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Integer Points on A^4+20002*B^4=C^4+20002*D^4

[2026.06.08]A^4+20002*B^4=C^4+20002*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=20002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、8個の整数解
    546^4+20002*229^4=2666^4+20002*123^4
    2633^4+20002*563^4=6727^4+20002*149^4
    4713^4+20002*1424^4=16831^4+20002*594^4
    7052^4+20002*1237^4=14872^4+20002*373^4
    7981^4+20002*253^4=8021^4+20002*1^4
    12923^4+20002*3053^4=34819^4+20002*1961^4
    41997^4+20002*5396^4=66931^4+20002*216^4
    47724^4+20002*7577^4=72288^4+20002*6841^4
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 = 365/2, 30, 130, 30277/166, 493/4, 10, 2482/13, 28496/305, 28496/305
となる。
{gp2cによる計算]
gp > ss(20002,100000)
1:546^4+20002*229^4=2666^4+20002*123^4
2:2633^4+20002*563^4=6727^4+20002*149^4
3:4713^4+20002*1424^4=16831^4+20002*594^4
4:7052^4+20002*1237^4=14872^4+20002*373^4
5:7981^4+20002*253^4=8021^4+20002*1^4
6:12923^4+20002*3053^4=34819^4+20002*1961^4
7:41997^4+20002*5396^4=66931^4+20002*216^4
8:47724^4+20002*7577^4=72288^4+20002*6841^4
8 solutions,
time = 40h, 21min, 13,351 ms.
%1 = 8
{Pari/GPによる計算]
gp > uu(20002,546,229,2666,123)
[u, -1; 2*u - 365, 1; 4664*u + 925709, 1]
[u, -1; 2*u - 365, 1; 4664*u - 925709, 1]
[u, -1; 2*u + 365, 1; 4664*u + 925709, 1]
time = 1 ms.
%1 = [365/2, -925709/4664]
gp > uu(20002,2633,563,6727,149)
[u - 130, 1; u, -1; 147384*u + 26092609, 1]
[u - 130, 1; u, -1; 147384*u - 26092609, 1]
[u, -1; u + 130, 1; 147384*u + 26092609, 1]
time = 1 ms.
%2 = [130, -26092609/147384]
gp > uu(20002,4713,1424,16831,594)
[u, -1; 166*u + 30277, 1; 5045*u - 786356, 1]
[u, -1; 166*u - 30277, 1; 5045*u - 786356, 1]
[u, -1; 166*u + 30277, 1; 5045*u + 786356, 1]
time = 1 ms.
%3 = [-30277/166, 786356/5045]
gp > uu(20002,7052,1237,14872,373)
[u, -1; 4*u - 493, 1; 86940*u + 16931693, 1]
[u, -1; 4*u - 493, 1; 86940*u - 16931693, 1]
[u, -1; 4*u + 493, 1; 86940*u + 16931693, 1]
time = 1 ms.
%4 = [493/4, -16931693/86940]
gp > uu(20002,7981,253,8021,1)
[u - 10, 1; u, -1; 32004*u + 64016401, 1]
[u - 10, 1; u, -1; 32004*u - 64016401, 1]
[u, -1; u + 10, 1; 32004*u + 64016401, 1]
time = 1 ms.
%5 = [10, -64016401/32004]
gp > uu(20002,12923,3053,34819,1961)
[u, -1; 13*u - 2482, 1; 210588*u + 53052565, 1]
[u, -1; 13*u - 2482, 1; 210588*u - 53052565, 1]
[u, -1; 13*u + 2482, 1; 210588*u + 53052565, 1]
time = 1 ms.
%6 = [2482/13, -53052565/210588]
gp > uu(20002, 41997, 5396, 66931, 216)
[u, -1; 305*u - 28496, 1; 47656*u + 10235257, 1]
[u, -1; 305*u - 28496, 1; 47656*u - 10235257, 1]
[u, -1; 305*u + 28496, 1; 47656*u + 10235257, 1]
time = 1 ms.
%7 = [28496/305, -10235257/47656]
gp > uu(20002, 47724,7577,72288,6841)
[u, -1; 36*u - 10001, 1; 73692*u + 52105105, 1]
[u, -1; 36*u - 10001, 1; 73692*u - 52105105, 1]
[u, -1; 36*u + 10001, 1; 73692*u + 52105105, 1]
time = 1 ms.
