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Integer Points on A^4+20003*B^4=C^4+20003*D^4


[2026.06.10]A^4+20003*B^4=C^4+20003*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=20003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、9個の整数解
    73^4+20003*7^4=93^4+20003*3^4
    73^4+20003*7^4=93^4+20003*3^4
    257^4+20003*64^4=739^4+20003*38^4
    1617^4+20003*342^4=4027^4+20003*172^4
    4945^4+20003*16394^4=89605^4+20003*16208^4
    8778^4+20003*5787^4=67582^4+20003*2981^4
    16163^4+20003*487^4=16207^4+20003*371^4
    19813^4+20003*1171^4=20857^4+20003*593^4
    29309^4+20003*4184^4=50537^4+20003*2038^4
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 = 83, 415/29, 1385/6. 2158/17, 4097/24. 12035/289. 40006/221, 40006/257, 100015/548
となる。
{gp2cによる計算]
gp > sss(20003,100000)
1:73^4+20003*7^4=93^4+20003*3^4
2:203^4+20003*59^4=701^4+20003*19^4
3:257^4+20003*64^4=739^4+20003*38^4
4:1617^4+20003*342^4=4027^4+20003*172^4
5:4945^4+20003*16394^4=89605^4+20003*16208^4
6:8778^4+20003*5787^4=67582^4+20003*2981^4
7:16163^4+20003*487^4=16207^4+20003*371^4
8:19813^4+20003*1171^4=20857^4+20003*593^4
9:29309^4+20003*4184^4=50537^4+20003*2038^4
9 solutions,
time = 9h, 32min, 1,522 ms.
%1 = 9
{Pari/GPによる計算]
gp > uu(20003,73,7,93,3)
[u - 83, 1; u, -1; 20*u + 6989, 1]
[u - 83, 1; u, -1; 20*u - 6989, 1]
[u, -1; u + 83, 1; 20*u + 6989, 1]
time = 1 ms.
%1 = [83, -6989/20]
gp > uu(20003,203,59,701,19)
[u, -1; 24*u + 4097, 1; 65*u - 9379, 1]
[u, -1; 24*u - 4097, 1; 65*u - 9379, 1]
[u, -1; 24*u + 4097, 1; 65*u + 9379, 1]
time = 1 ms.
%2 = [-4097/24, 9379/65]
gp > uu(20003,257,64,739,38)
[u, -1; 6*u + 1385, 1; 221*u - 40006, 1]
[u, -1; 6*u - 1385, 1; 221*u - 40006, 1]
[u, -1; 6*u + 1385, 1; 221*u + 40006, 1]
time = 1 ms.
%3 = [-1385/6, 40006/221]
gp > uu(20003,1617,342,4027,172)
[u, -1; 170*u + 36637, 1; 257*u - 40006, 1]
[u, -1; 170*u - 36637, 1; 257*u - 40006, 1]
[u, -1; 170*u + 36637, 1; 257*u + 40006, 1]
time = 1 ms.
%4 = [-36637/170, 40006/257]
gp > uu(20003,4945,16394,89605,16208)
[u, -1; 186*u + 247025, 1; 16301*u - 21517750, 1]
[u, -1; 186*u - 247025, 1; 16301*u - 21517750, 1]
[u, -1; 186*u + 247025, 1; 16301*u + 21517750, 1]
time = 1 ms.
%5 = [-247025/186, 21517750/16301]
gp > uu(20003,8778,5787,67582,2981)
[u, -1; 548*u - 100015, 1; 22448*u + 4237573, 1]
[u, -1; 548*u - 100015, 1; 22448*u - 4237573, 1]
[u, -1; 548*u + 100015, 1; 22448*u + 4237573, 1]
time = 1 ms.
%6 = [100015/548, -4237573/22448]
gp > uu(20003,16163,487,16207,371)
[u, -1; 29*u - 415, 1; 1716*u + 9032921, 1]
[u, -1; 29*u - 415, 1; 1716*u - 9032921, 1]
[u, -1; 29*u + 415, 1; 1716*u + 9032921, 1]
time = 1 ms.
%7 = [415/29, -9032921/1716]
gp > uu(20003,19813,1171,20857,593)
[u, -1; 289*u - 12035, 1; 1764*u + 1431781, 1]
[u, -1; 289*u - 12035, 1; 1764*u - 1431781, 1]
[u, -1; 289*u + 12035, 1; 1764*u + 1431781, 1]
time = 1 ms.
%8 = [12035/289, -1431781/1764]
gp > uu(20003,29309,4184,50537,2038)
[u, -1; 17*u - 2158, 1; 392718*u + 100382525, 1]
[u, -1; 17*u - 2158, 1; 392718*u - 100382525, 1]
[u, -1; 17*u + 2158, 1; 392718*u + 100382525, 1]
time = 1 ms.
%9 = [2158/17, -100382525/392718]

■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883個のuが抽出される。
[MAGMAによる計算]
> PP(20003,1,200);
**u= 1/3 , ht= 49 ; ( -41/49 , -1581/7 ), ( 679/271 , -126637/271 )
**u= 1/35 , ht= 6907 ; ( 13393/3187 , 8145259/3187 ), ( -6907/6323 , -5540311/6323 )
**u= 1/36 , ht= 561 ; ( -337/561 , -107098/153 ), ( 561/337 , -1178078/1011 )
**u= 1/49 , ht= 2163 ; ( 2163/346 , 10734249/2422 ), ( -346/2163 , -3578083/5047 )
**u= 1/65 , ht= 66 ; ( -77431/439904 , -360141169/439904 ), ( 66 , 53221 )
**u= 1/81 , ht= 7998 ; ( 1279/7998 , -65611843/71982 ), ( 1278/7999 , -65618563/71991 )
**u= 1/105 , ht= 99733 ; ( 99733/11008 , -102824245/11008 ), ( 887284/100939 , 915128549/100939 )
**u= 1/120 , ht= 122197 ; ( 122197/58432 , 593551869/233728 ), ( -221096/12449 , -970401399/49796 )
**u= 1/128 , ht= 12736 ; ( -67784/4749 , -1230121453/75984 ), ( -10759/12736 , 301821433/203776 )
**u= 1/144 , ht= 19579 ; ( -6313/19579 , -7053653/5594 ), ( -27235/9377 , -69134907/18754 )
**u= 1/152 , ht= 3943 ; ( -3943/2138 , -22121263/8552 ), ( 244234/47641 , 1227240049/190564 )
**u= 1/179 , ht= 6533 ; ( 6533/1521 , 8974993/1521 ), ( 493319/762893 , -1215579193/762893 )
**u= 1/193 , ht= 61874 ; ( -205311/26786 , 287665947/26786 ), ( -61874/9729 , 29006921/3243 )
**u= 2/7 , ht= 113 ; ( 113/20 , 42941/40 ), ( -885/9944 , 3735683/19888 )
**u= 2/19 , ht= 3794 ; ( 3794/1089 , 2433397/2178 ), ( 6510/463 , 4023473/926 )
**u= 2/35 , ht= 13038 ; ( 13038/5197 , 11743911/10394 ), ( 134749/33594 , -38732877/22396 )
**u= 2/43 , ht= 1013 ; ( -95369/12250 , -12739267/3500 ), ( -626/1013 , 1104401/2026 )
**u= 2/63 , ht= 97772 ; ( -106168/27679 , -123166011/55358 ), ( -97772/4239 , -329580421/25434 )
**u= 2/67 , ht= 96598 ; ( 96598/12781 , 112803257/25562 ), ( 169202/66491 , -210461731/132982 )
**u= 2/163 , ht= 105282 ; ( 105282/92405 , 252943713/184810 ), ( -152930/250821 , 58938779/55738 )
**u= 2/179 , ht= 48639 ; ( 48639/18746 , 98635233/37492 ), ( -2777609/418794 , -1771760831/279196 )
**u= 2/187 , ht= 4690 ; ( -43966/19733 , -93204509/39466 ), ( 4690/1511 , 9529843/3022 )
**u= 4/3 , ht= 1013 ; ( 557/1013 , 200251/2026 ), ( 1837/1253 , 385181/2506 )
**u= 4/7 , ht= 2155 ; ( 9831/1859 , 2647311/3718 ), ( 2155/183 , -190753/122 )
**u= 4/23 , ht= 3193 ; ( -22111/54219 , -82357/318 ), ( 3193/2557 , -1961959/5114 )
**u= 4/27 , ht= 158987 ; ( 112523/158987 , -303650969/953922 ), ( 254059/207211 , 511098233/1243266 )
**u= 4/39 , ht= 6785 ; ( -1171/6785 , -4300197/13570 ), ( 34577/1739 , -21622249/3478 )
**u= 4/71 , ht= 49471 ; ( -34103/63451 , 60702341/126902 ), ( 41235/49471 , -54270731/98942 )
**u= 4/83 , ht= 25281 ; ( -25281/6425 , -23766039/12850 ), ( -119003/170667 , -2340319/4214 )
**u= 4/99 , ht= 2859 ; ( 2859/2485 , 11307949/14910 ), ( -729235/753301 , -1043272187/1506602 )
**u= 4/123 , ht= 180521 ; ( 152231/180521 , 261911593/361042 ), ( 255721/185119 , -350147093/370238 )
**u= 4/135 , ht= 1269179 ; ( -1269179/170569 , 212574563/48734 ), ( -1335509/383791 , -4843935355/2302746 )
**u= 4/143 , ht= 194795 ; ( 261341/113479 , 340734427/226958 ), ( -194795/163849 , -304410689/327698 )
**u= 4/159 , ht= 47845 ; ( 3998941/446593 , 724885777/127598 ), ( 47845/13423 , -62664259/26846 )
**u= 4/179 , ht= 180841 ; ( -180841/92313 , -20897623/14202 ), ( -11415755/953861 , -15327612703/1907722 )
**u= 4/191 , ht= 71997 ; ( 71997/26921 , -106238229/53842 ), ( -2101245/1398257 , -3488438017/2796514 )
**u= 5/2 , ht= 69 ; ( 69/7 , 4386/7 ), ( -133/564 , 24431/376 )
**u= 5/33 , ht= 7127 ; ( -40147/7796 , -10507395/7796 ), ( -716/7127 , 1840313/7127 )
**u= 5/34 , ht= 101256 ; ( 643/125559 , 4677812/17937 ), ( -74753/101256 , -21881853/67504 )
**u= 5/38 , ht= 59293 ; ( -5340627/34169 , -1472449176/34169 ), ( 59293/279 , 1816374/31 )
**u= 5/49 , ht= 653 ; ( 653/256 , -1537099/1792 ), ( -6626/5563 , -18960161/38941 )
**u= 5/66 , ht= 6361 ; ( 7386371/1615727 , 2747261294/1615727 ), ( -1757/6361 , 2397786/6361 )
**u= 5/81 , ht= 85948 ; ( -107443/216064 , -97130301/216064 ), ( -85948/19699 , 106479161/59097 )
**u= 5/94 , ht= 861869 ; ( 861869/9383 , -373747640/9383 ), ( -2574067/220271 , 1120251916/220271 )
**u= 5/118 , ht= 15409 ; ( -15409/2822 , 15221473/5644 ), ( -7678/351809 , 341923457/703618 )
**u= 5/162 , ht= 28091 ; ( -19447/28091 , -8335232/12039 ), ( 1652329/458272 , 5856593645/2749632 )
**u= 5/182 , ht= 3199786 ; ( -4122837/1326841 , -2613243630/1326841 ), ( -3199786/841017 , -1330817619/560678 )
**u= 9 , ht= 81 ; ( 81/68 , 10573/204 ), ( 90/59 , 10771/177 )
**u= 9/4 , ht= 743 ; ( -297/761 , 163408/2283 ), ( -279/743 , 158728/2229 )
**u= 9/16 , ht= 255 ; ( -2813/3075 , 1667152/9225 ), ( -255/143 , -116953/429 )
**u= 9/25 , ht= 1927 ; ( -1927/1602 , 1253063/4806 ), ( 522/2447 , 1251119/7341 )
**u= 9/35 , ht= 5753 ; ( -18079/33931 , 22747211/101793 ), ( -5753/617 , 3423301/1851 )
**u= 9/49 , ht= 4509 ; ( 26059/206970 , 1022237371/4346370 ), ( -1928/4509 , -24030913/94689 )
**u= 9/52 , ht= 61809 ; ( 61809/26735 , -48565522/80205 ), ( -337075/129807 , -260488669/389421 )
**u= 9/68 , ht= 5453 ; ( 153389/88381 , 145993048/265143 ), ( -55/5453 , -4497223/16359 )
**u= 9/91 , ht= 3439 ; ( -298433/174039 , -329585239/522117 ), ( -1293/3439 , -3505073/10317 )
**u= 9/115 , ht= 17251 ; ( 17251/11409 , -22181081/34227 ), ( 34143/793 , -2817455/183 )
**u= 9/152 , ht= 10927 ; ( -374437/2390 , 1846723691/28680 ), ( 10927/9137 , -17562178/27411 )
**u= 9/164 , ht= 31405 ; ( -16319/31405 , -45327097/94215 ), ( -31405/16319 , 45327097/48957 )
**u= 9/172 , ht= 292449 ; ( -1524267/305125 , -2038869224/915375 ), ( 292449/78481 , -397143616/235443 )
**u= 9/187 , ht= 1831671 ; ( -1831671/517049 , -2602849489/1551147 ), ( -5602755/447817 , -7686663319/1343451 )
**u= 9/196 , ht= 134187 ; ( 253593/187835 , 441846593/563505 ), ( -134187/95159 , -1612254779/1998339 )
**u= 13/8 , ht= 118 ; ( 118/21 , 37609/84 ), ( -10903/70519 , -5597762/70519 )
**u= 13/12 , ht= 82387 ; ( -115657/7469 , 11135623/7469 ), ( -82387/659 , -7916566/659 )
**u= 13/15 , ht= 2677 ; ( -2677/863 , 302147/863 ), ( 2891/389 , 313371/389 )
**u= 13/16 , ht= 1389 ; ( 7329/1441 , -75336/131 ), ( -55/1389 , 102817/926 )
**u= 13/21 , ht= 16271 ; ( -65123/145570 , -20270311/145570 ), ( -12520/16271 , 2609553/16271 )
**u= 13/24 , ht= 34789 ; ( -120989/18184 , -66499717/72736 ), ( 34789/6739 , -4815157/6739 )
**u= 13/29 , ht= 407 ; ( -407/318 , -77149/318 ), ( -18170/12747 , 1105101/4249 )
**u= 13/44 , ht= 5175 ; ( -4087/5175 , -1213258/5175 ), ( -14879/18209 , 4326442/18209 )
**u= 13/45 , ht= 41614 ; ( 34679/41614 , -10079149/41614 ), ( 42226/30261 , -28998203/90783 )
**u= 13/68 , ht= 37663 ; ( 18863959/1028987 , -4321086427/1028987 ), ( 37391/37663 , -12138836/37663 )
**u= 13/69 , ht= 486463 ; ( 486463/382618 , -142596541/382618 ), ( 600770/14201 , -138457141/14201 )
**u= 13/77 , ht= 2539 ; ( -1482149/178 , -360742919/178 ), ( 838/2539 , 650761/2539 )
**u= 13/80 , ht= 4049 ; ( -251/4049 , -2012875/8098 ), ( 33699/26509 , -10637021/26509 )
**u= 13/87 , ht= 5311 ; ( 3397/5311 , 1631057/5311 ), ( -208805/84791 , 58304987/84791 )
**u= 13/96 , ht= 22283 ; ( 377957/554555 , 182384301/554555 ), ( -9235/22283 , -13110513/44566 )
**u= 13/101 , ht= 437312 ; ( 1541037/929078 , -501601287/929078 ), ( -437312/368733 , 53150929/122911 )
**u= 13/108 , ht= 116963 ; ( -96931/116963 , -6255394/16709 ), ( -606319/193757 , 550439827/581271 )
**u= 13/111 , ht= 14671 ; ( -7567477/327287 , 2213498013/327287 ), ( 14671/3571 , -4412459/3571 )
**u= 13/125 , ht= 84443 ; ( 84443/22712 , -135586321/113560 ), ( -284156/40459 , 445042033/202295 )
**u= 13/133 , ht= 95937 ; ( 103359/201610 , 72472131/201610 ), ( -59096/95937 , -12014427/31979 )
**u= 13/141 , ht= 134491 ; ( 154507/207100 , 85101619/207100 ), ( -35884/134491 , -45845501/134491 )
**u= 13/164 , ht= 118741 ; ( 13873083/3717547 , -5101689744/3717547 ), ( 45505/118741 , 45168901/118741 )
**u= 13/176 , ht= 59167 ; ( 172921/75219 , -138779537/150438 ), ( 59167/48497 , 28151083/48497 )
**u= 13/197 , ht= 4509642 ; ( 165839711/43932762 , 66789850781/43932762 ), ( 4509642/647851 , 