 27257/12293 , 3380971/12293 ), ( -856937/3012805 , -354177079/3012805 )
**u= 101/160 , ht= 502499 ; ( 502499/21349 , -126671725/42698 ), ( 11806301/2371651 , -1516475335/2371651 )
**u= 101/188 , ht= 1130383 ; ( -328845/1130383 , 160699845/1130383 ), ( 3546454379/403096245 , 32481767210/26873083 )
**u= 106 , ht= 342 ; ( -342/31 , -4467/62 ), ( -487/18 , -3943/12 )
**u= 106/89 , ht= 349258 ; ( 349258/309297 , 85540783/618594 ), ( -664111/169182 , 41887723/112788 )
**u= 106/101 , ht= 5856948 ; ( -1509072500/35072743 , -294836378291/70145486 ), ( 5856948/1218607 , -89889107/187478 )
**u= 106/141 , ht= 70009 ; ( -70009/31892 , 2536373/9112 ), ( 522067/124888 , 123888219/249776 )
**u= 106/197 , ht= 583660 ; ( -583660/388141 , 14708635/59714 ), ( -3028688/1376055 , 60499751/183474 )
**u= 109/4 , ht= 6945 ; ( 18633/6629 , -53367/947 ), ( 721/6945 , 43792/2315 )
**u= 109/8 , ht= 9267 ; ( -5275/9267 , 289715/9267 ), ( 187269/29090 , 4127653/23272 )
**u= 109/9 , ht= 13433 ; ( -15943/8151 , 1541053/24453 ), ( -13433/5981 , 423783/5981 )
**u= 109/41 , ht= 124807 ; ( 124807/13123 , -7699421/13123 ), ( 29782979/14616987 , 678677049/4872329 )
**u= 109/60 , ht= 213943 ; ( -213943/114857 , 18024490/114857 ), ( -473849/309001 , 41994220/309001 )
**u= 109/84 , ht= 674197 ; ( 38443/674197 , -59314942/674197 ), ( -4357387/19673291 , -1769759362/19673291 )
**u= 109/96 , ht= 595837 ; ( 10067849/1086689 , 1901597021/2173378 ), ( 595837/121315 , -57096671/121315 )
**u= 109/104 , ht= 33217 ; ( -401276/688033 , 311374327/2752132 ), ( -33217/15208 , -14281807/60832 )
**u= 109/161 , ht= 638255 ; ( 638255/453749 , 95224217/453749 ), ( -9548939/4847359 , 1302185017/4847359 )
**u= 113/18 , ht= 55171 ; ( -157190/11857 , -12577341/23714 ), ( 55171/3230 , 4418073/6460 )
**u= 113/22 , ht= 45337 ; ( -45337/11731 , -2066212/11731 ), ( 234153/99469 , -11236306/99469 )
**u= 113/30 , ht= 165902 ; ( -125273/165902 , 1948745/30164 ), ( -208714/261301 , -34478103/522602 )
**u= 113/70 , ht= 190928 ; ( -81973/190928 , 32726053/381856 ), ( -89636/216329 , -36878063/432658 )
**u= 113/78 , ht= 75031 ; ( 782857/49163 , -65200438/49163 ), ( -37162/75031 , -732619/7898 )
**u= 113/94 , ht= 278485 ; ( -840755/99706 , 154514575/199412 ), ( 23161/278485 , 5099948/55697 )
**u= 113/142 , ht= 2038 ; ( -678217/1367435 , 34239830/273487 ), ( -2038/731 , -485683/1462 )
**u= 113/190 , ht= 43241 ; ( -43241/10966 , -11575045/21932 ), ( -31453799/4372981 , 4120021172/4372981 )
**u= 117/8 , ht= 21269 ; ( -23989/76381 , -6311161/229143 ), ( -21269/6638 , -7025729/79656 )
**u= 117/20 , ht= 7493 ; ( 603509/39349 , 74990762/118047 ), ( -2443/7493 , 977380/22479 )
**u= 117/25 , ht= 1185 ; ( -1185/757 , -975281/11355 ), ( -54047/9447 , -7616513/28341 )
**u= 117/97 , ht= 7489957 ; ( 7489957/1026255 , 413214049/615753 ), ( 36771835/6301419 , 10196642335/18904257 )
**u= 117/100 , ht= 68961 ; ( -68961/58961 , 3596740/25269 ), ( -98875/1036869 , 1445110982/15553035 )
**u= 117/112 , ht= 67757 ; ( -1555649/300891 , -930617461/1805346 ), ( 67757/63081 , 54374107/378486 )
**u= 117/113 , ht= 739631 ; ( 5446021/1583003 , 1672929673/4749009 ), ( -739631/41425 , -218525531/124275 )
**u= 117/140 , ht= 50993 ; ( -24943/50993 , -18638770/152979 ), ( 676439/619669 , 301205761/1859007 )
**u= 117/152 , ht= 38755 ; ( -38755/13172 , -56014291/158064 ), ( -11296/52525 , -73521787/630300 )
**u= 117/172 , ht= 397503 ; ( -397503/324145 , -37332940/194487 ), ( -857367/192329 , -319781194/576987 )
**u= 117/185 , ht= 1632223 ; ( -38242301/1029869 , 14439068653/3089607 ), ( -1632223/404923 , -634737865/1214769 )
**u= 121/36 , ht= 1315 ; ( -1213/19325 , 2325434/42515 ), ( -1315/467 , 837900/5137 )
**u= 121/48 , ht= 733 ; ( 665/733 , 1372115/16126 ), ( -14797/88585 , 24898847/389774 )
**u= 121/56 , ht= 25 ; ( 25/24 , 103789/1056 ), ( 24/25 , 103789/1100 )
**u= 121/84 , ht= 2405 ; ( 2405/2033 , -2887729/22363 ), ( -93143/57863 , -7734888/48961 )
**u= 121/120 , ht= 2953 ; ( -15979/82091 , 91664176/903001 ), ( -932/2953 , 13575391/129932 )
**u= 121/141 , ht= 45145 ; ( 45145/6887 , 54254417/75757 ), ( 6887/45145 , 54254417/496595 )
**u= 121/144 , ht= 24021 ; ( -24263/15787 , 69508507/347314 ), ( 24021/15545 , -206117137/1025970 )
**u= 121/152 , ht= 57771 ; ( -272945/54159 , 49034980/85107 ), ( 57771/36605 , -16872737/80531 )
**u= 121/164 , ht= 425 ; ( -54819/57443 , -101738766/631873 ), ( 425/179 , 590885/1969 )
**u= 121/196 , ht= 1869 ; ( 3819/10309 , -1767132/13013 ), ( 1037/1869 , -6985842/47971 )
**u= 122/33 , ht= 17771 ; ( 2294/17771 , -1865429/35542 ), ( 17771/2294 , 50417/124 )
**u= 122/37 , ht= 54977 ; ( 40436/54977 , 7521289/109954 ), ( -16164039/105268 , -1781787073/210536 )
**u= 122/73 , ht= 6245746 ; ( -36156191/7343310 , 87853327/225948 ), ( 6245746/1076227 , 981083389/2152454 )
**u= 122/97 , ht= 422773 ; ( 612714978/1570067 , -109321755651/3140134 ), ( 422773/157578 , -26835527/105052 )
**u= 122/133 , ht= 75532 ; ( -15201612/6898289 , -3487764009/13796578 ), ( -75532/4909 , 15814633/9818 )
**u= 122/177 , ht= 64562 ; ( -968473/21370 , -46696321/8548 ), ( -31313/64562 , -17294681/129124 )
**u= 122/189 , ht= 105575 ; ( 105575/7876 , 79102805/47256 ), ( 146161/23540 , -22123643/28248 )
**u= 125/4 , ht= 309 ; ( -53/309 , 28694/1545 ), ( 577/111 , -3584/37 )
**u= 125/9 , ht= 2051 ; ( 1383/2051 , 997973/30765 ), ( 10487/8829 , -5525071/132435 )
**u= 125/16 , ht= 7129 ; ( 4313/7129 , -1492336/35645 ), ( 8793/7049 , -21239/371 )
**u= 125/24 , ht= 21727 ; ( -21727/2591 , -4793232/12955 ), ( 32507/41869 , 11617347/209345 )
**u= 125/32 , ht= 