%8 = [10001/36, -52105105/73692]

■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685個のuが抽出される。
[MAGMAによる計算]
> PP(20002,1,200);
**u= 1/14 , ht= 1517 ; ( 1517/744 , 421487/496 ), ( -16287/13361 , 7882626/13361 )
**u= 1/22 , ht= 15625 ; ( -29533/8373 , -4799628/2791 ), ( -1722/15625 , -14747031/31250 )
**u= 1/52 , ht= 7151 ; ( 13205/5899 , 10429736/5899 ), ( -7151/1951 , 5345402/1951 )
**u= 1/54 , ht= 9346 ; ( -1793/9346 , 41960843/56076 ), ( 9346/1793 , 41960843/10758 )
**u= 1/66 , ht= 14539 ; ( 87836/18677 , 145914601/37354 ), ( -6773/14539 , -1861566/2077 )
**u= 1/70 , ht= 4001 ; ( 529/4001 , -3376778/4001 ), ( -4001/529 , 3376778/529 )
**u= 1/76 , ht= 8349 ; ( 1075/8349 , 2446314/2783 ), ( -1614777/214885 , -1420210809/214885 )
**u= 1/82 , ht= 111913 ; ( -108175/111913 , 140952452/111913 ), ( 191291/102075 , -196350014/102075 )
**u= 1/86 , ht= 10597 ; ( 10597/9699 , 4440888/3233 ), ( -185958/106021 , -397038309/212042 )
**u= 1/90 , ht= 6631 ; ( -49649/40599 , -182540620/121797 ), ( 3921/6631 , 21925732/19893 )
**u= 1/100 , ht= 18279 ; ( 8119/18281 , 20003828/18281 ), ( 8117/18279 , -100005941/91395 )
**u= 1/106 , ht= 36059 ; ( -292426/265823 , -813785911/531646 ), ( 36059/22597 , -43814614/22597 )
**u= 1/114 , ht= 57659 ; ( 1149796/102193 , -2465094353/204386 ), ( 57659/47000 , 158857037/94000 )
**u= 1/118 , ht= 5709 ; ( -3020845/68867 , 3282494326/68867 ), ( -2846/5709 , 13859557/11418 )
**u= 1/126 , ht= 8116 ; ( 4480053/70984 , 30178378069/425904 ), ( 8116/6577 , -23453187/13154 )
**u= 1/130 , ht= 203059 ; ( -2185988/663753 , 1736602497/442502 ), ( 203059/57771 , 34388944/8253 )
**u= 1/132 , ht= 18397 ; ( 18397/4493 , 21758854/4493 ), ( 9029/18709 , 23868438/18709 )
**u= 1/136 , ht= 3300 ; ( 3300/427 , -15522821/1708 ), ( 47851/75272 , 416090131/301088 )
**u= 1/140 , ht= 35919 ; ( -11041/35919 , 14821558/11973 ), ( -490779/179489 , 618345123/179489 )
**u= 1/142 , ht= 5715 ; ( -5715/2024 , -14450093/4048 ), ( 2689184/517081 , 6526795001/1034162 )
**u= 1/166 , ht= 2395 ; ( -2395/1506 , -2430187/1004 ), ( 246318/60979 , 653910519/121958 )
**u= 1/174 , ht= 9031 ; ( -9031/1172 , 24026407/2344 ), ( 1937104/370147 , -5203152581/740294 )
**u= 1/180 , ht= 40057 ; ( 291659/218389 , -488865632/218389 ), ( 40057/2103 , -161456426/6309 )
**u= 1/198 , ht= 4927 ; ( -2146139/170549 , -3029557206/170549 ), ( 4113/4927 , 27094586/14781 )
**u= 2/33 , ht= 6277 ; ( -1115/6277 , 2589773/6277 ), ( 19073/4399 , -7951281/4399 )
**u= 2/65 , ht= 3899 ; ( -3899/1301 , 2343365/1301 ), ( -129991/213209 , -142364465/213209 )
**u= 2/113 , ht= 51401 ; ( 2068535/93833 , -1556521687/93833 ), ( -4393/51401 , -5539873/7343 )
**u= 2/137 , ht= 24103 ; ( 23479/325153 , -269826167/325153 ), ( -24103/17545 , 24675419/17545 )
**u= 4 , ht= 100 ; ( -100 , 9997/2 ), ( -7/100 , -10021/200 )
**u= 4/25 , ht= 601 ; ( 601/576 , -138749/384 ), ( -18432/9649 , 52014747/96490 )
**u= 4/97 , ht= 256 ; ( -282680/97119 , -2973689/1962 ), ( -256/35 , 254489/70 )
**u= 4/145 , ht= 5632 ; ( 12865477/5408612 , -1527840611/983384 ), ( -5632/1357 , 6976255/2714 )
**u= 4/169 , ht= 4272 ; ( 1864/28113 , -158725569/243646 ), ( -4272/3671 , -7322751/7342 )
**u= 4/177 , ht= 686924 ; ( 5379521/2890016 , 8124804373/5780032 ), ( 168491/686924 , 134432879/196264 )
**u= 4/193 , ht= 53932 ; ( -48937/53932 , -101176877/107864 ), ( 2616124/1403505 , -4124631893/2807010 )
**u= 8/17 , ht= 7277 ; ( -9505/1409 , -2801567/2818 ), ( -7277/3195 , -2317199/6390 )
**u= 8/33 , ht= 138355 ; ( 138355/14011 , -56490249/28022 ), ( -92683/156005 , 73712997/312010 )
**u= 8/113 , ht= 946267 ; ( 1117031/8223 , 279899177/5482 ), ( -946267/108475 , 715969717/216950 )
**u= 8/129 , ht= 18743 ; ( 3298483/1320397 , 2853582101/2640794 ), ( 10583/18743 , 17287519/37486 )
**u= 8/153 , ht= 4461 ; ( 1333/4461 , 12217361/26766 ), ( 18544677/30181 , -48662404477/181086 )
**u= 8/161 , ht= 258287 ; ( 489695/43081 , -441082133/86162 ), ( -28937/258287 , 233201197/516574 )
**u= 8/177 , ht= 90743 ; ( -5545/90743 , 85529479/181486 ), ( -90743/5545 , 85529479/11090 )
**u= 9/2 , ht= 1 ; ( 0 , 283/6 ), ( 39963/283 , -19982/3 )
**u= 9/4 , ht= 25 ; ( 25/7 , -5192/21 ), ( 4453/12791 , -2708659/38373 )
**u= 9/34 , ht= 24461 ; ( 1193656/112813 , 1398300793/676878 ), ( 24461/17168 , 34852729/103008 )
**u= 9/38 , ht= 12783 ; ( -333595/336459 , 292087042/1009377 ), ( -1847/12783 , -7962188/38349 )
**u= 9/40 , ht= 3169 ; ( -3169/41 , 2004517/123 ), ( -43177/6298 , 110391685/75576 )
**u= 9/56 , ht= 18149 ; ( -5637/18149 , -14222206/54447 ), ( -230499/1243 , 172500901/3729 )
**u= 9/62 , ht= 18471 ; ( 18471/3064 , 29487109/18384 ), ( -1184156/71685 , 1868318983/430110 )
**u= 9/68 , ht= 24097 ; ( -43325/37731 , -47378152/113193 ), ( 11517/24097 , -22024913/72291 )
**u= 9/86 , ht= 1021783 ; ( -5346859/1258242 , -10188339571/7549452 ), ( -1021783/199761 , 965549768/599283 )
**u= 9/88 , ht= 32686 ; ( 71755/29581 , 72811303/88743 ), ( 32686/19475 , -142775903/233700 )
**u= 9/94 , ht= 341903 ; ( 128711/341903 , -354215924/1025709 ), ( -921157/137941 , 903098206/413823 )
**u= 9/104 , ht= 155853 ; ( 195510/61981 , -836686781/743772 ), ( -155853/93019 , 185104921/279057 )
**u= 9/140 , ht= 653117 ; ( 4558293/566473 , -5435204261/1699419 ), ( 653117/159597 , -795556174/478791 )
**u= 9/146 , ht= 120213 ; ( -34969/250023 , -305059742/750069 ), ( -120213/62141 , -163521172/186423 )
**u= 9/154 , ht= 28487 ; ( -344222/41801 , 860655781/250806 ), ( 28487/4694 , 71659877/28164 )
**u= 9/170 , ht= 24091 ; ( -24091/189 , 31413356/567 ), ( 955513/11673 , 1245991874/35019 )
**u= 9/184 , ht= 139259 ; ( 479953/306464 , 3089921053/3677568 ), ( -94628/139259 , -913583399/1671108 )
**u= 9/188 , ht= 74219 ; ( -74219/3531 , -9262196/963 ), ( -1122979/191721 , 1562107921/575163 )
**u= 9/194 , ht= 5340 ; ( -39791848/23486199 , -128721401981/140917194 ), ( 3109/5340 , 17213887/32040 )
**u= 9/196 , ht= 30599 ; ( -30599/24487 , -384088184/514227 ), ( -111173/16105 , 1100923037/338205 )
**u= 9/200 , ht= 27492 ; ( -35456/38775 , 31620799/49500 ), ( -517/27492 , 155553503/329904 )
**u= 10 , ht= 691 ; ( 759/1613 , -56465/1613 ), ( 691/23 , -953 )
**u= 10/9 , ht= 981 ; ( -49321/3243 , 14067587/9729 ), ( 981/767 , 354425/2301 )
**u= 10/41 , ht= 12811 ; ( -15101/35847 , -875181/3983 ), ( -63/12811 , 2594187/12811 )
**u= 10/57 , ht= 9347 ; ( -88121/52973 , 24548573/52973 ), ( -7639/9347 , 2882171/9347 )
**u= 10/73 , ht= 438713 ; ( 438713/72691 , 120155755/72691 ), ( -559279/430747 , 190741115/430747 )
**u= 10/81 , ht= 35227 ; ( -35227/5239 , -30409655/15717 ), ( 37181/13327 , -33725071/39981 )
**u= 10/121 , ht= 32363 ; ( -44821/16343 , -182555387/179773 ), ( 32363/7679 , 1652947/1097 )
**u= 10/129 , ht= 90979 ; ( -90979/72937 , -41883007/72937 ), ( -370937/327211 , -177663909/327211 )
**u= 10/169 , ht= 16901 ; ( 5633/16901 , -95212607/219713 ), ( -42661/37783 , -23428297/37783 )
**u= 10/193 , ht= 110183 ; ( -273167/59709 , -40948863/19903 ), ( 110183/26421 , 49780037/26421 )
**u= 16/9 , ht= 5 ; ( 5/4 , -1921/16 ), ( -679/572 , 266363/2288 )
**u= 16/73 , ht= 110516 ; ( -110516/54391 , 105246251/217564 ), ( 661289/60900 , -567425483/243600 )
**u= 16/89 , ht= 5849 ; ( -5849/1488 , -1897993/1984 ), ( 24747/6680 , -24183087/26720 )
**u= 16/97 , ht= 7764 ; ( 3128864/141361 , 3084869257/565444 ), ( -6301/7764 , 9848477/31056 )
**u= 16/105 , ht= 1627 ; ( -1627/508 , 1746641/2032 ), ( -110576/44999 , 122335885/179996 )
**u= 16/121 , ht= 11729 ; ( 2912/11729 , -13294259/46916 ), ( 2896/11745 , -13307107/46980 )
**u= 16/145 , ht= 584852 ; ( 1624424/263331 , 220188763/117036 ), ( -147387/584852 , -726309789/2339408 )
**u= 16/153 , ht= 7003 ; ( -7003/1648 , -8899297/6592 ), ( 2053/280088 , -346476257/1120352 )
**u= 16/169 , ht= 4252 ; ( -4515/3364 , -95158987/174928 ), ( 4252/1589 , 10959499/11804 )
**u= 18 , ht= 45 ; ( 45/29 , -3773/87 ), ( 29/45 , -3773/135 )
**u= 18/73 , ht= 4035 ; ( 4035/53 , -2438083/159 ), ( -19461/6235 , 12346729/18705 )
**u= 18/137 , ht= 34069 ; ( 11875/34069 , -29862397/102207 ), ( -193723/103019 , 