1773666529/647851 )
**u= 16/23 , ht= 4547 ; ( 44101/7761 , 21476473/31044 ), ( 407/4547 , -2189521/18188 )
**u= 16/59 , ht= 7447 ; ( 7447/4615 , 6729991/18460 ), ( 29093/28739 , 31413761/114956 )
**u= 16/91 , ht= 167239 ; ( -109713/167239 , 190816101/668956 ), ( -425497/85689 , 138026219/114252 )
**u= 16/131 , ht= 48307 ; ( -109277/254379 , 316901869/1017516 ), ( 10579/48307 , 56604409/193228 )
**u= 16/179 , ht= 4659 ; ( 4659/5 , 6233787/20 ), ( -5331427/962571 , 2416277577/1283428 )
**u= 16/191 , ht= 1123349 ; ( 32256577/374475 , 44585816083/1497900 ), ( -1123349/944417 , -2028409459/3777668 )
**u= 17/6 , ht= 727 ; ( -727/188 , 89213/376 ), ( 9187/6313 , 662326/6313 )
**u= 17/14 , ht= 1251 ; ( 20421/62935 , -6005022/62935 ), ( 1094/1251 , -100549/834 )
**u= 17/18 , ht= 3391 ; ( 153542/21751 , -31915879/43502 ), ( 1307/3391 , 373974/3391 )
**u= 17/21 , ht= 7244 ; ( -841/7244 , -810607/7244 ), ( 237884/6941 , -26452997/6941 )
**u= 17/69 , ht= 82573 ; ( 620611/67760 , -125783813/67760 ), ( -46802/82573 , 269343/1163 )
**u= 17/94 , ht= 85985 ; ( 8117139/1285454 , -3865300893/2570908 ), ( -85985/28131 , 7091766/9377 )
**u= 17/126 , ht= 417645 ; ( 1893811/567654 , 3229709633/3405924 ), ( -417645/171503 , -368773876/514509 )
**u= 17/133 , ht= 243611 ; ( 990169/9028 , -276988111/9028 ), ( 162082/243611 , 81849041/243611 )
**u= 17/149 , ht= 68499 ; ( 68499/37954 , -3312273/5422 ), ( -5722738/813933 , 570469949/271311 )
**u= 17/162 , ht= 241019 ; ( 6703198/18708771 , 110435781607/336757878 ), ( 241019/54377 , -228833288/163131 )
**u= 17/178 , ht= 104277 ; ( -42929/104277 , 36492494/104277 ), ( -9766667/679886 , -6336425437/1359772 )
**u= 17/189 , ht= 28603 ; ( -28603/20432 , -35164049/61296 ), ( -2281532/25297 , -2282514851/75891 )
**u= 18/31 , ht= 13648 ; ( -37741/9100 , -30570997/54600 ), ( 5557/13648 , 11603897/81888 )
**u= 18/59 , ht= 50382 ; ( -154887/21818 , 169923779/130908 ), ( 47845/50382 , -75480403/302292 )
**u= 18/83 , ht= 2229294 ; ( -4754782/154419 , -6129790399/926514 ), ( -2229294/296977 , -2897838959/1781862 )
**u= 18/91 , ht= 10079 ; ( 10079/4346 , -14808619/26076 ), ( 1236194/170311 , -1683597539/1021866 )
**u= 18/103 , ht= 197141 ; ( 19672/197141 , -40625333/168978 ), ( 438655/48388 , 633455263/290328 )
**u= 18/107 , ht= 177638 ; ( 177638/26661 , -262792027/159966 ), ( -336190/180513 , -558255779/1083078 )
**u= 18/127 , ht= 214072 ; ( -186359/214072 , -452376199/1284432 ), ( 426548/30689 , -681614311/184134 )
**u= 25/9 , ht= 2664 ; ( -1427/2664 , 544081/7992 ), ( -4596/2555 , -4733437/38325 )
**u= 25/27 , ht= 12137 ; ( 12893/4903 , 21503671/73545 ), ( -12137/5077 , -20510161/76155 )
**u= 25/41 , ht= 10121 ; ( 64151/178582 , 24302467/178582 ), ( 5078/10121 , -1450211/10121 )
**u= 25/44 , ht= 1957 ; ( -549/1957 , 269670/1957 ), ( 94715/44401 , 69393221/222005 )
**u= 25/48 , ht= 8867 ; ( 8867/3917 , -6716421/19585 ), ( 11351/11669 , -22558741/116690 )
**u= 25/51 , ht= 5549 ; ( -14689/15265 , -15129963/76325 ), ( -5549/4345 , 5033467/21725 )
**u= 25/64 , ht= 12751 ; ( 15/12751 , -10201611/63755 ), ( -14017/5835 , 16196459/38900 )
**u= 25/83 , ht= 225767 ; ( 194111/273459 , -305540557/1367295 ), ( 169493/225767 , 257215259/1128835 )
**u= 25/91 , ht= 5469 ; ( 219255/255317 , -321063369/1276585 ), ( 1517/5469 , -360965/1823 )
**u= 25/104 , ht= 61 ; ( 35/61 , 71726/305 ), ( 10567209/6181087 , -2497120045/6181087 )
**u= 25/107 , ht= 2019 ; ( 2019/1591 , 2659179/7955 ), ( 29627/4161 , 2063303/1387 )
**u= 25/113 , ht= 5326 ; ( 40967/21192 , -49033811/105960 ), ( -5326/4957 , -1546985/4957 )
**u= 25/116 , ht= 139065 ; ( 1934805/37567 , -2084395419/187835 ), ( 139065/134579 , -208444703/672895 )
**u= 25/164 , ht= 17677 ; ( -247289/167877 , -76558619/167877 ), ( 14889/17677 , 5919971/17677 )
**u= 25/177 , ht= 580189 ; ( 580189/495406 , -1015074611/2477030 ), ( 457178/5597671 , 1494514101/5597671 )
**u= 26/3 , ht= 108263 ; ( 108263/29128 , -7604767/58256 ), ( 215432/57167 , -15168057/114334 )
**u= 26/7 , ht= 586 ; ( 586/551 , 83477/1102 ), ( -551/586 , -83477/1172 )
**u= 26/19 , ht= 25645 ; ( -1696/25645 , 4394639/51290 ), ( 188723/67636 , -34279367/135272 )
**u= 26/27 , ht= 7765 ; ( 3464/7765 , 5199223/46590 ), ( 15703/59392 , -37563917/356352 )
**u= 26/31 , ht= 36325 ; ( 1446866/189903 , -318703321/379806 ), ( 36325/1402 , 7939501/2804 )
**u= 26/55 , ht= 89101 ; ( 285554/73859 , 85803569/147718 ), ( -70806/89101 , -33108013/178202 )
**u= 26/67 , ht= 1157484 ; ( -11835561/53276 , 3800186457/106552 ), ( 510079/1157484 , 135376759/771656 )
**u= 26/111 , ht= 2064638 ; ( -2064638/36185 , -853389689/72370 ), ( 3035699/291298 , -1260337853/582596 )
**u= 26/115 , ht= 2332367 ; ( -2332367/1507828 , -1168287671/3015656 ), ( 2496129/916144 , 1118495707/1832288 )
**u= 26/163 , ht= 2708832 ; ( 404827/2708832 , -1371666467/5417664 ), ( 15087911/7966604 , 8544758291/15933208 )
**u= 26/167 , ht= 117565 ; ( 1386642/138985 , 100918509/39710 ), ( -117565/25554 , -20328973/17036 )
**u= 26/199 , ht= 1865351 ; ( 1865351/1706350 , -1398918511/3412700 ), ( 25985942/9588541 , -15327082667/19177082 )
**u= 29/7 , ht= 99695 ; ( -249659/21059 , 12305113/21059 ), ( -99695/3947 , -4904353/3947 )
**u= 29/13 , ht= 30 ; ( 17/30 , -2309/30 ), ( 133840/56867 , -9737941/56867 )
**u= 29/15 , ht= 253 ; ( -1651799/1934011 , -182935799/1934011 ), ( -253/223 , 24257/223 )
**u= 29/28 , ht= 33001 ; ( 33001/1641 , 3246866/1641 ), ( -267689/69799 , 27185456/69799 )
**u= 29/37 , ht= 430091 ; ( -430091/37854 , 48771247/37854 ), ( -560650/16417 , 63360563/16417 )
**u= 29/40 , ht= 311707 ; ( -311707/30427 , 36784412/30427 ), ( -7128657/2274473 , 878875271/2274473 )
**u= 29/48 , ht= 103889 ; ( 404309/57209 , 105074783/114418 ), ( 43835/103889 , 29015489/207778 )
**u= 29/55 , ht= 153117 ; ( -153117/36127 , -21667041/36127 ), ( -2282023/212733 , 11691037/7879 )
**u= 29/63 , ht= 85321 ; ( 139813/28807 , 21041533/28807 ), ( -56485/85321 , 15082767/85321 )
**u= 29/71 , ht= 42307 ; ( -386859/309935 , 77567937/309935 ), ( 42307/5529 , 117133/97 )
**u= 29/72 , ht= 18336523 ; ( 169236631/55687895 , -28074806364/55687895 ), ( -5984853/18336523 , -9118392602/55009569 )
**u= 29/103 , ht= 169349 ; ( -1131631/234149 , -217800781/234149 ), ( 63465/169349 , 34085617/169349 )
**u= 29/124 , ht= 309499 ; ( -109807/309499 , -67912439/309499 ), ( 206169/725977 , 156066314/725977 )
**u= 29/152 , ht= 309561 ; ( 454641/40841 , 104512647/40841 ), ( 277159/309561 , 31711094/103187 )
**u= 29/165 , ht= 353267 ; ( -353267/125728 , -89449069/125728 ), ( 1216942/1386397 , 440057383/1386397 )
**u= 29/168 , ht= 593653 ; ( 593653/446933 , 178865138/446933 ), ( -4802783/402592 , -4640469843/1610368 )
**u= 29/175 , ht= 67161 ; ( -4272479/31765 , 5248246693/158825 ), ( -67161/17161 , 17029585/17161 )
**u= 29/180 , ht= 1300909 ; ( -16436479/5130351 , 12870260396/15391053 ), ( -1278141/1300909 , 1363178600/3902727 )
**u= 29/181 , ht= 270121 ; ( 1282413/1619254 , 39698019/124558 ), ( 59618/270121 , -69112901/270121 )
**u= 29/184 , ht= 188118 ; ( 33029/188118 , 192453563/752472 ), ( 188118/33029 , 192453563/132116 )
**u= 29/189 , ht= 72109 ; ( -72109/19378 , -19063167/19378 ), ( -2720096/809563 , -2173703143/2428689 )
**u= 29/196 , ht= 18383 ; ( 14609/18383 , -42734278/128681 ), ( 190747/228439 , 541626583/1599073 )
**u= 29/197 , ht= 26973 ; ( 3141861/2183456 , -90662193/198496 ), ( -16958/26973 , 2768237/8991 )
**u= 36/7 , ht= 1143 ; ( 1143/275 , -310879/1650 ), ( -10743/1787 , -2883403/10722 )
**u= 36/11 , ht= 3215 ; ( -23107/61021 , -21646771/366126 ), ( 3215/817 , 1100527/4902 )
**u= 36/19 , ht= 3437 ; ( 3437/2715 , 1909291/16290 ), ( 22549/7197 , 10318907/43182 )
**u= 36/55 , ht= 1401729 ; ( 1401729/86459 , -1041592673/518754 ), ( 27952361047/1331228157 , 20755299372449/7987368942
)
**u= 36/71 , ht= 136033 ; ( -7283369/762493 , -6171039047/4574958 ), ( -88469/136033 , 136741519/816198 )
**u= 36/91 , ht= 43059 ; ( -43059/4027 , -41257847/24162 ), ( 125843/46491 , 127986793/278946 )
**u= 36/127 , ht= 791653 ; ( 791653/349185 , -975150619/2095110 ), ( 7091145/1476221 , -8163269467/8857326 )
**u= 36/139 , ht= 19171 ; ( 19171/4437 , -23201441/26622 ), ( -54085/128283 , -164147989/769698 )
**u= 36/163 , ht= 4551 ; ( -4551/3599 , -7408183/21594 ), ( -1644365/62571 , -2101064149/375426 )
**u= 37/2 , ht= 16792 ; ( -16792/1111 , 775913/2222 ), ( -42501/23497 , 1134296/23497 )
**u= 37/9 , ht= 437 ; ( 437/32 , 21603/32 ), ( -4842/1843 , 40361/291 )
**u= 37/18 , ht= 7232 ; ( -7232/139 , -1008999/278 ), ( -25927/20364 , 13798289/122184 )
**u= 37/26 , ht= 99327 ; ( 47318/136745 , 24262481/273490 ), ( -99327/6182 , 16686953/12364 )
**u= 37/57 , ht= 70358 ; ( 124207/31240 , 15897577/31240 ), ( 70358/7829 , 8787377/7829 )
**u= 37/58 , ht= 8634 ; ( -8634/6997 , 2783013/13994 ), ( 6997/8634 , 927671/5756 )
**u= 37/78 , ht= 181289 ; ( 7633243/308536 , -2218554691/617072 ), ( 181289/92603 , 29559402/92603 )
**u= 37/89 , ht= 69118 ; ( 17587/69118 , 11062217/69118 ), ( 2074100/343673 , -326091679/343673 )
**u= 37/94 , ht= 874417 ; ( -767763/874417 , 185487918/874417 ), ( 902239/309099 , -50675208/103033 )
**u= 37/113 , ht= 441309 ; ( 441309/17672 , 77189901/17672 ), ( -881590/255909 , 53479171/85303 )
**u= 37/114 , ht= 14741 ; ( -37420456/1333445 , 13146120849/2666890 ), ( -14741/11264 , -6513343/22528 )
**u= 37/122 , ht= 1973013 ; ( 1973013/490655 , -33564348/44605 ), ( -37744157/15272346 , 4929419583/10181564 )
**u= 37/134 , ht= 102970 ; ( -2025949/731241 , -58560688/104463 ), ( 102970/87511 , -51437099/175022 )
**u= 37/138 , ht= 546521 ; ( 546521/149035 , -109409028/149035 ), ( -890785/533674 , 401118411/1067348 )
**u= 37/150 , ht= 386585 ; ( -386585/100213 , 402081712/501065 ), ( -330517/543617 , -128108022/543617 )
**u= 37/166 , ht= 126722 ; ( -104433/126722 , 69568683/253444 ), ( 129598/148107 , 27792371/98738 )
**u= 37/182 , ht= 187766 ; ( -306667/60154 , 138631637/120308 ), ( -187766/112093 , 97007773/224186 )
**u= 37/193 , ht= 999028 ; ( -2265390213/70695490 , -517683534951/70695490 ), ( 999028/508881 , -85361077/169627 )
**u= 40/3 , ht= 74 ; ( 74/73 , 11387/292 ), ( 202/251 , -35263/1004 )
**u= 40/27 , ht= 458 ; ( 458/431 , 206695/1724 ), ( 427/12826 , 12652423/153912 )
**u= 40/39 , ht= 34124 ; ( 34124/32447 , 18599537/129788 ), ( -22693/118256 , 2503263/24896 )
**u= 40/87 , ht= 35299 ; ( 35299/14912 , -22606893/59648 ), ( -210007/4216 , 123921241/16864 )
**u= 40/119 , ht= 54076 ; ( -107087/385216 , -275870735/1540864 ), ( 30057/54076 , 42687505/216304 )
**u= 40/179 , ht= 16154 ; ( -158662/9061 , 134483351/36244 ), ( -15887/16154 , 19173227/64616 )
**u= 41/3 , ht= 4991 ; ( -4991/4609 , 183713/4609 ), ( 10331/15161 , -495457/15161 )
**u= 41/16 , ht= 12175 ; ( 124551/6005 , 15577977/12010 ), ( 12175/11749 , 2114077/23498 )
**u= 41/19 , ht= 31099 ; ( -166111/21837 , -11404799/21837 ), ( -8307/31099 , 2191219/31099 )
**u= 41/33 , ht= 868 ; ( -7765181/384214 , -697530953/384214 ), ( -868/593 , 94319/593 )
**u= 41/36 , ht= 41443 ; ( -41443/2819 , 3892526/2819 ), ( 319709/62157 , -91566688/186471 )
**u= 41/52 , ht= 255485 ; ( 411187/92765 , -47473976/92765 ), ( -255485/1473 , 9591807/491 )
**u= 41/56 , ht= 139295 ; ( -487107/1027381 , 