11101 ; ( 6618/11101 , 5234967/88808 ), ( -19933/14946 , -16818081/199280 )
**u= 125/41 , ht= 19947 ; ( -19947/2849 , 824523/2035 ), ( -190141/47577 , 18722557/79295 )
**u= 125/48 , ht= 67027 ; ( -108529/22597 , -13744131/45194 ), ( 67027/40579 , -24291599/202895 )
**u= 125/49 , ht= 1067 ; ( 4079/2413 , 1484293/12065 ), ( 939/1067 , -3116221/37345 )
**u= 125/52 , ht= 18431 ; ( 80621/128617 , 48981569/643085 ), ( -18431/12353 , 7159498/61765 )
**u= 125/57 , ht= 91817 ; ( -91817/38941 , -4812679/27815 ), ( 448847/121211 , -157075611/606055 )
**u= 125/96 , ht= 643 ; ( 57713/17161 , 402989/1310 ), ( -131/643 , 287676/3215 )
**u= 125/97 , ht= 2937 ; ( -749/2937 , 1335767/14685 ), ( 15077/10421 , -8076973/52105 )
**u= 125/132 , ht= 9637 ; ( 9637/4049 , -5373607/20245 ), ( -4906813/472319 , -2534204341/2361595 )
**u= 125/177 , ht= 46873 ; ( 263059/120043 , -172129781/600215 ), ( 46873/7579 , 28265823/37895 )
**u= 130/21 , ht= 10258 ; ( 7249/10258 , -1010539/20516 ), ( -61822/191 , -4976869/382 )
**u= 130/29 , ht= 3553 ; ( -1338357/24106 , 126447339/48212 ), ( 3553/2334 , 133951/1556 )
**u= 130/33 , ht= 629 ; ( 508/629 , 81521/1258 ), ( -187588/95651 , -21234003/191302 )
**u= 130/41 , ht= 22963 ; ( -837487/838716 , -133195645/1677432 ), ( 22963/7916 , -2730085/15832 )
**u= 130/53 , ht= 173798 ; ( -53201/173798 , 23225899/347596 ), ( -752117/806666 , -140914603/1613332 )
**u= 130/69 , ht= 21247 ; ( 145726/38707 , -21979515/77414 ), ( 6826/21247 , -3253171/42494 )
**u= 130/77 , ht= 362707 ; ( -362707/257066 , 68463323/514132 ), ( 4411378/3490381 , 866318539/6980762 )
**u= 130/89 , ht= 2450736 ; ( -727748/2917069 , -497805437/5834138 ), ( -2450736/810313 , 427391003/1620626 )
**u= 130/93 , ht= 2942518 ; ( -3622702/1136201 , -642564069/2272402 ), ( 2942518/1733611 , 578040597/3467222 )
**u= 130/113 , ht= 1312617 ; ( 907108/3941271 , -754541309/7882542 ), ( -1312617/113104 , -245801141/226208 )
**u= 130/149 , ht= 14980411 ; ( -35682014/3060817 , 7672069729/6121634 ), ( -14980411/2676238 , 3260114747/5352476 )
**u= 130/197 , ht= 756346 ; ( -1459697/57006 , 359839021/114012 ), ( 756346/290047 , 199542995/580094 )
**u= 136/5 , ht= 92 ; ( 2596/3199 , 317341/12796 ), ( 92/57 , -2791/76 )
**u= 136/41 , ht= 4374 ; ( 643/4374 , 971731/17496 ), ( 14311/18202 , 5087591/72808 )
**u= 136/69 , ht= 418736 ; ( 1086748/1302649 , 483603627/5210596 ), ( -418736/4303 , -9183431/1324 )
**u= 136/101 , ht= 5785 ; ( -128117/144684 , 66651791/578736 ), ( 5785/836 , 106105/176 )
**u= 136/185 , ht= 1503441 ; ( 1503441/855494 , -807415257/3421976 ), ( -12349069/3407094 , -1993208215/4542792 )
**u= 137/8 , ht= 1399 ; ( -1119/1399 , -43383/1399 ), ( 11545/18768 , 707975/25024 )
**u= 137/12 , ht= 8843 ; ( -8843/4693 , 295902/4693 ), ( 916019/42745 , 27241939/42745 )