132892809/1027381 ), ( -139295/10047 , -5441034/3349 )
**u= 41/57 , ht= 27497 ; ( 27497/1022 , 3244573/1022 ), ( -317956/886739 , 111079509/886739 )
**u= 41/72 , ht= 70268 ; ( -867377/187623 , 352827719/562869 ), ( -11057/70268 , 37707897/281072 )
**u= 41/83 , ht= 23661 ; ( 23661/5899 , 3469803/5899 ), ( 1352705/387747 , -66743759/129249 )
**u= 41/100 , ht= 29551 ; ( -28651/29551 , -6428587/29551 ), ( 41007/5311 , -32291084/26555 )
**u= 41/112 , ht= 455303 ; ( 114613/455303 , -155211229/910606 ), ( 1538725/3641903 , 1306994773/7283806 )
**u= 41/120 , ht= 233123 ; ( 410207/43847 , 70582987/43847 ), ( 233123/31133 , -40239844/31133 )
**u= 41/136 , ht= 12821 ; ( -7477/12821 , -2703346/12821 ), ( 1262971/446512 , -975975239/1786048 )
**u= 41/139 , ht= 494623 ; ( -5480227/3806085 , 1228630369/3806085 ), ( 337275/494623 , 110239069/494623 )
**u= 41/153 , ht= 15403 ; ( -532669/408264 , 388968289/1224792 ), ( 14262/15403 , 12166283/46209 )
**u= 41/156 , ht= 108233 ; ( -2566649/14838115 , -2937543641/14838115 ), ( -108233/88951 , 27329162/88951 )
**u= 41/171 , ht= 414365 ; ( -185909/519827 , 16107791/74261 ), ( 414365/67083 , -257194469/201249 )
**u= 41/172 , ht= 1948037 ; ( -62179483/3110555 , -1821780542/444365 ), ( -1948037/713077 , -424920328/713077 )
**u= 41/192 , ht= 177095 ; ( 264197/153113 , -264338623/612452 ), ( 177095/122939 , 186624547/491756 )
**u= 45 , ht= 2 ; ( 1/2 , 103/6 ), ( -208/109 , -10801/327 )
**u= 45/17 , ht= 51651 ; ( 51651/22972 , -10423009/68916 ), ( 65764/13863 , -12395693/41589 )
**u= 45/38 , ht= 78747 ; ( 16683/395201 , 109058416/1185603 ), ( 78747/24559 , 22742674/73677 )
**u= 45/46 , ht= 99424 ; ( -34873/99424 , -63922049/596544 ), ( -28399/140237 , 43401706/420711 )
**u= 45/97 , ht= 37264 ; ( 163181/593182 , -270996901/1779546 ), ( 37264/3217 , 16475461/9651 )
**u= 45/154 , ht= 203789 ; ( -5607131/15716633 , 9261560830/47149899 ), ( 203789/84593 , 122464532/253779 )
**u= 45/161 , ht= 10771 ; ( -10771/3558 , 6437317/10674 ), ( 1206036/983557 , 883160071/2950671 )
**u= 45/178 , ht= 783071 ; ( 24777988/1542491 , 29627350385/9254946 ), ( -783071/233843 , -487650034/701529 )
**u= 49/9 , ht= 74 ; ( 7/74 , 22317/518 ), ( 468/421 , 566633/8841 )
**u= 49/17 , ht= 1247 ; ( 1247/402 , 540157/2814 ), ( -4370/4861 , -2695223/34027 )
**u= 49/25 , ht= 159 ; ( 159/112 , 486237/3920 ), ( -2018/1407 , -410057/3283 )
**u= 49/27 , ht= 1361 ; ( 1361/929 , -856279/6503 ), ( -6151/5179 , 4178571/36253 )
**u= 49/33 , ht= 21617 ; ( 28781/16 , 16533757/112 ), ( 14060/21617 , -14814383/151319 )
**u= 49/44 , ht= 5383 ; ( -8567/18621 , -13597597/130347 ), ( -5383/169 , 274832/91 )
**u= 49/72 , ht= 1204 ; ( -312536/39521 , 1069298021/1106588 ), ( 799/1204 , 4904829/33712 )
**u= 49/75 , ht= 311365 ; ( 28069/311365 , -1353833063/10897775 ), ( 18036341/1939975 , 78556825349/67899125 )
**u= 49/89 , ht= 34753 ; ( -187553/40020 , -180932681/280140 ), ( -12506/34753 , -34846523/243271 )
**u= 49/104 , ht= 24169 ; ( 24169/15 , 24649357/105 ), ( -24875/18038 , 125350781/505064 )
**u= 49/115 , ht= 164433 ; ( 164433/34253 , -180132723/239771 ), ( -4387799/1496889 , -28090723/59199 )
**u= 49/116 , ht= 6655 ; ( -5873/6655 , 9560332/46585 ), ( 1559271/852947 , 1914373801/5970629 )
**u= 49/123 , ht= 11453 ; ( 19489/29575 , -39284519/207025 ), ( 11453/7763 , 15346059/54341 )
**u= 49/129 , ht= 187778 ; ( 148181/187778 , 271703249/1314446 ), ( 1259204/164483 , 75918097/60599 )
**u= 49/131 , ht= 1079 ; ( 107179/63687 , -10977331/34293 ), ( 1079/817 , 1549171/5719 )
**u= 49/132 , ht= 21047 ; ( 11201/21101 , 27449211/147707 ), ( -21047/12877 , 28350137/90139 )
**u= 49/144 , ht= 21561 ; ( 758261/555073 , -2255485083/7771022 ), ( -21561/19207 , -103959074/403347 )
**u= 49/155 , ht= 49987 ; ( 169521/84679 , 235935225/592753 ), ( -35913/49987 , -76635235/349909 )
**u= 49/169 , ht= 13180 ; ( 7497/22454 , -400095879/2043314 ), ( -13180/10641 , -95432487/322777 )
**u= 49/171 , ht= 187191 ; ( -1899347/109029 , -7463966233/2289609 ), ( -187191/81317 , -800709977/1707657 )
**u= 49/195 , ht= 13171 ; ( -13171/6641 , 20599519/46487 ), ( -21703397/393613 , -30314449511/2755291 )
**u= 50/11 , ht= 146 ; ( 3/146 , -13707/292 ), ( -12311/5766 , 2126307/19220 )
**u= 50/39 , ht= 147683 ; ( 524768/242603 , -510617869/2426030 ), ( 147683/24248 , -132191541/242480 )
**u= 50/43 , ht= 4110 ; ( -4110/2347 , 4389381/23470 ), ( 3755/8958 , 3002697/29860 )
**u= 50/47 , ht= 1263680 ; ( 9419835/472868 , -9144797253/4728680 ), ( -1263680/41181 , 408653383/137270 )
**u= 50/67 , ht= 10975 ; ( -10975/4234 , -13617991/42340 ), ( 24513/13190 , -32225587/131900 )
**u= 50/111 , ht= 1763 ; ( 1652/1763 , -3600097/17630 ), ( 6487544/145781 , 9669437101/1457810 )
**u= 50/123 , ht= 521 ; ( -422/521 , -210333/1042 ), ( 3918301/418090 , -6180989237/4180900 )
**u= 50/139 , ht= 47291 ; ( -297046/53379 , -503244013/533790 ), ( 2054/47291 , -4154203/24890 )
**u= 50/151 , ht= 539437 ; ( -779079/778928 , 382930929/1557856 ), ( 539437/362184 , 75281571/241456 )
**u= 50/159 , ht= 385433 ; ( -385433/91408 , 706441241/914080 ), ( 23770252/1596589 , 8497446257/3193178 )
**u= 50/163 , ht= 330041 ; ( -310325/750642 , 1466684611/7506420 ), ( -330041/17638 , 119360209/35276 )
**u= 50/187 , ht= 260710 ; ( -1683987/354670 , 114771243/122300 ), ( 260710/8757 , 56056913/9730 )
**u= 52/3 , ht= 911 ; ( 3011/1847 , -169123/3694 ), ( 911/83 , 44271/166 )
**u= 52/27 , ht= 68021 ; ( -68021/32215 , 32540599/193290 ), ( -358885/58049 , -52401577/116098 )
**u= 52/35 , ht= 1923 ; ( -939/14519 , 2387601/29038 ), ( 1923/863 , 345887/1726 )
**u= 52/71 , ht= 130149 ; ( -257877/550387 , -142055403/1100774 ), ( 11579/130149 , 10179253/86766 )
**u= 52/83 , ht= 169373 ; ( 169373/27111 , -43344461/54222 ), ( -1890745/23899 , 477828281/47798 )
**u= 52/95 , ht= 67043 ; ( -113369/889831 , 242510095/1779662 ), ( 16123/67043 , -18641551/134086 )
**u= 52/123 , ht= 47647 ; ( -22981663/1843099 , -7092268471/3686198 ), ( -47647/23611 , 16358087/47222 )
**u= 52/167 , ht= 3257 ; ( -853782901/239609853 , 317854540109/479219706 ), ( 919/3257 , 1213027/6514 )
**u= 52/179 , ht= 127417 ; ( 629379/1076879 , 462874047/2153758 ), ( -127417/19363 , 47827021/38726 )
**u= 52/199 , ht= 1647777 ; ( 1647777/1535825 , 881372523/3071650 ), ( -2165365/208797 , 283731601/139198 )
**u= 53 , ht= 31 ; ( 1471/140 , 18847/140 ), ( 24/31 , 529/31 )
**u= 53/9 , ht= 2714 ; ( 41/2714 , -111957/2714 ), ( -5384/4741 , 295677/4741 )
**u= 53/14 , ht= 1647 ; ( -48591/12155 , -2573694/12155 ), ( 968/1647 , 21817/366 )
**u= 53/22 , ht= 16999 ; ( 587/16999 , -1096102/16999 ), ( -5981/26793 , 589534/8931 )
**u= 53/30 , ht= 31 ; ( -4921/23189 , -1783748/23189 ), ( -31/29 , -3194/29 )
**u= 53/42 , ht= 125261 ; ( -207889/272843 , -30537986/272843 ), ( -125261/89758 , 27438321/179516 )
**u= 53/86 , ht= 309190 ; ( -844401/572621 , -18567372/81803 ), ( -309190/135903 , 28683851/90602 )
**u= 53/102 , ht= 10792669 ; ( -39033419/12293185 , -5677599372/12293185 ), ( 2581210/10792669 , 3079139369/21585338 )
**u= 53/113 , ht= 482 ; ( -899403/1073464 , 204503541/1073464 ), ( 482/309 , -27869/103 )
**u= 53/118 , ht= 460247 ; ( 324173/1203214 , -371900683/2406428 ), ( 86010/460247 , 139736111/920494 )
**u= 53/130 , ht= 1188233 ; ( -74405484/2551669 , -23321403669/5103338 ), ( 1188233/114147 , -62322256/38049 )
**u= 53/153 , ht= 185101 ; ( -11242553/14709766 , 3145889739/14709766 ), ( -129858/185101 , -115260331/555303 )
**u= 53/158 , ht= 4771623 ; ( 4771623/1253944 , -1703802267/2507888 ), ( -48793291/9134817 , 2857220082/3044939 )
**u= 53/185 , ht= 144933 ; ( -144933/448 , -27080019/448 ), ( 230968/1389003 , -87696897/463001 )
**u= 53/190 , ht= 145453 ; ( -1658729/121571 , -314926610/121571 ), ( -145453/110287 , 34563908/110287 )
**u= 58/7 , ht= 8635 ; ( -7746/8635 , 806199/17270 ), ( 773050/842109 , -26473393/561406 )
**u= 58/11 , ht= 541 ; ( 541/460 , -61849/920 ), ( 38840/21207 , -1285359/14138 )
**u= 58/15 , ht= 20582 ; ( -20582/2833 , -2112521/5666 ), ( -216431/36994 , 22341895/73988 )
**u= 58/59 , ht= 144952 ; ( 1503376/151391 , -304805519/302782 ), ( -144952/100547 , 35587993/201094 )
**u= 58/71 , ht= 25057 ; ( -25057/7378 , -825767/2108 ), ( -7325290/291377 , 1622381873/582754 )
**u= 58/95 , ht= 1086383 ; ( -1086383/594262 , -316980413/1188524 ), ( 1826202/2297 , 467480581/4594 )
**u= 58/99 , ht= 98908 ; ( -189836/67681 , -52665331/135362 ), ( 98908/3087 , -77577179/18522 )
**u= 58/155 , ht= 821257 ; ( -1772159/1229044 , 705166331/2458088 ), ( -294712/821257 , 6634801/38198 )
**u= 58/179 , ht= 229600 ; ( -164021/229600 , 99147997/459200 ), ( 1042872/1799863 , -56224949/276902 )
**u= 61/5 , ht= 36 ; ( -31/36 , 1361/36 ), ( 36/31 , -1361/31 )
**u= 61/28 , ht= 15873 ; ( -160407/123107 , -13699959/123107 ), ( -8683/15873 , -31431/407 )
**u= 61/45 , ht= 17674 ; ( 21447/16562 , 6982631/49686 ), ( -17674/14769 , -5935247/44307 )
**u= 61/56 , ht= 17482 ; ( -1433219/331677 , -140958542/331677 ), ( -17482/1921 , -6741113/7684 )
**u= 61/69 , ht= 1707893 ; ( -1707893/461996 , -188183067/461996 ), ( -43128230/461209 , 4587616923/461209 )
**u= 61/87 , ht= 120607 ; ( -120607/10175 , -14455549/10175 ), ( 708029/149489 , 86427819/149489 )
**u= 61/101 , ht= 247390 ; ( 335099/437450 , -5454757/33650 ), ( -247390/243653 , 44683327/243653 )
**u= 61/104 , ht= 862635 ; ( 186059/1165245 , -1176253/8895 ), ( 862635/169267 , -114793859/169267 )
**u= 61/111 , ht= 1033 ; ( 959/1033 , 190153/1033 ), ( -10750943/12118831 , 2185501353/12118831 )
**u= 61/116 , ht= 599427 ; ( 599427/34855 , -82806213/34855 ), ( -2598793/817961 , -375735782/817961 )
**u= 61/143 , ht= 719 ; ( 4837797/195055 , -741368373/195055 ), ( 719/711 , -51611/237 )
**u= 61/149 , ht= 7440761 ; ( 7440761/1363498 , -1182356341/1363498 ), ( 41907300/3225857 , -6569542591/3225857 )
**u= 61/159 , ht= 23461 ; ( -3230627/696991 , 533618283/696991 ), ( -23461/17243 , 4701083/17243 )
**u= 61/164 , ht= 12577 ; ( -12577/3905 , -2159486/3905 ), ( -6591483/1635659 , 1113652796/1635659 )
**u= 61/172 , ht= 368361 ; ( -8652231/1415101 , 1472281911/1415101 ), ( -188231/368361 , -7718624/40929 )
**u= 61/173 , ht= 430147 ; ( -430147/345412 , 92910971/345412 ), ( -1469624/1117591 , -310951103/1117591 )
**u= 61/192 , ht= 645509 ; ( 645509/90569 , 462607539/362276 ), ( 68465779/433831 , -48591647699/1735324 )
**u= 64/11 , ht= 8903 ; ( -55553/21601 , 19757839/172808 ), ( -799/8903 , -2962411/71224 )
**u= 64/47 , ht= 317 ; ( 519071/87235 , -360858847/697880 ), ( -223/317 , -265727/2536 )
**u= 64/95 , ht= 13913 ; ( -14453/21103 , -24932165/168824 ), ( -13913/347 , -13566103/2776 )
**u= 64/123 , ht= 13273 ; ( -13273/913 , -14756261/7304 ), ( -1180003/370253 , 105515953/227848 )
**u= 64/131 , ht= 193711 ; ( -193711/122785 , -262518811/982280 ), ( -23687/293071 , -336552167/2344568 )
**u= 64/135 , ht= 492403 ; ( 492403/51247 , 1725752185/1229928 ), ( 1157233/744883 , -4797513065/17877192 )
**u= 64/155 , ht= 77269 ; ( -460307/17093 , 573513145/136744 ), ( 77269/389 , -96207973/3112 )
**u= 64/159 , ht= 711841 ; ( -5436349/821255 , -6933247027/6570040 ), ( -711841/110507 , -908420547/884056 )
**u= 64/183 , ht= 672953 ; ( 672953/166165 , 937762877/1329320 ), ( 1556249/1424611 , 