**u= 137/37 , ht= 148963 ; ( 16923107/1272161 , 882114839/1272161 ), ( -148963/8067 , -2586481/2689 )
**u= 137/40 , ht= 6261 ; ( 56311/102191 , -6308953/102191 ), ( 6261/4531 , -417863/4531 )
**u= 137/88 , ht= 52157 ; ( -721993/85462 , 233183471/341848 ), ( -15317/52157 , 4358759/52157 )
**u= 137/180 , ht= 297849 ; ( -43339/550561 , -63336915/550561 ), ( 297849/152759 , 115168424/458277 )
**u= 137/197 , ht= 26323779 ; ( -26323779/10120433 , -3383607063/10120433 ), ( 112498063/5097147 , -4503758029/1699049 )
**u= 144 , ht= 17 ; ( -17/3 , -283/36 ), ( -51/377 , -1481/4524 )
**u= 144/25 , ht= 1869 ; ( 1869/353 , -4753933/21180 ), ( 1663/4611 , -12252917/276660 )
**u= 144/89 , ht= 148123 ; ( 148123/383 , 139803629/4596 ), ( 565463/291877 , 600668681/3502524 )
**u= 144/145 , ht= 112827 ; ( -115149/162671 , 240118891/1952052 ), ( -112827/50617 , 148988011/607404 )
**u= 144/181 , ht= 11975 ; ( 11975/5097 , -17518363/61164 ), ( 4812069/611059 , -6529489931/7332708 )
**u= 145/6 , ht= 583 ; ( -13301/18353 , -463402/18353 ), ( -544/583 , -32425/1166 )
**u= 145/18 , ht= 2773 ; ( 999/2773 , -312098/8319 ), ( 105953/12829 , -3768220/12829 )
**u= 145/22 , ht= 322401 ; ( 322401/48787 , 12694554/48787 ), ( 2034157/1488711 , -32756832/496237 )
**u= 145/102 , ht= 67496 ; ( 3303137/432341 , -21502780/33257 ), ( 67496/20717 , -11850059/41434 )
**u= 145/122 , ht= 42608 ; ( 42608/18209 , 8504723/36418 ), ( 3834251/146548 , -704314441/293096 )
**u= 145/138 , ht= 7997 ; ( -164044/153563 , 43865597/307126 ), ( 7997/5536 , -1898723/11072 )
**u= 145/158 , ht= 299381 ; ( 1589313/9331 , 165989346/9331 ), ( 254962/299381 , 82139927/598762 )
**u= 145/186 , ht= 1635469 ; ( 382852/1635469 , -380683057/3270938 ), ( -622630451/3674608 , -141115546713/7349216 )
**u= 145/194 , ht= 316907 ; ( 48120382/6445981 , 11237246677/12891962 ), ( -316907/73049 , 37637750/73049 )
**u= 146/9 , ht= 2057 ; ( 7521/784 , 1119787/4704 ), ( -2057/744 , 327671/4464 )
**u= 146/21 , ht= 29098 ; ( -9437/29098 , -2323781/58196 ), ( -259643/344170 , 32707609/688340 )
**u= 146/61 , ht= 108898 ; ( 547286/158545 , -14737955/63418 ), ( 108898/40613 , -15034601/81226 )
**u= 146/81 , ht= 7844 ; ( -63920/60689 , -39411571/364134 ), ( -7844/1183 , 3547319/7098 )
**u= 146/105 , ht= 3659599 ; ( 4166732/1265897 , 738967563/2531794 ), ( 3659599/17324 , 621062685/34648 )
**u= 146/165 , ht= 1288994 ; ( -9957989/1030846 , 2129616421/2061692 ), ( -154921/1288994 , -276172437/2577988 )
**u= 146/177 , ht= 219257 ; ( -47600/219257 , 49434345/438514 ), ( 506149/90140 , 22654969/36056 )
**u= 146/181 , ht= 55831 ; ( 55831/51306 , -16893967/102612 ), ( 51306/55831 , -16893967/111662 )
**u= 148/5 , ht= 28983 ; ( -32441/7019 , -1196219/14038 ), ( 28983/11213 , -1161343/22426 )
**u= 148/33 , ht= 3997 ; ( -3997/3373 , -494141/6746 ), ( -66757/3643 , -6320381/7286 )
**u= 148/93 , ht= 3618793 ; ( 3618793/134765 , 114877713/53906 ), ( 12437765/3791617 , 2062671965/7583234 )
**u= 148/133 , ht= 11189 ; ( 5793/11189 , -2390127/22378 ), ( -11189/5793 , -796709/3862 )
**u= 148/177 , ht= 2575273 ; ( -238832717/122311003 , 58718997663/244622006 ), ( -2575273/1319081 , 633188247/2638162 )
**u= 149/10 , ht= 11914 ; ( 175574/306589 , 18366539/613178 ), ( 8229/11914 , 749149/23828 )
**u= 149/90 , ht= 510119 ; ( -381266/510119 , 99045659/1020238 ), ( 1068547/1119127 , -120320128/1119127 )
**u= 149/114 , ht= 1111541 ; ( 1923299/38879 , 168346536/38879 ), ( 1111541/81608 , -195087003/163216 )
**u= 149/162 , ht= 111093 ; ( 74945/111093 , 125826500/999837 ), ( 75293/410320 , 52225055/492384 )
**u= 149/174 , ht= 247685 ; ( 5002711/1159573 , 555228202/1159573 ), ( 37232/247685 , -54160609/495370 )
**u= 149/190 , ht= 1529719 ; ( -1529719/532541 , 183003104/532541 ), ( 9010987/3941407 , 1111226156/3941407 )
**u= 153/14 , ht= 4545 ; ( 4545/2098 , -907265/12588 ), ( -228642/30955 , 8403743/37146 )
**u= 153/38 , ht= 30671 ; ( -370757/220353 , -64503986/661059 ), ( -30671/14445 , 5072672/43335 )
**u= 153/62 , ht= 194226 ; ( -138869/194226 , -91247257/1165356 ), ( 522238/625251 , -311310689/3751506 )
**u= 153/110 , ht= 150597 ; ( -25184333/1838982 , -12852695483/11033892 ), ( 18047/150597 , -38600410/451791 )
**u= 153/122 , ht= 18249 ; ( -13211472/2497187 , -655207229/1362102 ), ( 18249/3589 , 4985156/10767 )
**u= 153/182 , ht= 81472 ; ( 43161/81472 , 60366661/488832 ), ( -675587/291561 , 240887228/874683 )
**u= 157/32 , ht= 1919 ; ( 159814/62979 , 8863711/71976 ), ( 119/1919 , -173593/3838 )
**u= 157/33 , ht= 12757 ; ( 12757/12581 , 821863/12581 ), ( 26120359/7106287 , -1242313357/7106287 )
**u= 157/48 , ht= 258133 ; ( -258133/112811 , -15581808/112811 ), ( 325457/197551 , 21064896/197551 )
**u= 157/68 , ht= 3438109 ; ( 3438109/2150733 , 267018044/2150733 ), ( 469398517/2966389 , -30912967994/2966389 )
**u= 157/92 , ht= 77255 ; ( -77255/32179 , 6409399/32179 ), ( -32556887/1269627 , 831869229/423209 )
**u= 157/105 , ht= 164587 ; ( 879323/69227 , 72167045/69227 ), ( 11917/164587 , -13501427/164587 )
**u= 157/113 , ht= 14609 ; ( 14609/4149 , -1289033/4149 ), ( 148065/17663 , -12657805/17663 )
**u= 157/128 , ht= 57994 ; ( 57994/43979 , 105204017/703664 ), ( 334007/98262 , -167755063/524064 )
**u= 157/160 , ht= 12996 ; ( -12996/8359 , -12485691/66872 ), ( -9220783/15473943 , 404301169/3438654 )
**u= 157/168 , ht= 443479 ; ( 443479/151159 , 48491418/151159 ), ( -524081/176048 , 228883843/704192 )
**u= 157/185 , ht= 228871 ; ( 228871/12051 , 24891331/12051 ), ( -9565041/7338779 , 1309392977/7338779 )
**u= 157/188 , ht= 6231397 ; ( -10763359/3220235 , -246005275/644047 ), ( -6231397/437237 , -683937424/437237 )
**u= 162/5 , ht= 511 ; ( 426/511 , -2897/126 ), ( 146/639 , 202429/11502 )
**u= 162/53 , ht= 1241 ; ( -1038/1241 , -1666649/22338 ), ( -36834/27031 , 47066941/486558 )
**u= 162/65 , ht= 47304 ; ( -3386419/426936 , 3893133211/7684848 ), ( 34291/47304 , 66647087/851472 )
**u= 162/73 , ht= 123011 ; ( 6959/272752 , -329888293/4909536 ), ( -83068/123011 , -179436713/2214198 )
**u= 162/77 , ht= 15146 ; ( -1605/15146 , 18912893/272628 ), ( 2417403/263482 , 3019594391/4742676 )
**u= 162/85 , ht= 163298 ; ( 310342/42167 , -408533065/759006 ), ( 163298/63967 , -228802309/1151406 )
**u= 162/137 , ht= 69355 ; ( -69355/39012 , 131785471/702216 ), ( 1947687/268204 , 3256236959/4827672 )
**u= 162/193 , ht= 21848 ; ( -733/21848 , -42971957/393264 ), ( 5834467048/196366733 , -11475620742757/3534601194 )
**u= 169/5 , ht= 3647 ; ( -6511/2639 , 1539245/34307 ), ( 1807/3647 , 894707/47411 )
**u= 169/21 , ht= 479 ; ( 479/395 , -56915/1027 ), ( 131/751 , 348999/9763 )
**u= 169/36 , ht= 1103 ; ( -25915/30843 , -72557866/1202877 ), ( -1103/1003 , -895023/13039 )
**u= 169/44 , ht= 1821 ; ( -1193/1821 , 1445023/23673 ), ( 235789/117561 , 58299159/509431 )
**u= 169/60 , ht= 25793 ; ( 563/25793 , 19998377/335309 ), ( 25793/563 , 19998377/7319 )
**u= 169/64 , ht= 11127 ; ( -143331/204343 , -61475733/817372 ), ( -6683/11127 , 13851147/192868 )
**u= 169/96 , ht= 357781 ; ( -357781/12115 , -10797239/4846 ), ( 373490/40981 , -2946890515/4262024 )
**u= 169/109 , ht= 9267 ; ( 1183065/32243 , -1236189171/419159 ), ( 4907/9267 , -3651023/40157 )
**u= 169/141 , ht= 713 ; ( -713/697 , -1184593/9061 ), ( 1966327/1873079 , -3226378303/24350027 )
**u= 169/164 , ht= 158737 ; ( -506323/333603 , 776899564/4336839 ), ( -158737/64303 , 219447794/835939 )
**u= 169/200 , ht= 10604 ; ( 10604/105 , 300093193/27300 ), ( -18487/8863 , 29009401/115219 )
**u= 173 , ht= 155 ; ( 155/117 , -1135/117 ), ( -35611/48005 , -68435/9601 )
**u= 173/8 , ht= 619 ; ( 12726/9367 , -1355187/37468 ), ( 619/42 , -18003/56 )
**u= 173/12 , ht= 1187 ; ( 1187/17 , -31119/17 ), ( 10153/11413 , 402279/11413 )
**u= 173/17 , ht= 2847 ; ( -2847/773 , 92319/773 ), ( -18707/22773 , -307957/7591 )
**u= 173/24 , ht= 120755 ; ( 1210315/273299 , -46184745/273299 ), ( 68428/120755 , -4135097/96604 )
**u= 173/40 , ht= 68252 ; ( 68252/24113 , -13924933/96452 ), ( -137849/230644 , 51696877/922576 )
**u= 173/48 , ht= 110989 ; ( 110989/67921 , -13713337/135842 ), ( 180601/14201 , -9550436/14201 )
**u= 173/49 , ht= 7427 ; ( 15355/14259 , 7810345/99813 ), ( 7427/3365 , 607963/4711 )
**u= 173/73 , ht= 27187 ; ( -27187/177 , -1766753/177 ), ( -120329/91955 , 9843049/91955 )
**u= 173/108 , ht= 161515 ; ( -161515/134597 , -49861004/403791 ), ( -400163/103363 , 98022206/310089 )
**u= 173/112 , ht= 59967 ; ( 16907/99083 , -8092256/99083 ), ( 59967/29167 , -5368481/29167 )
**u= 173/120 , ht= 713386 ; ( 153241/1567399 , -131238992/1567399 ), ( 713386/39361 , 238156525/157444 )
**u= 173/145 , ht= 37287 ; ( 37287/14957 , 3679881/14957 ), ( -139579/570519 , 17932763/190173 )
**u= 173/172 , ht= 447083 ; ( -17192421/1977301 , 1726429512/1977301 ), ( 447083/32625 , 14907337/10875 )
**u= 173/180 , ht= 124167 ; ( 124087/124167 , -53745688/372501 ), ( 2567721/1415411 , -897698563/4246233 )
**u= 173/192 , ht= 84311 ; ( -1022323/368573 , 3882843/12494 ), ( 84311/16825 , -18124151/33650 )
**u= 178/57 , ht= 379 ; ( -379/248 , -51283/496 ), ( 147311/64604 , -18217571/129208 )
**u= 178/73 , ht= 347151 ; ( -347151/296416 , 58495617/592832 ), ( 1519745/355176 , 66675215/236784 )
**u= 178/93 , ht= 60971 ; ( -60971/47374 , 11167777/94748 ), ( -1987583/8608558 , -1277796441/17217116 )
**u= 178/117 , ht= 406570 ; ( 118618/3029227 , -491847377/6058454 ), ( 172827/406570 , 215005379/2439420 )
**u= 178/125 , ht= 17126458 ; ( -20318346/70300189 , 61357728441/703001890 ), ( -17126458/5474997 , -5025351703/18249990
)
**u= 178/153 , ht= 3378552 ; ( -3378552/1108921 , -1979022599/6653526 ), ( -72342328/33957057 , 44479048427/203742342 )
**u= 178/161 , ht= 2787100 ; ( -2787100/758799 , 549703865/1517598 ), ( 10858324/1282065 , 138723997/170942 )
**u= 178/173 , ht= 347737 ; ( -6023999/229710 , -18295595/7068 ), ( 347737/33050 , 68910569/66100 )
**u= 180/17 , ht= 165581 ; ( 276753/225311 , 65800999/1351866 ), ( 165581/9147 , 30679247/54882 )
**u= 180/37 , ht= 1127 ; ( 258983/159971 , 11832697/137118 ), ( -1127/979 , -406343/5874 )
**u= 180/41 , ht= 17631 ; ( 69217/1239 , -19824769/7434 ), ( 3527/17631 , 5149193/105786 )
**u= 180/61 , ht= 60343 ; ( -60343/28379 , -23300423/170274 ), ( -978089/1220717 , 546622069/7324302 )
**u= 180/137 , ht= 15930581 ; ( 9629403/15930581 , -9749200703/95583486 ), ( 73190943/53707351 , 47545466575/322244106 )
**u= 180/157 , ht= 209779 ; ( 250501/210937 , 183602737/1265622 ), ( -209779/89777 , -127933537/538662 )
**u= 181/4 , ht= 153 ; ( 166989/19325 , -474516/3865 ), ( 11/153 , -721/51 )
**u= 181/49 , ht= 18637 ; ( 3687/18637 , 6925227/130459 ), ( -199367/44169 , 24811651/103061 )
**u= 181/121 , ht= 338751 ; ( 338751/87695 , 62972235/192929 ), ( -362635/407007 , -54503325/497453 )
**u= 181/128 , ht= 14611 ; ( 6290/14611 , 21415475/233776 ), ( -905259/613805 , -73621159/491044 )
**u= 181/184 , ht= 602818 ; ( 759705/3960454 , 1627267197/15841816 ), ( 602818/79959 , -81793697/106612 )
**u= 185/12 , ht= 213607 ; ( 802771/249007 , 21312593/249007 ), ( -192179/213607 , 7317247/213607 )
**u= 185/16 , ht= 5037 ; ( -104981953/4421889 , 3081426838/4421889 ), ( 5037/881 , -301861/1762 )
**u= 185/21 , ht= 