2854354313/11396888 )
**u= 64/195 , ht= 216349 ; ( -216349/200821 , 412238461/1606568 ), ( 10075789/92309 , -14071757221/738472 )
**u= 65/21 , ht= 886 ; ( 189977/4694 , 10799787/4694 ), ( 886/283 , 52881/283 )
**u= 65/22 , ht= 53 ; ( 4117/344 , -480635/688 ), ( -31/53 , 3572/53 )
**u= 65/42 , ht= 356324 ; ( -356324/169537 , 63441261/339074 ), ( -212323/889204 , -146976999/1778408 )
**u= 65/61 , ht= 107 ; ( -107/76 , 12715/76 ), ( -136164/72517 , 14946181/72517 )
**u= 65/82 , ht= 6999 ; ( 922761/33307 , -103716528/33307 ), ( -6999/1013 , -794378/1013 )
**u= 65/98 , ht= 273894 ; ( -273894/947 , 67266567/1894 ), ( 966401/1694598 , -1117903881/7908124 )
**u= 65/114 , ht= 499106 ; ( 14707913/5141261 , 2063525734/5141261 ), ( -49177/499106 , 132845113/998212 )
**u= 65/166 , ht= 6996881 ; ( 6996881/2202552 , -2344662259/4405104 ), ( -14753536/303577 , -4716823595/607154 )
**u= 65/198 , ht= 10204583 ; ( 10204583/3523216 , -3768662973/7046432 ), ( -12634868/1985859 , -13394627237/11915154 )
**u= 73/8 , ht= 6117 ; ( -162488/20929 , -21651811/83716 ), ( -6117/836 , 43117/176 )
**u= 73/20 , ht= 33777 ; ( -560629/19131 , 29354188/19131 ), ( 33777/9553 , -1837978/9553 )
**u= 73/24 , ht= 4601 ; ( 4601/2390 , 1189061/9560 ), ( 15266/26669 , 7047543/106676 )
**u= 73/27 , ht= 5989 ; ( -348329/181477 , -71658103/544431 ), ( -4315/5989 , 448927/5989 )
**u= 73/32 , ht= 227 ; ( 76/227 , -126817/1816 ), ( -50885/21059 , -7293491/42118 )
**u= 73/36 , ht= 12567 ; ( -9329/16465 , 1329118/16465 ), ( 12567/8473 , -3193456/25419 )
**u= 73/76 , ht= 2802087 ; ( -28017063/872105 , -2860229256/872105 ), ( 935593/2802087 , 100480504/934029 )
**u= 73/80 , ht= 93157 ; ( -93157/34943 , 20832445/69886 ), ( 16665767/3548297 , -1783924537/3548297 )
**u= 73/81 , ht= 271537 ; ( -271537/64350 , -264571141/579150 ), ( -31652796/1080581 , 30028245571/9725229 )
**u= 73/83 , ht= 353871 ; ( 29507/353871 , -37867619/353871 ), ( -4449459/1851403 , -513922349/1851403 )
**u= 73/105 , ht= 158906 ; ( -425213/69382 , -51674127/69382 ), ( 158906/150121 , -26219411/150121 )
**u= 73/123 , ht= 341369 ; ( 188357/341369 , 50613161/341369 ), ( 460865/158039 , 63247339/158039 )
**u= 73/129 , ht= 309011 ; ( 1668241/11347240 , 1524765311/11347240 ), ( -48964/309011 , -41593107/309011 )
**u= 73/131 , ht= 232283 ; ( -82117/232283 , 33006443/232283 ), ( -665215/2094693 , -98144863/698231 )
**u= 73/132 , ht= 3087461 ; ( 1193723/3087461 , 445157938/3087461 ), ( 3375071/326597 , 456003493/326597 )
**u= 73/153 , ht= 119237 ; ( 119237/79304 , 20733021/79304 ), ( 635720/929021 , -162982869/929021 )
**u= 73/179 , ht= 325917 ; ( -378450257/4918697 , -59270874119/4918697 ), ( 325917/284747 , 67775069/284747 )
**u= 73/185 , ht= 266464 ; ( -182659/266464 , 51432779/266464 ), ( 554102/699717 , -47365757/233239 )
**u= 73/200 , ht= 114057 ; ( -1106129/366737 , 964512028/1833685 ), ( 114057/71209 , 111289102/356045 )
**u= 81 , ht= 112 ; ( 537/196 , 49481/1764 ), ( -112/111 , -15793/999 )
**u= 81/4 , ht= 555 ; ( -373/555 , -134266/4995 ), ( 535/717 , 178411/6453 )
**u= 81/25 , ht= 2413 ; ( -33301/6582 , 84848867/296190 ), ( 576/2413 , -6201001/108585 )
**u= 81/35 , ht= 701 ; ( 74723/19073 , -45622145/171657 ), ( -701/461 , 496423/4149 )
**u= 81/49 , ht= 4881 ; ( 439/4962 , 24412243/312606 ), ( 358/4881 , -23981323/307503 )
**u= 81/56 , ht= 1951 ; ( 1951/1552 , 7462867/55872 ), ( -155/5459 , -4086881/49131 )
**u= 81/97 , ht= 47680 ; ( 441/47680 , -46965703/429120 ), ( 117858/234937 , -258886037/2114433 )
**u= 81/115 , ht= 13807 ; ( -91521/66719 , 121465003/600471 ), ( 10173/13807 , -18392659/124263 )
**u= 81/137 , ht= 140218 ; ( -29821919/4124626 , 35240285701/37121634 ), ( 140218/134327 , 227295853/1208943 )
**u= 81/139 , ht= 351 ; ( 73/351 , 422713/3159 ), ( -1112403/862231 , -1659473311/7760079 )
**u= 81/160 , ht= 65529 ; ( 13394/65529 , -676871699/4718088 ), ( -175773/29827 , -225533855/268443 )
**u= 81/163 , ht= 57975 ; ( -24503/57975 , 80363333/521775 ), ( -57975/24503 , -80363333/220527 )
**u= 81/169 , ht= 16901 ; ( -111/16982 , 287025103/1986894 ), ( -30/16901 , 285647131/1977417 )
**u= 81/172 , ht= 35421 ; ( 5321/35421 , -46979209/318789 ), ( -13445/52431 , 70992427/471879 )
**u= 81/188 , ht= 21011 ; ( 51583/75365 , 125231387/678285 ), ( 2399/21011 , -28998053/189099 )
**u= 81/193 , ht= 75340 ; ( 7355853/6341638 , -13493487929/57074742 ), ( -75340/123 , -104674123/1107 )
**u= 82/3 , ht= 4138 ; ( -9203/4954 , 395721/9908 ), ( 1585/4138 , 167153/8276 )
**u= 82/31 , ht= 40 ; ( 155172/38423 , -19656423/76846 ), ( -40/9 , -1681/6 )
**u= 82/87 , ht= 170693 ; ( 167764/170693 , 49308319/341386 ), ( 2336495/536204 , -493894397/1072408 )
**u= 82/103 , ht= 42153 ; ( -1072311/314128 , -250476783/628256 ), ( -3940/42153 , 3163469/28102 )
**u= 82/131 , ht= 1937226 ; ( 1937226/1552741 , 627648309/3105482 ), ( -1951526/319359 , 166643007/212906 )
**u= 82/143 , ht= 29335 ; ( -63416428/3171665 , -16771208617/6343330 ), ( 29335/24616 , -10114951/49232 )
**u= 83 , ht= 9 ; ( -9 , 81 ), ( -1/9 , -9 )
**u= 83/3 , ht= 2 ; ( -1/2 , 43/2 ), ( -2 , -43 )
**u= 83/9 , ht= 23 ; ( -13/23 , 871/23 ), ( 23/13 , 67 )
**u= 83/12 , ht= 163 ; ( 131/163 , -7954/163 ), ( 1127/29 , -42921/29 )
**u= 83/16 , ht= 157 ; ( 227/13 , 9977/13 ), ( 145/157 , 18767/314 )
**u= 83/27 , ht= 500 ; ( 439/500 , 113861/1500 ), ( 500/439 , 113861/1317 )
**u= 83/28 , ht= 6733 ; ( 4561/7213 , -495757/7213 ), ( -6733/111 , -130411/37 )
**u= 83/33 , ht= 1015 ; ( 817/1015 , 82167/1015 ), ( 21473/9659 , -1484917/9659 )
**u= 83/36 , ht= 147 ; ( 2123/827 , -150048/827 ), ( -47/147 , 30491/441 )
**u= 83/42 , ht= 1466 ; ( 6863/2054 , 1019189/4108 ), ( -1466/1117 , -262239/2234 )
**u= 83/55 , ht= 692 ; ( -289671/47996 , 23902185/47996 ), ( 692/417 , 21925/139 )
**u= 83/58 , ht= 431 ; ( -61229/7530 , 10314133/15060 ), ( 83/431 , 36692/431 )
**u= 83/62 , ht= 13917 ; ( -13917/7345 , -1360134/7345 ), ( 48215/111156 , 6981567/74104 )
**u= 83/64 , ht= 367 ; ( 367/65 , -65459/130 ), ( -601/1449 , -91837/966 )
**u= 83/71 , ht= 201 ; ( -201/154 , 23421/154 ), ( -1970/2787 , 105227/929 )
**u= 83/98 , ht= 41 ; ( -3/41 , 31272/287 ), ( -1259/633 , -357316/1477 )
**u= 83/102 , ht= 15451 ; ( 15451/2905 , -248994/415 ), ( -18655/8234 , 4521419/16468 )
**u= 83/112 , ht= 313 ; ( -557/829 , 116027/829 ), ( 313/61 , 74093/122 )
**u= 83/118 , ht= 6571 ; ( -6571/863 , -790268/863 ), ( -3461/6687 , 299282/2229 )
**u= 83/135 , ht= 217 ; ( 217/82 , 29587/82 ), ( 2/473 , 180985/1419 )
**u= 83/138 , ht= 9341 ; ( -146491/4550 , -37798927/9100 ), ( 6167/9341 , 1443388/9341 )
**u= 83/143 , ht= 14151 ; ( -14151/5362 , -283779/766 ), ( -17794/5067 , 809557/1689 )
**u= 83/150 , ht= 26530 ; ( 26530/7849 , -37195819/78490 ), ( -291715/10937 , 196234902/54685 )
**u= 83/190 , ht= 7339 ; ( 147963/106447 , 27580092/106447 ), ( 7339/5809 , 1416238/5809 )
**u= 85/7 , ht= 2047 ; ( -669/2047 , -61989/2047 ), ( -5909/6483 , 83887/2161 )
**u= 85/8 , ht= 5774 ; ( -199586/77493 , 26220613/309972 ), ( -1843/5774 , -742115/23096 )
**u= 85/12 , ht= 9931 ; ( -6737/9931 , 451143/9931 ), ( -49177/31961 , 2204990/31961 )
**u= 85/32 , ht= 1646 ; ( -1646/583 , -857081/4664 ), ( 16203/19661 , -3126559/39322 )
**u= 85/44 , ht= 40153 ; ( -40153/1449 , -2890744/1449 ), ( 212649/32843 , -15483545/32843 )
**u= 85/48 , ht= 65833 ; ( -61691/65833 , 13560691/131666 ), ( 759229/225673 , 119058279/451346 )
**u= 85/56 , ht= 1847 ; ( -41/1847 , 149974/1847 ), ( 299122/50241 , -32829857/66988 )
**u= 85/69 , ht= 126283 ; ( -1568881/1121498 , 173765637/1121498 ), ( 108964/126283 , -15028931/126283 )
**u= 85/77 , ht= 218956 ; ( -38147/218956 , 21155869/218956 ), ( 759184/395513 , 81482825/395513 )
**u= 85/84 , ht= 162073 ; ( 162073/132841 , 20833584/132841 ), ( -576377/374461 , 68333975/374461 )
**u= 85/92 , ht= 4321 ; ( -403953/13681 , 42052086/13681 ), ( 4321/3677 , -590321/3677 )
**u= 85/93 , ht= 274829 ; ( 274829/26798 , -28885047/26798 ), ( 1850146/835423 , -212358867/835423 )
**u= 85/96 , ht= 787 ; ( 115313/16516 , 99044035/132128 ), ( 787/371 , -92473/371 )
**u= 85/104 , ht= 37744 ; ( 37744/5543 , -16880069/22172 ), ( 66727/60409 , -9957026/60409 )
**u= 85/144 , ht= 154031 ; ( -154031/39073 , 504507/953 ), ( 1823721/112537 , 713531323/337611 )
**u= 85/151 , ht= 138591 ; ( -94926447/731741 , -12653430081/731741 ), ( 10597/138591 , -6175723/46197 )
**u= 85/197 , ht= 2573401 ; ( -28253503/1046504 , 4304514905/1046504 ), ( 1568898/2573401 , 458870227/2573401 )
**u= 89/16 , ht= 14827 ; ( -14827/5985 , 1355243/11970 ), ( 236803/1151 , 20098123/2302 )
**u= 89/27 , ht= 7969 ; ( -27079/145229 , -3488407/62241 ), ( 5755/7969 , -41647/613 )
**u= 89/28 , ht= 1037721 ; ( -10409459/1366529 , -588774709/1366529 ), ( -1037721/111049 , 58556744/111049 )
**u= 89/32 , ht= 40633 ; ( 60792/1877 , -29172393/15016 ), ( -40633/10833 , 1681449/7222 )
**u= 89/59 , ht= 188883 ; ( -188883/39565 , -15712977/39565 ), ( 43305881/1796313 , -1176486927/598771 )
**u= 89/73 , ht= 74520 ; ( 218923/97232 , 21695693/97232 ), ( -74520/6703 , -6777001/6703 )
**u= 89/75 , ht= 11737 ; ( -3145061/4042111 , 470188151/4042111 ), ( -11737/5945 , 6039431/29725 )
**u= 89/76 , ht= 17215 ; ( -23799/137305 , 1839816/19615 ), ( -17215/8323 , 1767151/8323 )
**u= 89/81 , ht= 67280 ; ( 7251/67280 , -58107077/605520 ), ( -2102452/1009519 , -222517969/1009519 )
**u= 89/84 , ht= 16075 ; ( -3559/16075 , -1599671/16075 ), ( 972287/56561 , 94627508/56561 )
**u= 89/97 , ht= 38579 ; ( 102311/113466 , 15951227/113466 ), ( 18694/38579 , -4475767/38579 )
**u= 89/121 , ht= 10939 ; ( -10939/6670 , -16433933/73370 ), ( -250538/56439 , -9982447/18813 )
**u= 89/147 , ht= 247265 ; ( -55733/247265 , -228045451/1730855 ), ( -959585/768607 , -1106134279/5380249 )
**u= 89/153 , ht= 648943 ; ( 5408959/417776 , 711352599/417776 ), ( 124850/648943 , 86652021/648943 )
**u= 89/169 , ht= 1973561 ; ( 1973561/177408 , -507132517/329472 ), ( -177408/1973561 , -3549927619/25656293 )
**u= 89/187 , ht= 4157 ; ( -4566383/958345 , -676375883/958345 ), ( -2575/4157 , 708857/4157 )
**u= 89/192 , ht= 367439 ; ( 963527/740483 , -713992441/2961932 ), ( -367439/84991 , 110795811/169982 )
**u= 89/193 , ht= 51340 ; ( 3436131/1370534 , -544806969/1370534 ), ( -51340/16017 , -138953/281 )
**u= 97/5 , ht= 2832 ; ( 127/2832 , 64967/2832 ), ( 44296/11881 , -1049761/11881 )
**u= 97/14 , ht= 94883 ; ( 339587/52503 , -13040492/52503 ), ( -1178/94883 , -7201799/189766 )
**u= 97/21 , ht= 133970 ; ( -61309/133970 , 6858121/133970 ), ( 511736/8261 , 23828147/8261 )
**u= 97/30 , ht= 4061 ; ( 4061/674 , 457787/1348 ), ( 155899/77839 , -9692834/77839 )
**u= 97/66 , ht= 94121 ; ( -183626/363569 , -67202741/727138 ), ( -48109/94121 , 8719558/94121 )
**u= 97/82 , ht= 80225 ; ( 362319/465557 , -54244842/465557 ), ( -80225/1587 , 2459474/529 )
**u= 97/98 , ht= 189419 ; ( -189419/54903 , 138768062/384321 ), ( -12750435/863129 , -8992617514/6041903 )
**u= 97/110 , ht= 422561 ; ( -7604711/754739 , -813852830/754739 ), ( -189094/422561 , 98603659/845122 )
**u= 97/138 , ht= 782636 ; ( -11082332/4764485 , -261624747/866270 ), ( -640013/782636 , 241195647/1565272 )
**u= 97/150 , ht= 10900 ; ( -3675163/273455 , -327367292/195325 ), ( -10900/4769 , 14796381/47690 )
**u= 97/190 , ht= 407937 ; ( -407937/3997 , -57099804/3997 ), ( 3284578/9813 , -306490199/6542 )
**u= 98/31 , ht= 13757 ; ( -11293/23600 , 20605379/330400 ), ( -13757/8208 , -4205419/38304 )
**u= 98/39 , ht= 6823 ; ( 6823/4132 , 7044789/57848 ), ( 60284/76651 , 86127807/1073114 )
**u= 98/67 , ht= 13106 ; ( 13106/9473 , -18720509/132622 ), ( 259865/60998 , 309029509/853972 )
**u= 98/95 , ht= 31772 ; ( -17997/31772 , 50337045/444808 ), ( 20008/136533 , 63405735/637154 )
**u= 98/103 , ht= 1007 ; ( 5073/34376 , 49878021/481264 ), ( -80/1007 , -76313/742 )
**u= 98/111 , ht= 70883 ; ( 70883/64852 , -143157277/907928 ), ( -537667/12412 , 801394913/173768 )
**u= 98/139 , ht= 261319 ; ( 2701458/291073 , 4530585489/4075022 ), ( 261319/246726 , -199756211/1151388 )
**u= 98/187 , ht= 685582 ; ( -487469/685582 , 1626964991/9598148 ), ( -7451494/199237 , -14416833737/2789318 )
**u= 98/199 , ht= 28617 ; ( 1247767/1404380 , 3748114933/19661320 ), ( -28617/1432 , 57166801/20048 )
**u= 100/23 , ht= 4603 ; ( 178809/118411 , 102840963/1184110 ), ( 4603/1383 , -768697/4610 )
**u= 100/31 , ht= 8789 ; ( 23193/2299 , -2594853/4598 ), ( -8789/945 , 547037/1050 )
**u= 100/47 , ht= 86081 ; ( 86081/19341 , -60483769/193410 ), ( -266867/117593 , -199959427/1175930 )
**u= 100/59 , ht= 122721 ; ( -122721/48497 , -20271933/96994 ), ( 84535/194511 , 54303663/648370 )
**u= 100/103 , ht= 295633 ; ( 957789/91369 , 976513287/913690 ), ( -295633/84483 , -104025017/281610 )
**u= 100/131 , ht= 3891 ; ( -3891/563 , -900021/1126 ), ( -131237/3955 , 150289261/39550 )
**u= 100/143 , ht= 12449 ; ( -2770123/2361 , -662558891/4722 ), ( 9589/12449 , -18792437/124490 )
**u= 100/183 , ht= 57067 ; ( 57067/42793 , 96499659/427930 ), ( 9355523/392183 , 12668076271/3921830 )
**u= 101/4 , ht= 2357 ; ( -6697/1609 , 135118/1609 ), ( -1855/2357 , 59471/2357 )
**u= 101/12 , ht= 11189 ; ( -11189/1129 , 386989/1129 ), ( -15037/23047 , -947937/23047 )
**u= 101/13 , ht= 19067 ; ( -19067/2240 , -687803/2240 ), ( 24120/43139 , -1771901/43139 )
**u= 101/24 , ht= 13100 ; ( -9901/13100 , -3202447/52400 ), ( -24340801/28877183 , -1841028428/28877183 )
**u= 101/39 , ht= 251 ; ( 9493193/1001021 , -593125223/1001021 ), ( -217/251 , 20619/251 )
**u= 101/40 , ht= 14091 ; ( -616/18181 , 4580341/72724 ), ( 14091/7156 , 3978947/28624 )
**u= 101/45 , ht= 87397 ; ( -87397/488 , -5833467/488 ), ( 142518/33703 , 29331503/101109 )
**u= 101/52 , ht= 56629 ; ( -56629/16379 , 4229836/16379 ), ( 1161339/360071 , -87256409/360071 )
**u= 101/55 , ht= 31723 ; ( -171861/19891 , -12766869/19891 ), ( 30513/31723 , -3248333/31723 )
**u= 101/63 , ht= 506449 ; ( 944781/1600009 , 440303899/4800027 ), ( -506449/224981 , -43772943/224981 )
**u= 101/64 , ht= 62677 ; ( -9429/62677 , -5046096/62677 ), ( 788587/238251 , 21861533/79417 )
**u= 101/69 , ht= 328874 ; ( 2633083/1166074 , -238030617/1166074 ), ( 328874/83303 , 28044637/83303 )
**u= 101/76 , ht= 309013 ; ( 41220199/3401763 , 3587928233/3401763 ), ( -309013/46021 , 27104252/46021 )
**u= 101/87 , ht= 4057343 ; ( 4057343/1728527 , -409335457/1728527 ), ( 8086571/751601 , -753838121/751601 )
**u= 101/116 , ht= 242707 ; ( 2517087/829055 , 284025558/829055 ), ( -242707/40461 , -8790674/13487 )
**u= 101/127 , ht= 737935 ; ( -269051/737935 , -8007677/67085 ), ( 1450097/1531863 , -78850079/510621 )
**u= 101/136 , ht= 2701203 ; ( -2701203/829901 , -327929274/829901 ), ( 16187609/1369617 , -628431057/456539 )
**u= 101/143 , ht= 422585 ; ( -141693/422585 , -53038833/422585 ), ( 27755603/4406277 , 1114754603/1468759 )
**u= 101/157 , ht= 8140254 ; ( -9948917/2394580 , -1275915989/2394580 ), ( 8140254/6657389 , -1311202759/6657389 )
**u= 101/160 , ht= 92543 ; ( -92543/42857 , -25674095/85714 ), ( 2346109/705184 , 2466924205/5641472 )
**u= 101/180 , ht= 237219 ; ( 333889/119919 , -142093696/359757 ), ( 237219/168269 , -116488520/504807 )
**u= 101/183 , ht= 4247659 ; ( 4247659/28411 , 571812283/28411 ), ( 255239/6371099 , 858334797/6371099 )
**u= 101/196 , ht= 334221 ; ( 748141/186483 , 751913714/1305381 ), ( 334221/198383 , 379029793/1388681 )
**u= 109/2 , ht= 3088 ; ( -3088/207 , -77437/414 ), ( -3883/1372 , 117847/2744 )
**u= 109/18 , ht= 38357 ; ( 12891/67495 , 8384906/202485 ), ( 38357/33063 , -6174904/99189 )
**u= 109/22 , ht= 294099 ; ( 17256993/48482 , 1549747173/96964 ), ( 101681/294099 , 358318/7541 )
**u= 109/33 , ht= 64633 ; ( 64633/20188 , 532169/2884 ), ( -369544/111539 , -21245819/111539 )
**u= 109/38 , ht= 266721 ; ( -4505141/237050 , 532673419/474100 ), ( 266721/217739 , -20332016/217739 )
**u= 109/65 , ht= 1192758 ; ( -5388073/1817298 , -62731075/259614 ), ( 1192758/944897 , -117518317/944897 )
**u= 109/78 , ht= 503989 ; ( -503989/83039 , -43209964/83039 ), ( -11927684/131311 , -2018367533/262622 )
**u= 109/90 , ht= 159779 ; ( -87586/159779 , -33117103/319558 ), ( 14524151/429566 , -2640991997/859132 )
**u= 109/122 , ht= 639223 ; ( -157262/639223 , 139299637/1278446 ), ( 6273625/376229 , 664975954/376229 )
**u= 109/129 , ht= 105623 ; ( -105623/41050 , -12328587/41050 ), ( 733318/133339 , 81091807/133339 )
**u= 109/137 , ht= 54762 ; ( -21547/54762 , -6598121/54762 ), ( -1813680/760687 , 220511953/760687 )
**u= 109/150 , ht= 44291 ; ( 42274157/21278543 , -427101042/1636811 ), ( -44291/7543 , 26355082/37715 )
**u= 109/170 , ht= 976181 ; ( -176249/976181 , -123892204/976181 ), ( -2430051/3423746 , 1048730213/6847492 )
**u= 109/177 , ht= 536671 ; ( 536671/254474 , -75692157/254474 ), ( -816814/1656289 , 235348317/1656289 )
**u= 109/193 , ht= 1745547 ; ( 1745547/513242 , -242120817/513242 ), ( 3905290/817227 , -176986373/272409 )
**u= 109/198 , ht= 1142671 ; ( 1651439/1450469 , 296261354/1450469 ), ( 106086/1142671 , -928078651/6856026 )
**u= 113 , ht= 1307 ; ( -5779/664 , 33659/664 ), ( -282/1307 , -8137/1307 )
**u= 113/4 , ht= 2765 ; ( -20513/7567 , 57872/1081 ), ( 2765/2369 , 68719/2369 )
**u= 113/9 , ht= 2146 ; ( -2019/2146 , -249541/6438 ), ( -6062/32127 , 2758057/96381 )
**u= 113/16 , ht= 76867 ; ( -951219/683875 , -44069427/683875 ), ( -76867/9087 , 1944203/6058 )
**u= 113/19 , ht= 13531 ; ( 153073/141863 , -8557901/141863 ), ( -6571/13531 , 616513/13531 )
**u= 113/20 , ht= 96211 ; ( -54621/96211 , 4656684/96211 ), ( 46727/232407 , 3322168/77469 )
**u= 113/25 , ht= 38 ; ( 740601/6189560 , 1466882691/30947800 ), ( -38/5 , 9019/25 )
**u= 113/27 , ht= 37151 ; ( -239917/73745 , -36796303/221235 ), ( -37151/14063 , 5827571/42189 )
**u= 113/33 , ht= 53509 ; ( -179341/41660 , -9947819/41660 ), ( 42718/53509 , 3700137/53509 )
**u= 113/36 , ht= 74239 ; ( 74239/44831 , -4894854/44831 ), ( 4759/120201 , -20366078/360603 )
**u= 113/43 , ht= 50587 ; ( 6431/50587 , 3146447/50587 ), ( 16232775/3697037 , 1027233293/3697037 )
**u= 113/51 , ht= 298249 ; ( 1439653/1354687 , 132812857/1354687 ), ( 298249/267649 , -26924109/267649 )
**u= 113/75 , ht= 18469 ; ( -630517/257521 , 25222101/117055 ), ( -18469/1993 , -7567901/9965 )
**u= 113/83 , ht= 55623 ; ( 55623/179 , 4767261/179 ), ( 19516075/2849343 , 563511721/949781 )
**u= 113/88 , ht= 108263 ; ( -108263/9470 , -38362973/37880 ), ( -661035/509629 , 73664579/509629 )
**u= 113/91 , ht= 693001 ; ( 693001/146819 , 63572129/146819 ), ( -4936689/3661121 , -551593889/3661121 )
**u= 113/99 , ht= 4280853 ; ( 11405621/1586801 , -1077901647/1586801 ), ( -4280853/49627 , 1202280587/148881 )
**u= 113/144 , ht= 390419 ; ( -3691717/624107 , 422683901/624107 ), ( 390419/333071 , -115873269/666142 )
**u= 113/153 , ht= 292452 ; ( -847427/312636 , 315331463/937908 ), ( -292452/151639 , 115006859/454917 )
**u= 113/155 , ht= 54100267 ; ( 560016981/44050259 , 65795181531/44050259 ), ( 54100267/5476587 , 2123033817/1825529 )
**u= 113/160 , ht= 22006 ; ( 22006/16289 , -26064851/130312 ), ( -39019/46611 , -4822501/31074 )
**u= 113/163 , ht= 5875479 ; ( -3628073/5875479 , 829423981/5875479 ), ( -28915323/136819 , -3473163887/136819 )
**u= 113/172 , ht= 82385 ; ( -15827/82385 , 10350941/82385 ), ( 2891585/117827 , -357074191/117827 )
**u= 113/192 , ht= 56483 ; ( -55399/56483 , -5893477/32276 ), ( -7890281/3655147 , 4534350163/14620588 )
**u= 113/195 , ht= 4603 ; ( -4603/3953 , 797105/3953 ), ( 25663/11233 , -3680309/11233 )
**u= 116/3 , ht= 26099 ; ( 26099/13265 , -920727/26530 ), ( -8281/44069 , -1391191/88138 )
**u= 116/39 , ht= 123953 ; ( -191041/73847 , 23749857/147694 ), ( 56431/123953 , -15792859/247906 )
**u= 116/43 , ht= 222967 ; ( -222967/84483 , 29032217/168966 ), ( -2250523/71047 , -274249843/142094 )
**u= 116/67 , ht= 634345 ; ( -471267/634345 , -120127317/1268690 ), ( 2194823/10683 , -111221027/7122 )
**u= 116/79 , ht= 1032267 ; ( 390629/1032267 , 182187227/2064534 ), ( 1464027/576541 , 259728607/1153082 )
**u= 116/127 , ht= 1386107 ; ( 1386107/1170715 , -379712597/2341430 ), ( 2252461/465571 , -481377083/931142 )
**u= 116/167 , ht= 8289605 ; ( -23674203/5449595 , -5830075497/10899190 ), ( -8289605/947499 , 667464267/631666 )
**u= 116/199 , ht= 220243 ; ( -13180163/1406139 , 3472440989/2812278 ), ( -220243/139549 , -68305417/279098 )
**u= 117/4 , ht= 2459 ; ( 47439/2701 , -2579287/8103 ), ( -2459/939 , 148328/2817 )
**u= 117/16 , ht= 115327 ; ( -190701/35561 , 42991663/213366 ), ( -115327/5583 , -12827357/16749 )
**u= 117/20 , ht= 255121 ; ( 14270147/1450567 , 1777756811/4351701 ), ( 255121/58831 , 32501726/176493 )
**u= 117/23 , ht= 90073 ; ( -614773/139271 , -7618349/37983 ), ( -82621/90073 , -16257989/270219 )
**u= 117/37 , ht= 50800 ; ( -10731/50800 , 8762023/152400 ), ( -938414/45897 , 158556137/137691 )
**u= 117/40 , ht= 4633 ; ( -207/4633 , -19847/339 ), ( -101967/66743 , -21380557/200229 )
**u= 117/44 , ht= 5393 ; ( -5393/3253 , -1158679/9759 ), ( -237821/19271 , -43907057/57813 )
**u= 117/53 , ht= 35291 ; ( -97931/868312 , 176470007/2604936 ), ( -10978/35291 , -7462357/105873 )
**u= 117/55 , ht= 213783 ; ( 2730267/68597 , -561739961/205791 ), ( -213783/144817 , -53118085/434451 )
**u= 117/79 , ht= 821553 ; ( -472783/821553 , 233689361/2464659 ), ( 1405671/1549609 , -515786857/4648827 )
**u= 117/88 , ht= 13865 ; ( -393623/62316 , -414762781/747792 ), ( 13865/9993 , -4447058/29979 )
**u= 117/95 , ht= 649081 ; ( 1793521/3105281 , 969488713/9315843 ), ( 649081/2321 , 175485337/6963 )
**u= 117/119 , ht= 3401 ; ( -28619/84431 , 26975023/253293 ), ( -3401/2549 , 1286017/7647 )
**u= 117/125 , ht= 3058 ; ( -67/3058 , 948565/9174 ), ( 68672/136127 , 236405807/2041905 )
**u= 117/128 , ht= 19319 ; ( -3593/19319 , 6166561/57957 ), ( -104749/85682 , -679479559/4112736 )
**u= 117/140 , ht= 3282581 ; ( 411461/3282581 , -1085757812/9847743 ), ( -4108171/1553009 , 1441399591/4659027 )
**u= 117/151 , ht= 413869 ; ( -39945671/15402171 , -14591904139/46206513 ), ( -210969/413869 , -158331271/1241607 )
**u= 117/164 , ht= 26268709 ; ( 99478229/1273471 , -35337915013/3820413 ), ( 26268709/11133559 , 10134426673/33400677 )
**u= 121/8 , ht= 565 ; ( 772/397 , -978941/17468 ), ( 565/427 , -200638/4697 )
**u= 121/9 , ht= 701 ; ( -701/450 , -748351/14850 ), ( 780/619 , 897181/20427 )
**u= 121/16 , ht= 8709 ; ( -16221/7927 , 14431839/174394 ), ( 5893/8709 , 1401416/31933 )
**u= 121/20 , ht= 3451 ; ( 2421/3451 , 269424/5423 ), ( 8267/423 , 411733/517 )
**u= 121/32 , ht= 