36191 ; ( -36191/21083 , -1410541/21083 ), ( 206471/24317 , -7021327/24317 )
**u= 185/28 , ht= 8787 ; ( -6071/8787 , -415883/8787 ), ( 729/29497 , 88250/2269 )
**u= 185/36 , ht= 581 ; ( 103/581 , -26059/581 ), ( -527585431/93331287 , 689176298/2718387 )
**u= 185/61 , ht= 145367 ; ( -29238361/13873193 , 1859041909/13873193 ), ( -145367/76071 , -3142439/25357 )
**u= 185/92 , ht= 392809 ; ( -69279/724037 , -51323799/724037 ), ( 392809/55047 , 9329564/18349 )
**u= 185/168 , ht= 49061 ; ( -15276214/1038623 , 5839300465/4154492 ), ( 49061/5038 , 18809781/20152 )
**u= 185/173 , ht= 752393 ; ( 23241459/10193213 , -2455399695/10193213 ), ( -752393/238551 , -25456495/79517 )
**u= 185/192 , ht= 3156167 ; ( 29736223/1041061 , 1732991125/594892 ), ( 3156167/2024201 , 764364741/4048402 )
**u= 185/196 , ht= 4076733 ; ( -4076733/627859 , 2973456372/4395013 ), ( -6813137/1304619 , -1666949873/3044111 )
**u= 193/2 , ht= 67 ; ( -9066/2297 , -146457/4594 ), ( 23/67 , 574/67 )
**u= 193/30 , ht= 147017 ; ( -18332459/21212906 , 2212251601/42425812 ), ( -147017/15163 , -5835984/15163 )
**u= 193/42 , ht= 91933 ; ( -223561/85283 , -11163516/85283 ), ( -91933/14807 , -334488/1139 )
**u= 193/50 , ht= 64447 ; ( 780169/206981 , 41091562/206981 ), ( -4189/64447 , -16438498/322235 )
**u= 193/110 , ht= 10091 ; ( -2482059/426946 , 380441619/853892 ), ( -10091/2911 , -793352/2911 )
**u= 193/138 , ht= 12348625 ; ( 101896459/411909695 , -7180209170/82381939 ), ( 12348625/4548923 , 1113410570/4548923 )
**u= 193/158 , ht= 718405 ; ( -247518349/17759306 , 44927696663/35518612 ), ( -81786/718405 , -130905407/1436810 )
**u= 193/182 , ht= 32041 ; ( 245475/264832 , 70168887/529664 ), ( 32041/243 , -115306/9 )
**u= 196/5 , ht= 12001 ; ( 15847/807 , 3411733/11298 ), ( -12001/6929 , 243017/7462 )
**u= 196/69 , ht= 265423 ; ( -265423/23273 , 221395039/325822 ), ( -1392653/147565 , -1164140311/2065910 )
**u= 196/89 , ht= 790075 ; ( 1453503/57805 , -274574085/161854 ), ( -790075/25743 , 248744135/120134 )
**u= 196/129 , ht= 143671 ; ( -331955/179401 , 428777263/2511614 ), ( 143671/653 , 163275633/9142 )
**u= 197/14 , ht= 16484 ; ( 573255/61477 , 15303150/61477 ), ( 16484/1665 , 19735/74 )
**u= 197/18 , ht= 14317 ; ( -14317/4817 , -455756/4817 ), ( 57571/8404 , 3529269/16808 )
**u= 197/70 , ht= 116593 ; ( -116593/91762 , -17697217/183524 ), ( -20896006/5007809 , 2563540961/10015618 )
**u= 197/82 , ht= 1320587 ; ( -7814336/1152159 , -1019609393/2304318 ), ( 1320587/459833 , -90275138/459833 )
**u= 197/186 , ht= 281359 ; ( 309730/96977 , -63104765/193954 ), ( -281359/14195 , 5477800/2839 )
528
>

■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。

[2026.06.16追記]u=904/5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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