679 ; ( -656/679 , -4273007/59752 ), ( 29145/14816 , 148001681/1303808 )
**u= 121/35 , ht= 3839 ; ( 4079/41669 , 24778661/458359 ), ( 3699/3839 , -3154117/42229 )
**u= 121/36 , ht= 1993 ; ( -22609/1437 , -40769951/47421 ), ( -1831/1993 , 1623924/21923 )
**u= 121/43 , ht= 15091 ; ( 24243/15145 , 18744123/166595 ), ( 15091/4431 , 3438691/16247 )
**u= 121/51 , ht= 517 ; ( 6869/19687 , 14893297/216557 ), ( 517/211 , 398847/2321 )
**u= 121/52 , ht= 108191 ; ( -457719/24919 , -330534954/274109 ), ( -108191/879 , 26012208/3223 )
**u= 121/59 , ht= 50807 ; ( 50807/9861 , -39752953/108471 ), ( 117629/39277 , -95271467/432047 )
**u= 121/60 , ht= 2763581 ; ( 7941581/994781 , -6199449160/10942591 ), ( 2763581/56621 , -2141487944/622831 )
**u= 121/65 , ht= 72708 ; ( 140121/279494 , -252099447/3074434 ), ( 72708/6733 , 58879295/74063 )
**u= 121/72 , ht= 5624 ; ( -5624/4175 , -23774627/183700 ), ( -8357/12228 , -150817259/1614096 )
**u= 121/81 , ht= 466 ; ( -47897/21826 , -142121333/720258 ), ( 466/403 , -1663577/13299 )
**u= 121/92 , ht= 274751 ; ( 274751/227739 , -342316529/2505129 ), ( -563209/36171 , 180462613/132627 )
**u= 121/105 , ht= 109372 ; ( -223121/19184 , 229484227/211024 ), ( 109372/907 , -112088929/9977 )
**u= 121/107 , ht= 85171 ; ( 57541/341305 , 358068493/3754355 ), ( 29531/85171 , 93251699/936881 )
**u= 121/113 , ht= 320 ; ( -19/320 , -340799/3520 ), ( 320570/99881 , 356963351/1098691 )
**u= 121/124 , ht= 62853 ; ( 62853/9749 , 70830768/107239 ), ( 136115/107049 , -64282021/392513 )
**u= 121/131 , ht= 16819 ; ( -3421/72297 , 82848257/795267 ), ( -4927/16819 , 20060353/185009 )
**u= 121/147 , ht= 10519 ; ( -4639/10519 , -13939979/115709 ), ( 441953/53 , -3751221609/4081 )
**u= 121/152 , ht= 1178751 ; ( 2538607/173582 , 12549195823/7637608 ), ( 1178751/106049 , 1459263584/1166539 )
**u= 121/156 , ht= 308587 ; ( 308587/33559 , 387720871/369149 ), ( 498185/249131 , 695757451/2740441 )
**u= 121/179 , ht= 8829 ; ( -970687/355345 , -1383067051/3908795 ), ( 8829/7397 , -15411367/81367 )
**u= 122/3 , ht= 148 ; ( 148/19 , -4489/38 ), ( -3236/2473 , 129121/4946 )
**u= 122/31 , ht= 20722 ; ( -17798/79573 , 8223973/159146 ), ( 20722/13313 , 2483693/26626 )
**u= 122/39 , ht= 43097 ; ( -43097/18458 , 5301031/36916 ), ( -898309/157694 , -103164713/315388 )
**u= 122/55 , ht= 189774 ; ( 682489/29354 , 91729231/58708 ), ( 189774/211 , 25489093/422 )
**u= 122/83 , ht= 185276 ; ( -49863/185276 , -31655241/370552 ), ( 830908/1122819 , 76813359/748546 )
**u= 122/115 , ht= 1112 ; ( 893587/647188 , 214257235/1294376 ), ( -337/1112 , 225635/2224 )
**u= 122/127 , ht= 34228882 ; ( -147195907/11033266 , 30122218553/22066532 ), ( -34228882/2881661 , -7010043769/5763322
)
**u= 122/175 , ht= 433142 ; ( 14197814/224813 , -17007572279/2248130 ), ( -387559/433142 , 696161333/4331420 )
**u= 125 , ht= 77 ; ( -77/26 , -2237/130 ), ( -222/11 , 13253/55 )
**u= 125/14 , ht= 543 ; ( -543/451 , 576/11 ), ( -6929/3123 , -424514/5205 )
**u= 125/33 , ht= 722 ; ( 29/722 , 37143/722 ), ( 29932/54979 , 16080009/274895 )
**u= 125/34 , ht= 67 ; ( -6089/14107 , -4007932/70535 ), ( 67/16 , -35939/160 )
**u= 125/41 , ht= 11206 ; ( 591249/109048 , 172139067/545240 ), ( -11206/3867 , 226365/1289 )
**u= 125/49 , ht= 826 ; ( -723/826 , 2405733/28910 ), ( 3214/1803 , -2692271/21035 )
**u= 125/58 , ht= 44587 ; ( -75186/24313 , -53825007/243130 ), ( -44587/16254 , -10777343/54180 )
**u= 125/94 , ht= 29604 ; ( 23447/29604 , 6550279/59208 ), ( -987284/157683 , -289034853/525610 )
**u= 125/97 , ht= 9986 ; ( 164009/64972 , 77703803/324860 ), ( 9986/577 , -881245/577 )
**u= 125/106 , ht= 14051 ; ( 2338/14051 , -13118447/140510 ), ( 14051/2338 , -13118447/23380 )
**u= 125/114 , ht= 24623 ; ( -8701/24623 , 12471026/123115 ), ( 24623/8701 , 12471026/43505 )
**u= 125/129 , ht= 2323 ; ( 48481/19112 , -26471219/95560 ), ( -1676/2323 , -1455079/11615 )
**u= 125/174 , ht= 33739 ; ( -26407/33739 , -25276588/168695 ), ( -1127501/369073 , 699917424/1845365 )
**u= 125/177 , ht= 1933268 ; ( 1047029/1933268 , -1308221457/9666340 ), ( 2974976/1403747 , 1957361751/7018735 )
**u= 136/31 , ht= 360025 ; ( -19418082/6041179 , 3882396699/24164716 ), ( 360025/37302 , -23053581/49736 )
**u= 136/59 , ht= 115649 ; ( -3075932/118859 , -810946723/475436 ), ( 115649/27208 , 31306943/108832 )
**u= 136/63 , ht= 723934 ; ( 723934/11859 , 591318281/142308 ), ( -1213935/345266 , 1030966733/4143192 )
**u= 136/95 , ht= 400674 ; ( -101514/433919 , -148999359/1735676 ), ( -42581/400674 , 44902631/534232 )
**u= 136/99 , ht= 444155 ; ( -2529279/38831612 , -39846207383/465979344 ), ( 444155/85764 , 463195891/1029168 )
**u= 136/147 , ht= 93407 ; ( 620224/144091 , -264811441/576364 ), ( 93407/55792 , -316749781/1562176 )
**u= 136/167 , ht= 119665 ; ( -24642/119665 , -4166103/36820 ), ( 2841293/39958 , -1259643073/159832 )
**u= 136/171 , ht= 489476 ; ( -110027/489476 , -17310887/150608 ), ( 2398877/1357784 , 1236459003/5431136 )
**u= 137 , ht= 1551 ; ( 1551/16 , 3297/16 ), ( 2674/8881 , 36973/8881 )
**u= 137/4 , ht= 7921 ; ( 9497/1797 , 160597/1797 ), ( 7295/7921 , 183572/7921 )
**u= 137/9 , ht= 1119 ; ( -1119/440 , 92069/1320 ), ( 122602/16647 , -9567319/49941 )
**u= 137/27 , ht= 19669 ; ( -27307/395 , -1211697/395 ), ( 19669/5197 , 903719/5197 )
**u= 137/28 , ht= 12153 ; ( -12153/1891 , -555753/1891 ), ( 80237/33449 , 3931909/33449 )
**u= 137/35 , ht= 1025197 ; ( -4970451/2351821 , -277885683/2351821 ), ( -1025197/324693 , 18125303/108231 )
**u= 137/43 , ht= 887637 ; ( -775861/887637 , -66055159/887637 ), ( 752102297/12771391 , -42155683829/12771391 )
**u= 137/48 , ht= 68971 ; ( -101047/184685 , 24926757/369370 ), ( 27017/68971 , 8768437/137942 )
**u= 137/60 , ht= 57751 ; ( 57751/32471 , 4384582/32471 ), ( -379153/240707 , 29724979/240707 )
**u= 137/91 , ht= 288383 ; ( -186489/364919 , -33403323/364919 ), ( 211865/288383 , -29166077/288383 )
**u= 137/99 , ht= 2446635 ; ( -36154961/603145 , -3073954433/603145 ), ( -2446635/1385897 , -717167123/4157691 )
**u= 137/108 , ht= 7138699 ; ( 2588008013/680033585 , -712777082093/2040100755 ), ( -7138699/298907 , -1903365824/896721
)
**u= 137/136 , ht= 342048 ; ( 964508/240495 , -396182627/961980 ), ( -304835/342048 , -60870571/456064 )
**u= 137/147 , ht= 37 ; ( 67777/24559 , -7467831/24559 ), ( 37 , 26841/7 )
**u= 137/156 , ht= 406613 ; ( -406613/226487 , -49669521/226487 ), ( 2486405/63179 , -265433036/63179 )
**u= 137/163 , ht= 1131681 ; ( 23818727/464035 , -2598719311/464035 ), ( 282833/1131681 , 42414923/377227 )
**u= 137/171 , ht= 1054295 ; ( 1285733/317873 , -147978241/317873 ), ( 1054295/1004401 , 162695451/1004401 )
**u= 137/177 , ht= 41162 ; ( 5415131/137050 , 615744351/137050 ), ( -41162/7159 , -4749333/7159 )
**u= 137/192 , ht= 343633 ; ( -274001/343633 , 52033488/343633 ), ( 19298473/626041 , -351695167/96314 )
**u= 137/195 , ht= 707737 ; ( -2249627/352843 , 271689131/352843 ), ( 707737/14707 , -84461851/14707 )
**u= 137/196 , ht= 407437 ; ( 130387/407437 , -51172534/407437 ), ( -7154677/1476727 , -6117195277/10337089 )
**u= 144/35 , ht= 44581 ; ( -121199/52751 , -78182755/633012 ), ( 13571/44581 , -27561803/534972 )
**u= 144/55 , ht= 38499 ; ( 324981/43601 , 243149857/523212 ), ( 13199/38499 , 30181165/461988 )
**u= 144/67 , ht= 138975 ; ( -156673707/8318885 , -128418489671/99826620 ), ( -138975/98473 , -139433719/1181676 )
**u= 144/95 , ht= 69487 ; ( 175451/376471 , 404875889/4517652 ), ( -21523/69487 , -70903993/833844 )
**u= 144/127 , ht= 401367 ; ( -401367/314405 , -574609279/3772860 ), ( -908655/39019 , 1025055547/468228 )
**u= 144/155 , ht= 621449 ; ( 621449/519769 , 1008712439/6237228 ), ( -14887481/211999 , 18538433491/2543988 )
**u= 144/167 , ht= 1432973 ; ( 1149073/1432973 , 2373837373/17195676 ), ( -1647203/855793 , -2399007563/10269516 )
**u= 144/179 , ht= 2326021 ; ( 2326021/117877 , 3116181121/1414524 ), ( -2921293/579589 , 3984963589/6955068 )
**u= 144/187 , ht= 4050799 ; ( 4050799/359079 , -794491603/615564 ), ( 16166353/766497 , -22133976193/9197964 )
**u= 144/191 , ht= 3382859 ; ( 3382859/98423 , 359805149/90852 ), ( -3481777/673619 , -4901572793/8083428 )
**u= 144/199 , ht= 207963 ; ( 2665971/207127 , 3772382431/2485524 ), ( -207963/191 , 293393879/2292 )
**u= 145/2 , ht= 2359 ; ( 2359/1997 , -35506/1997 ), ( 6071/518 , 160597/1036 )
**u= 145/6 , ht= 392 ; ( 288629/54047 , 841700/7721 ), ( -392/209 , -959/22 )
**u= 145/18 , ht= 3329 ; ( 4697454/560887 , 998589211/3365322 ), ( 3329/478 , -237375/956 )
**u= 145/26 , ht= 1844 ; ( 1844/1103 , 181921/2206 ), ( 403327/170029 , -18545938/170029 )
**u= 145/42 , ht= 196099 ; ( -497197/323399 , -31920052/323399 ), ( -196099/130367 , 12675804/130367 )
**u= 145/101 , ht= 757424 ; ( 588227/1176754 , 109809833/1176754 ), ( 757424/642587 , -82905907/642587 )
**u= 145/117 , ht= 357631 ; ( 357631/115558 , 33761965/115558 ), ( 1186988/401521 , 112571033/401521 )
**u= 145/138 , ht= 315751 ; ( 49619851/5202857 , 442500270/472987 ), ( 315751/65267 , 31457852/65267 )
**u= 145/149 , ht= 298299 ; ( -298299/54982 , 30749535/54982 ), ( -855956/164247 , -29453253/54749 )
**u= 145/173 , ht= 619592 ; ( -126211/619592 , 69073709/619592 ), ( -91022422/378841 , 9943319507/378841 )
**u= 145/182 , ht= 618959 ; ( 618959/317203 , -77925724/317203 ), ( -6691771/1407593 , 766183876/1407593 )
**u= 145/186 , ht= 22624 ; ( -22624/18637 , 6640131/37274 ), ( 32473211/2104247 , 3685926776/2104247 )
**u= 145/194 , ht= 178477 ; ( -35373/391039 , 45419790/391039 ), ( -10279/178477 , 1590748/13729 )
**u= 146/19 , ht= 57034 ; ( 57034/39695 , -5011289/79390 ), ( 23405/87042 , -2165081/58028 )
**u= 146/43 , ht= 215885 ; ( 949751/2246 , -103063801/4492 ), ( 215885/66202 , 24516721/132404 )
**u= 146/51 , ht= 93154 ; ( -17317/93154 , 238361/3964 ), ( -10999927/13285186 , 2038891631/26570372 )
**u= 146/71 , ht= 99 ; ( 99/28 , -14349/56 ), ( -5314760/800967 , -249918453/533978 )
**u= 146/103 , ht= 1919323 ; ( -1118504/1940905 , 376346677/3881810 ), ( 1919323/344968 , -327625087/689936 )
**u= 146/115 , ht= 164814 ; ( 163549/164814 , 41217119/329628 ), ( 28478/489973 , -87120857/979946 )
**u= 146/171 , ht= 296566 ; ( -296566/217555 , -79616143/435110 ), ( 4108663/1029846 , 2750705863/6179076 )
**u= 146/183 , ht= 276020 ; ( -183127/276020 , 74175809/552040 ), ( -4713379/30895676 , 6998399641/61791352 )
**u= 146/195 , ht= 986554 ; ( 2262739/26314 , -523071655/52628 ), ( -986554/220691 , 233686151/441382 )
**u= 149/7 , ht= 2349 ; ( -5727/12545 , 301137/12545 ), ( 2349/1291 , -58453/1291 )
**u= 149/24 , ht= 944 ; ( 3803/67939 , -2733751/67939 ), ( -944/263 , -157459/1052 )
**u= 149/28 , ht= 18745 ; ( 30959/32891 , 1958311/32891 ), ( -18745/3641 , 828382/3641 )
**u= 149/40 , ht= 39941 ; ( -53003/127677 , 651361/11607 ), ( 39941/3376 , 8310797/13504 )
**u= 149/47 , ht= 119943 ; ( -119943/27355 , 6908373/27355 ), ( -239737/26847 , 1505891/2983 )
**u= 149/52 , ht= 25895 ; ( 573/25895 , -1530567/25895 ), ( 163945/1104729 , -21989772/368243 )
**u= 149/56 , ht= 107391 ; ( -4399113/11553449 , 758056122/11553449 ), ( -81470/107391 , -11018721/143188 )
**u= 149/61 , ht= 124172 ; ( -1092683389/24946892 , 69927361229/24946892 ), ( -124172/11139 , 2659567/3713 )
**u= 149/64 , ht= 284089 ; ( -770011/103525 , 1241869/2525 ), ( -284089/14147 , 74583059/56588 )
**u= 149/93 , ht= 143744 ; ( 1856431/1629790 , 195178973/1629790 ), ( -143744/60329 , -12317449/60329 )
**u= 149/96 , ht= 285880 ; ( 65551/285880 , -188364549/2287040 ), ( -864719/1216177 , 119786962/1216177 )
**u= 149/109 , ht= 298937 ; ( 298937/46384 , 25874921/46384 ), ( -8136628/4421371 , -792116357/4421371 )
**u= 149/112 , ht= 3807653 ; ( -3807653/2888603 , 414393274/2888603 ), ( 22859525/2991509 , 1999040119/2991509 )
**u= 149/117 , ht= 262640 ; ( -777873/395962 , 232051487/1187886 ), ( 262640/259689 , 98195117/779067 )
**u= 149/119 , ht= 14438127 ; ( 14438127/4517825 , -104004051/347525 ), ( 15217865/38210331 , -1225264313/12736777 )
**u= 149/148 , ht= 251823 ; ( -5871319/950519 , -592807154/950519 ), ( -251823/3403 , -25102463/3403 )
**u= 149/165 , ht= 1760071 ; ( -121511017/6363488 , 12805100383/6363488 ), ( 1113106/1760071 , -219161901/1760071 )
**u= 149/173 , ht= 14495234 ; ( 25425879/3834476 , -2770857321/3834476 ), ( 14495234/2910981 , 531079397/970327 )
**u= 149/175 , ht= 199783 ; ( 3225343/147915 , -1749656321/739575 ), ( 61883/199783 , -22667573/199783 )
**u= 149/184 , ht= 16299499 ; ( 16299499/2134742 , 7307500873/8538968 ), ( 51664099/5334877 , 5772279004/5334877 )
**u= 149/199 , ht= 2146627 ; ( -41616857/3905985 , 4830963193/3905985 ), ( -932699/2146627 , 270501859/2146627 )
**u= 153/2 , ht= 299 ; ( -826/1587 , -5767/414 ), ( -299/57 , -11776/171 )
**u= 153/22 , ht= 112361 ; ( -128701/814208 , 187777091/4885248 ), ( -32608/112361 , -26593769/674166 )
**u= 153/26 , ht= 85584 ; ( -284259/118039 , -38044132/354117 ), ( 57569/85584 , 25505249/513504 )
**u= 153/50 , ht= 261 ; ( -1837554/42953 , -3151739219/1288590 ), ( -261/214 , 115781/1284 )
**u= 153/58 , ht= 63239 ; ( 2707/63239 , 1063132/17247 ), ( 1688093/233569 , -314853836/700707 )
**u= 153/61 , ht= 13577 ; ( -13577/11128 , -3325469/33384 ), ( 72592/143047 , -30385741/429141 )
**u= 153/62 , ht= 71738 ; ( 710217/813571 , 206261954/2440713 ), ( 71738/53541 , -34193989/321246 )
**u= 153/77 , ht= 28044 ; ( 7453/28044 , 150649/2052 ), ( -729964/1672983 , -388470941/5018949 )
**u= 153/106 , ht= 159548 ; ( -1113292/152083 , -561168457/912498 ), ( 115187/159548 , -98281513/957288 )
**u= 153/110 , ht= 137043 ; ( 137043/58558 , 75821519/351348 ), ( -2648622/224743 , 1352486653/1348458 )
**u= 153/154 , ht= 147695 ; ( 135452/147695 , -120642953/886170 ), ( 211835/57247 , 66051346/171741 )
**u= 153/166 , ht= 952487 ; ( 952487/591437 , -350373094/1774311 ), ( -149310923/9038303 , -46747363834/27114909 )
**u= 153/173 , ht= 9480 ; ( 29696957/46426230 , 17582428411/139278690 ), ( -9480/6263 , 3624851/18789 )
**u= 153/178 , ht= 900775 ; ( -2575078/630161 , -245111017/540138 ), ( -900775/355282 , 626711417/2131692 )
**u= 153/181 , ht= 131707 ; ( 69644783/3154252 , 22749571699/9462756 ), ( -32098/131707 , -44236219/395121 )
**u= 153/197 , ht= 369207 ; ( 369207/112910 , -131437703/338730 ), ( -42518040/1586911 , 14485159069/4760733 )
**u= 157/12 , ht= 22433 ; ( -11201/68353 , 1922822/68353 ), ( -22433/5999 , -644412/5999 )
**u= 157/15 , ht= 1451 ; ( -87521/72559 , 3512519/72559 ), ( -1451/349 , -46245/349 )
**u= 157/21 , ht= 52667 ; ( -299687/702832 , -27972969/702832 ), ( -6098/52667 , -1936561/52667 )
**u= 157/29 , ht= 7342 ; ( 63/7342 , -315813/7342 ), ( -55240/49411 , 3185797/49411 )
**u= 157/36 , ht= 5887 ; ( -7249313/3060227 , 376700971/3060227 ), ( -2655/5887 , 927506/17661 )
**u= 157/47 , ht= 170431 ; ( -170431/43907 , 418601/1909 ), ( 324553/165939 , 6649599/55313 )
**u= 157/48 , ht= 485 ; ( -5395207/29819399 , -1676121672/29819399 ), ( -485/79 , -54359/158 )
**u= 157/56 , ht= 6871 ; ( -63181/220891 , 13724707/220891 ), ( 6814/6871 , -2311903/27484 )
**u= 157/77 , ht= 2704626 ; ( 2266871/3850068 , 312931343/3850068 ), ( -2704626/1061213 , -203499169/1061213 )
**u= 157/80 , ht= 10921 ; ( 22286689/10328579 , -3506910961/20657158 ), ( 10921/10471 , 1080095/10471 )
**u= 157/87 , ht= 21322415 ; ( 31231453/154435 , -34700019/2305 ), ( -21322415/1933189 , 1594010601/1933189 )
**u= 157/95 , ht= 102383 ; ( -2556727/2430393 , 274420889/2430393 ), ( 102383/99913 , 11128841/99913 )
**u= 157/96 , ht= 18716 ; ( -136033/51392 , 90970767/411136 ), ( -18716/3629 , 11927861/29032 )
**u= 157/108 , ht= 7685 ; ( -6527/7685 , 2508977/23055 ), ( 73789/27133 , 6521418/27133 )
**u= 157/117 , ht= 736559 ; ( -736559/703178 , 12559189/100454 ), ( 1407840/17701 , -364673407/53103 )
**u= 157/119 , ht= 19701 ; ( 30730017/358837 , 2675652693/358837 ), ( 13415/19701 , -230577/2189 )
**u= 157/125 , ht= 18338 ; ( -155731/88284 , 79870663/441420 ), ( -18338/7933 , -8915029/39665 )
**u= 157/128 , ht= 26029 ; ( 26029/13363 , 754861/3818 ), ( -148309/126858 , 93990939/676576 )
**u= 157/143 , ht= 23144509 ; ( -3064908983/59271505 , -292574494481/59271505 ), ( -9628517/23144509 ,
2392497829/23144509 )
**u= 157/160 , ht= 178679 ; ( 21185949/760381 , 2140226661/760381 ), ( -178679/119841 , 14480711/79894 )
**u= 157/168 , ht= 64637 ; ( 2101/64637 , -6690481/64637 ), ( 9113086943/295327111 , -943279266841/295327111 )
**u= 157/173 , ht= 398914 ; ( 40333/398914 , -42092273/398914 ), ( -115087676/4334397 , -4030232119/1444799 )
**u= 157/180 , ht= 95173 ; ( -45927/95173 , 33948245/285519 ), ( 95173/45927 , -33948245/137781 )
**u= 160/3 , ht= 946 ; ( -277/946 , -114271/7568 ), ( -2578/1219 , 326199/9752 )
**u= 160/19 , ht= 5278 ; ( 2151/5278 , -1573347/42224 ), ( 51454/77157 , 8517053/205752 )
**u= 160/23 , ht= 77356 ; ( -203576/12923 , 61802293/103384 ), ( 77356/26687 , 24845753/213496 )
**u= 160/27 , ht= 135169 ; ( -427886/94153 , 431633213/2259672 ), ( -135169/16342 , -44786025/130736 )
**u= 160/87 , ht= 4952 ; ( 25669/112988 , 68362109/903904 ), ( -1399/4952 , 3035609/39616 )
**u= 160/99 , ht= 15758 ; ( -15339/15758 , 41518829/378192 ), ( 40837/212954 , -136452527/1703632 )
**u= 160/111 , ht= 42247 ; ( 2540953/112976 , 1694833611/903808 ), ( -42247/5816 , 28419637/46528 )
**u= 160/163 , ht= 240237 ; ( 1787247/201746 , -1452379725/1613968 ), ( -200026/240237 , -28048721/213544 )
**u= 162/7 , ht= 7064 ; ( 7064/3445 , -2916547/62010 ), ( 8476/2897 , -3387107/52146 )
**u= 162/11 , ht= 370 ; ( 249/370 , 209627/6660 ), ( -66658/7383 , 31624907/132894 )
**u= 162/35 , ht= 2166 ; ( -2166/529 , 1864723/9522 ), ( 659601/50426 , -553806515/907668 )
**u= 162/43 , ht= 331981 ; ( -163078/331981 , -343105939/5975658 ), ( -651710/72797 , 608385767/1310346 )
**u= 162/47 , ht= 20443 ; ( -6152/229365 , 222540241/4128570 ), ( 2724/20443 , 19991173/367974 )
**u= 162/95 , ht= 105676 ; ( -105676/81759 , 184180771/1471662 ), ( 219353/46548 , 309133771/837864 )
**u= 162/127 , ht= 377312 ; ( -377312/266705 , -736441589/4800690 ), ( 557252/112879 , 69711757/156294 )
**u= 162/131 , ht= 121974 ; ( 121974/64465 , 223322237/1160370 ), ( 168330/236057 , -469317841/4249026 )
**u= 162/143 , ht= 41664 ; ( 32707/41664 , 12797837/107136 ), ( 13993105/2226384 , 23964672851/40074912 )
**u= 162/151 , ht= 592 ; ( 163/592 , -1067177/10656 ), ( 592/163 , -1067177/2934 )
**u= 162/187 , ht= 137722 ; ( -1349653/882566 , -3118834727/15886188 ), ( 137722/98639 , 327634219/1775502 )
**u= 164/7 , ht= 3989 ; ( 3989/671 , -164971/1342 ), ( -235367/62077 , 10177963/124154 )
**u= 164/19 , ht= 637 ; ( 10047/12031 , -1067463/24062 ), ( -485/637 , 54479/1274 )
**u= 164/27 , ht= 57941 ; ( -90619/76835 , 9640597/153670 ), ( 57941/4541 , -14162983/27246 )
**u= 164/31 , ht= 46415 ; ( -13229/46415 , -4199501/92830 ), ( -169637/148609 , -19613513/297218 )
**u= 164/39 , ht= 1093811 ; ( 2181341/445217 , -217061281/890434 ), ( 1093811/285887 , -110314313/571774 )
**u= 164/55 , ht= 163349 ; ( 29721/163349 , -19235391/326698 ), ( -3179683/5010807 , -229110761/3340538 )
**u= 164/63 , ht= 50069 ; ( 22601/50069 , 6810803/100138 ), ( 44543/103637 , 13982479/207274 )
**u= 164/139 , ht= 10907 ; ( 10907/2715 , 2069639/5430 ), ( 3100491/1729117 , 50286071/266018 )
**u= 164/171 , ht= 950511 ; ( -950511/579217 , 682000399/3475302 ), ( 54990895/24667161 , 36928991843/148002966 )
**u= 164/187 , ht= 334207 ; ( -334207/2767 , 71381147/5534 ), ( -2118243/1541347 , -559514887/3082694 )
**u= 164/195 , ht= 324821 ; ( 3656759/1592489 , 869879801/3184978 ), ( 324821/11651 , 70890641/23302 )
**u= 166/53 , ht= 232 ; ( -14716/1025 , 1666793/2050 ), ( 232/73 , 27493/146 )
**u= 166/101 , ht= 5623 ; ( 5623/876 , -887803/1752 ), ( 32531/7768 , 5218363/15536 )
**u= 166/117 , ht= 15529 ; ( -33328/8805 , -17364559/52830 ), ( -1800/15529 , 7874863/93174 )
**u= 166/165 , ht= 25807 ; ( 8812/25807 , -5438063/51614 ), ( -444028/157067 , 93922591/314134 )
**u= 169/9 , ht= 589 ; ( -691/420 , -724861/16380 ), ( -522/589 , 707623/22971 )
**u= 169/11 , ht= 1335 ; ( 4309/12647 , 4452307/164411 ), ( 1335/1063 , -566777/13819 )
**u= 169/12 , ht= 2753 ; ( -1693/2753 , 1122153/35789 ), ( -2753/1693 , -1122153/22009 )
**u= 169/27 , ht= 14299 ; ( -14299/3599 , -22967023/140361 ), ( 85253/4877 , 133252991/190203 )
**u= 169/33 , ht= 161309 ; ( 161309/7430 , 92709589/96590 ), ( 389876/11531 , 224228047/149903 )
**u= 169/36 , ht= 5687 ; ( -5687/4761 , 13349968/185679 ), ( 6025/4423 , 4485592/57499 )
**u= 169/56 , ht= 1280 ; ( -1093/1280 , 5038823/66560 ), ( -16245/11587 , 14934367/150631 )
**u= 169/57 , ht= 23302 ; ( -10727/23302 , 19371447/302926 ), ( 342730/11161 , 258968823/145093 )
**u= 169/81 , ht= 4591 ; ( 8101/258 , 65649163/30186 ), ( -3402/4591 , 46286177/537147 )
**u= 169/83 , ht= 15877 ; ( -15877/2597 , -14656519/33761 ), ( 12285/22411 , 23284231/291343 )
**u= 169/89 , ht= 8070 ; ( -5369/8070 , -9145207/104910 ), ( -87790/974853 , 307794503/4224363 )
**u= 169/128 , ht= 2539 ; ( -1964/2539 , -58111403/528112 ), ( -2539/1964 , 58111403/408512 )
**u= 169/140 , ht= 188711 ; ( -909079/236611 , 1111521421/3075943 ), ( -66089/188711 , 236593840/2453243 )
**u= 169/147 , ht= 239 ; ( -179/239 , -2534447/21749 ), ( 825515/69551 , 7031764731/6329141 )
**u= 169/152 , ht= 66730 ; ( 13773/66730 , -336052221/3469960 ), ( -575381/95061 , 18437656/31687 )
**u= 169/163 , ht= 39801 ; ( -39801/13721 , 53752203/178373 ), ( -4541173/407493 , 1940555717/1765803 )
**u= 169/168 , ht= 39514 ; ( 1794811/79699 , -2328749343/1036087 ), ( 13495/39514 , 216494019/2054728 )
**u= 169/180 , ht= 20753 ; ( 40717/288557 , -391012574/3751241 ), ( -20753/9093 , -91203533/354627 )
**u= 169/200 , ht= 6541 ; ( -6541/2925 , 50668918/190125 ), ( 37025/77984 , -2441852503/20275840 )
**u= 170/3 , ht= 677 ; ( 3632/3169 , 126673/6338 ), ( -284/677 , 18377/1354 )
**u= 170/39 , ht= 67382 ; ( 36611/67382 , 7348473/134764 ), ( -615623/5686 , -59007865/11372 )
**u= 170/43 , ht= 2748 ; ( 1931/2748 , 337901/5496 ), ( 136421/95012 , -16725697/190024 )
**u= 170/87 , ht= 911 ; ( 742/911 , -168121/1822 ), ( -281278/63749 , 41271271/127498 )
**u= 170/99 , ht= 1541811 ; ( 5141464/356673 , -2359796515/2140038 ), ( -1541811/687328 , 773019367/4123968 )
**u= 170/167 , ht= 14916618 ; ( 14916618/7435519 , 472010583/2124434 ), ( 1615800998/98115519 , -106972789791/65410346 )
**u= 173/6 , ht= 7685 ; ( 431/14306 , 544073/28612 ), ( 7685/4367 , -166366/4367 )
**u= 173/10 , ht= 13 ; ( -2/13 , -49/2 ), ( -13/2 , -637/4 )
**u= 173/17 , ht= 13006 ; ( -136413/5044 , 4268373/5044 ), ( -13006/11187 , -179341/3729 )
**u= 173/25 , ht= 259 ; ( 259/40 , 49759/200 ), ( -40/259 , 49759/1295 )
**u= 173/33 , ht= 1963400 ; ( 2740453/230176 , 120038621/230176 ), ( -1963400/161173 , 86104883/161173 )
**u= 173/46 , ht= 2793 ; ( -2793/2152 , -363621/4304 ), ( 136903/634347 , 3717222/70483 )
**u= 173/58 , ht= 19666 ; ( 189469/342415 , 22663574/342415 ), ( 7457/19666 , 2435389/39332 )
**u= 173/70 , ht= 7991 ; ( 390217/83242 , -50757185/166484 ), ( -3069/7991 , 8926/131 )
**u= 173/73 , ht= 285922 ; ( -227449/285922 , -23735587/285922 ), ( 2423116/233043 , -52720927/77681 )
**u= 173/105 , ht= 451594 ; ( -238291/451594 , -39783937/451594 ), ( 21023816/806381 , -1639322189/806381 )
**u= 173/126 , ht= 620819 ; ( -620819/521756 , -138425853/1043512 ), ( -2193596/709759 , 393566737/1419518 )
**u= 173/138 , ht= 146674 ; ( -146674/90847 , -30820161/181694 ), ( 12937345/1383778 , -2324400961/2767556 )
**u= 173/142 , ht= 662847 ; ( 986377/868995 , 2905034/21195 ), ( 662847/612739 , -81787046/612739 )
**u= 173/166 , ht= 30409409 ; ( 30409409/1096050 , 5961721601/2192100 ), ( 378156486/119684111 , -77714901247/239368222
)
**u= 173/177 , ht= 4463578 ; ( 4151881/4463578 , 616657989/4463578 ), ( -5483668/4083359 , 691614489/4083359 )
**u= 178/3 , ht= 1058 ; ( -193826/99403 , -5357949/198806 ), ( 995/1058 , -37509/2116 )
**u= 178/55 , ht= 191127 ; ( 191127/169028 , 28366809/338056 ), ( -629152/751623 , -36324715/501082 )
**u= 178/83 , ht= 77317 ; ( 81383/186710 , -27820501/373420 ), ( 21790/77317 , 10970351/154634 )
**u= 178/143 , ht= 993880 ; ( 1332213/3590144 , 686528313/7180288 ), ( -993880/589047 , 1683943/9578 )
**u= 178/147 , ht= 530066 ; ( -6204254/131137 , 7895505271/1835918 ), ( 391967/530066 , 119827217/1060132 )
**u= 178/179 , ht= 32302 ; ( -3344146/1030435 , -701861677/2060870 ), ( -32302/97 , 6479197/194 )
**u= 178/191 , ht= 106820 ; ( -836945372/52993703 , 24821487607/15141058 ), ( 106820/80933 , -27767257/161866 )
**u= 181/2 , ht= 3263 ; ( -27101284/1425405 , 439324099/2850810 ), ( 3263/448 , 81769/896 )
**u= 181/9 , ht= 11063 ; ( 11063/4650 , -797153/13950 ), ( -4650/11063 , -797153/33189 )
**u= 181/17 , ht= 75240 ; ( 40747/75240 , 2626549/75240 ), ( 130804/6551 , 4025351/6551 )
**u= 181/22 , ht= 617499 ; ( -584237601/44884189 , 20395980624/44884189 ), ( -293521/617499 , 7937632/205833 )
**u= 181/26 , ht= 547 ; ( -719203/276649 , 29181304/276649 ), ( -105/547 , -21088/547 )
**u= 181/42 , ht= 96335 ; ( -292781/7225 , -14102352/7225 ), ( -96335/24709 , -4793072/24709 )
**u= 181/50 , ht= 8595 ; ( 262572/76055 , 143648439/760550 ), ( -319/8595 , -251078/4775 )
**u= 181/54 , ht= 63841 ; ( 2622419/254567 , 431648692/763701 ), ( -27595/63841 , 11395226/191523 )
**u= 181/57 , ht= 207667 ; ( -207667/132908 , 13835707/132908 ), ( 737068234/51938209 , -41478431389/51938209 )
**u= 181/65 , ht= 36032 ; ( 5677/36032 , -2186467/36032 ), ( -15174/99161 , 6010861/99161 )
**u= 181/66 , ht= 230741 ; ( 230741/67073 , -14508932/67073 ), ( 25236601/6256453 , 1570435726/6256453 )
**u= 181/89 , ht= 15710 ; ( 9391/15710 , -1283597/15710 ), ( -36207958/22645339 , -2995020689/22645339 )
**u= 181/94 , ht= 811263 ; ( 1164129/1958852 , 46923261/559672 ), ( 166376/811263 , -39786593/540842 )
**u= 181/113 , ht= 84461 ; ( 84461/16096 , -6793747/16096 ), ( -230070/391321 , 35869091/391321 )
**u= 181/114 , ht= 693673 ; ( -1989029/1594447 , 202323456/1594447 ), ( 693673/236476 , -116339237/472952 )
**u= 181/121 , ht= 38769 ; ( 38769/2050 , -34917681/22550 ), ( -292996/341997 , 135019613/1253989 )
**u= 181/122 , ht= 439645 ; ( 57663/439645 , 36408444/439645 ), ( 1335783/6278 , -5101517/292 )
**u= 181/129 , ht= 1677190 ; ( -1645721/1677190 , 198386323/1677190 ), ( 2578994/1224623 , -241048667/1224623 )
**u= 181/170 , ht= 373534 ; ( -876961/381309 , 92681068/381309 ), ( -297609/373534 , 92577715/747068 )
**u= 181/174 , ht= 63645133 ; ( -160573771/1095632 , 31489870317/2191264 ), ( -63645133/290471 , -6240930894/290471 )
**u= 181/177 , ht= 875030 ; ( -8197307/1340810 , -821435751/1340810 ), ( 875030/525809 , 100960363/525809 )
**u= 181/182 , ht= 2659081 ; ( 2659081/112030 , 533783249/224060 ), ( 59016158/7200667 , -11924746211/14401334 )
**u= 181/198 , ht= 365070 ; ( 553344991/1163779 , 57878031132/1163779 ), ( 365070/79411 , -234477091/476466 )
**u= 185 , ht= 3134 ; ( 637/3134 , 37735/3134 ), ( -78524/51157 , -896165/51157 )
**u= 185/3 , ht= 1189 ; ( 1307/409 , 16021/409 ), ( -1189/1103 , 20801/1103 )
**u= 185/4 , ht= 13811 ; ( -54341/26753 , 861167/26753 ), ( 13811/1087 , -214213/1087 )
**u= 185/16 , ht= 4781 ; ( 6693/119819 , -505881/17117 ), ( 4781/2157 , 51534/719 )
**u= 185/19 , ht= 13967 ; ( -13967/2069 , 451495/2069 ), ( 28829/231103 , -7447015/231103 )
**u= 185/27 , ht= 37319 ; ( -37319/19637 , 4830209/58911 ), ( -519923/57529 , -60024433/172587 )
**u= 185/43 , ht= 10243 ; ( 152669/38337 , 7586351/38337 ), ( 10243/10139 , -694897/10139 )
**u= 185/48 , ht= 3229 ; ( -214463/2635909 , 269537715/5271818 ), ( -3229/467 , 166259/467 )
**u= 185/49 , ht= 2039 ; ( -4513/63264 , -3265553/63264 ), ( 912/2039 , -114937/2039 )
**u= 185/67 , ht= 155333 ; ( 125539/427843 , 26839381/427843 ), ( -155333/10291 , -9370891/10291 )
**u= 185/73 , ht= 353284 ; ( -405457/410026 , 36225605/410026 ), ( 353284/261157 , -2123137/20089 )
**u= 185/81 , ht= 3388 ; ( -59/3388 , -672785/10164 ), ( -1308314/106117 , -260616691/318351 )
**u= 185/84 , ht= 539633 ; ( 629513/1097561 , -85270472/1097561 ), ( 539633/448031 , -47266140/448031 )
**u= 185/96 , ht= 9974 ; ( 533/9974 , -5757063/79792 ), ( -352318/56771 , -15822251/34936 )
**u= 185/97 , ht= 77471 ; ( -532059/214162 , 41530815/214162 ), ( 64052/77471 , -7279151/77471 )
**u= 185/124 , ht= 690717 ; ( 832783/148181 , -69252455/148181 ), ( 212189/690717 , 19719642/230239 )
**u= 185/129 , ht= 995644 ; ( -1172947/1116674 , -10403437/85898 ), ( 995644/24317 , -83175885/24317 )
**u= 185/139 , ht= 492893 ; ( -492893/83679 , 43336993/83679 ), ( -36177561/17483 , -3136264409/17483 )
**u= 185/144 , ht= 1309369 ; ( 1309369/442183 , 121934700/442183 ), ( -3094601/479453 , -552624967/958906 )
**u= 185/163 , ht= 1803331 ; ( 710077/2542099 , 35396557/363157 ), ( -376103/1803331 , -172921427/1803331 )
**u= 185/172 , ht= 4138943 ; ( 6419437/4491919 , -755514997/4491919 ), ( -4138943/125539 , 399312209/125539 )
**u= 193/3 , ht= 24617 ; ( -24617/13609 , -330859/13609 ), ( -65471/103 , -896811/103 )
**u= 193/16 , ht= 1659 ; ( -133647/2353 , 3834918/2353 ), ( 1595/1659 , 22086/553 )
**u= 193/24 , ht= 68077 ; ( 176609/99950 , 28602559/399800 ), ( -63917/68077 , -3292836/68077 )
**u= 193/36 , ht= 2631 ; ( 1891/2631 , -419933/7893 ), ( -2631/1891 , 419933/5673 )
**u= 193/44 , ht= 298875 ; ( -168073/343075 , -18247727/343075 ), ( -298875/134987 , 15664588/134987 )
**u= 193/56 , ht= 4435 ; ( -2061/4435 , -263502/4435 ), ( 33387796/3868551 , -804960379/1719356 )
**u= 193/57 , ht= 169963 ; ( -169963/82870 , 10275009/82870 ), ( 209180/69397 , -11980779/69397 )
**u= 193/60 , ht= 58897 ; ( -58897/44693 , 4122369/44693 ), ( 9215011/35869 , 513973798/35869 )
**u= 193/97 , ht= 594500 ; ( 84183/594500 , 42573951/594500 ), ( 1417016/842793 , 38965701/280931 )
**u= 193/104 , ht= 229591 ; ( -229591/120655 , -19039651/120655 ), ( -205713/491087 , -39084886/491087 )
**u= 193/120 , ht= 255319 ; ( -26358011/2954749 , 2091421531/2954749 ), ( -51841/255319 , 20543503/255319 )
**u= 193/123 , ht= 118171 ; ( 8098553/17539951 , -1542466799/17539951 ), ( -60425/118171 , 10595931/118171 )
**u= 193/132 , ht= 2021483 ; ( 2021483/1234895 , 195913481/1234895 ), ( -8117887/450299 , -672471781/450299 )
**u= 193/137 , ht= 4776459 ; ( -4776459/518180 , 404799483/518180 ), ( 26153962/2532879 , -246012651/281431 )
**u= 193/144 , ht= 69743 ; ( 54353/69743 , -7638306/69743 ), ( 1723955/133393 , 22980697/20522 )
**u= 193/171 , ht= 244201 ; ( 244201/39469 , -23285519/39469 ), ( 743931/69431 , -211009853/208293 )
**u= 193/177 , ht= 710899 ; ( -358247/906064 , 93314703/906064 ), ( 88862/710899 , 68612399/710899 )
**u= 193/195 , ht= 404653 ; ( -404653/125747 , 42595415/125747 ), ( 1660577/12914807 , -1308909189/12914807 )
**u= 196/3 , ht= 829 ; ( 373/1165 , 229557/16310 ), ( -149/829 , -130493/11606 )
**u= 196/23 , ht= 27281 ; ( -63293/6903 , 30482369/96642 ), ( -15269/27281 , 14980247/381934 )
**u= 196/27 , ht= 335 ; ( -2447/985 , 4108337/41370 ), ( 335/311 , -712681/13062 )
**u= 196/47 , ht= 10957 ; ( 45469/35975 , -39747797/503650 ), ( 10487/10957 , 10398443/153398 )
**u= 196/83 , ht= 20941 ; ( -20941/4125 , 1767611/5250 ), ( -4628525/1552651 , -4448545981/21737114 )
**u= 196/87 , ht= 261389 ; ( -651641/560965 , 802046523/7853510 ), ( -261389/33599 , -245861949/470386 )
**u= 196/95 , ht= 32717 ; ( -1092277/293583 , -1102386283/4110162 ), ( 8777/32717 , -33015581/458038 )
**u= 196/107 , ht= 162647 ; ( -216273/307385 , -388817379/4303390 ), ( 162647/29967 , -57033801/139846 )
**u= 196/155 , ht= 183531 ; ( 41973109/600261 , 52263274027/8403654 ), ( -183531/6251 , -12034369/4606 )
**u= 196/179 , ht= 161491 ; ( -101997/161491 , 255569103/2260874 ), ( 434903/11031 , 194036257/51478 )
**u= 196/191 , ht= 1393 ; ( 91817/97165 , 184769027/1360310 ), ( 1011/1393 , 2378921/19502 )
**u= 197 , ht= 3263 ; ( -533/4752 , -57941/4752 ), ( 1948/3263 , 587/251 )
**u= 197/18 , ht= 60505 ; ( -229292/28341 , 41783363/170046 ), ( -60505/47644 , -4659371/95288 )
**u= 197/22 , ht= 57174 ; ( 140529/209378 , 16867719/418756 ), ( -57174/10663 , 3894703/21326 )
**u= 197/30 , ht= 58579 ; ( -58579/18119 , -2390868/18119 ), ( -136996/1331 , 10704983/2662 )
**u= 197/33 , ht= 48137 ; ( -48137/46276 , 2733017/46276 ), ( -114368/69871 , 5487951/69871 )
**u= 197/34 , ht= 11689 ; ( -61/11689 , -486056/11689 ), ( -73867/28847 , 3296702/28847 )
**u= 197/42 , ht= 3902 ; ( 4742186/375257 , 439088629/750514 ), ( 59/3902 , -360207/7804 )
**u= 197/49 , ht= 31746 ; ( 14369/31746 , -12169427/222222 ), ( -77744/94551 , 2034963/31517 )
**u= 197/54 , ht= 252469 ; ( 252469/197599 , 50355832/592797 ), ( 9937639/3206539 , -546894546/3206539 )
**u= 197/78 , ht= 26849 ; ( -546937/1866500 , 244822343/3733000 ), ( -15947/26849 , 1964966/26849 )
**u= 197/86 , ht= 4660629 ; ( -4660629/200110 , -616405347/400220 ), ( 29958223/1210413 , -660472038/403471 )
**u= 197/126 , ht= 12979833 ; ( 16064219/161220 , 7708873739/967320 ), ( 12979833/4880147 , 3327410948/14640441 )
**u= 197/130 , ht= 597981 ; ( -280156/597981 , -107298715/1195962 ), ( 985921/5860661 , -482785432/5860661 )
**u= 197/154 , ht= 818717 ; ( 818717/611895 , 90376036/611895 ), ( 9704325/1368074 , 1733187341/2736148 )
**u= 197/166 , ht= 1905511 ; ( 10188483/27330722 , 5355524913/54661444 ), ( -1905511/526389 , -60495886/175463 )
**u= 197/169 , ht= 83707 ; ( 91979437/4279680 , -110874536657/55635840 ), ( -36500/83707 , -109960327/1088191 )
**u= 197/174 , ht= 5253905 ; ( -6897073/1721843 , 668121088/1721843 ), ( 5253905/3391757 , -587773526/3391757 )
883
>
また、小さい整点から求めたuについても調べる。

■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。


[参考文献]


Last Update: 2026.06.18
H.Nakao

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