181604657/309057 )
**u= 18/169 , ht= 4797 ; ( -4427/4797 , 78009383/187083 ), ( -74457/28633 , 3859679/4521 )
**u= 20 , ht= 198 ; ( 101/198 , 101/4 ), ( 99/202 , 99/4 )
**u= 20/33 , ht= 182386 ; ( -182386/111083 , -54864791/222166 ), ( -81639131/3390682 , -20992795479/6781364 )
**u= 20/57 , ht= 251282 ; ( -251282/142531 , -97545387/285062 ), ( 286253/24566 , -97010539/49132 )
**u= 20/89 , ht= 40231 ; ( 96787/251826 , 37942767/167884 ), ( 40231/2762 , 17014303/5524 )
**u= 25/4 , ht= 303 ; ( -303/179 , -14071/179 ), ( -217/755 , 156991/3775 )
**u= 25/6 , ht= 7283 ; ( -16835/4798 , -8573111/47980 ), ( 6395/7283 , -2374072/36415 )
**u= 25/16 , ht= 3857 ; ( 107681/5405 , -86252327/54050 ), ( -3601/3857 , -60308/551 )
**u= 25/24 , ht= 425 ; ( -425/166 , 894127/3320 ), ( 166/425 , 894127/8500 )
**u= 25/26 , ht= 17575 ; ( -17575/2289 , -3012488/3815 ), ( 21981/15485 , -721626/4075 )
**u= 25/36 , ht= 6635 ; ( 34931/8467 , -4313272/8467 ), ( 2847/6635 , 12996571/99525 )
**u= 25/42 , ht= 4817 ; ( -87553/28703 , -59714734/143515 ), ( 4817/1807 , -3334386/9035 )
**u= 25/46 , ht= 64459 ; ( -64459/9555 , 14732608/15925 ), ( 425593 , -57733556 )
**u= 25/48 , ht= 11659 ; ( -11659/10487 , -4345999/20974 ), ( -158359/9287 , -21981815/9287 )
**u= 25/52 , ht= 3097 ; ( -72087/39817 , 59387996/199085 ), ( 263/3097 , -2241416/15485 )
**u= 25/54 , ht= 6671 ; ( -21479/20047 , 4318288/20047 ), ( -6671/865 , 14830426/12975 )
**u= 25/82 , ht= 48049 ; ( -11032/48049 , -17857861/96098 ), ( 1347667/217133 , 176595298/155095 )
**u= 25/88 , ht= 15527 ; ( -777987/19766 , -584069629/79064 ), ( -15527/13689 , -3883798/13689 )
**u= 25/94 , ht= 975304 ; ( 108470437/9636440 , 211170554663/96364400 ), ( -975304/184703 , -384979745/369406 )
**u= 25/96 , ht= 8335 ; ( 62425/6766 , 492199567/270640 ), ( 8335/121 , -16335763/1210 )
**u= 25/114 , ht= 440909 ; ( 491680/62413 , 1058418067/624130 ), ( 135845/440909 , 70374804/314935 )
**u= 25/118 , ht= 638949 ; ( -5802129/946999 , 6386436484/4734995 ), ( -638949/83093 , 139991310/83093 )
**u= 25/138 , ht= 177671 ; ( -102814/177671 , -482310657/1776710 ), ( -3153689/1899802 , -1730100003/3799604 )
**u= 25/144 , ht= 24595 ; ( 11075/26637 , 103856693/399555 ), ( 14937/24595 , -207193711/737850 )
**u= 25/154 , ht= 7485 ; ( -7125/47501 , -59609648/237505 ), ( 7485/1787 , -9550176/8935 )
**u= 25/162 , ht= 4709 ; ( -56431/25199 , -235993786/377985 ), ( -4709/1712 , -2551085/3424 )
**u= 25/188 , ht= 261663 ; ( 11179/318747 , 29155642/106249 ), ( 261663/41233 , -363219702/206165 )
**u= 25/192 , ht= 411197 ; ( 319577/411197 , -2886603681/8223940 ), ( 1252381/576467 , 764183215/1152934 )
**u= 26/9 , ht= 2719 ; ( -17067/361 , -3012599/1083 ), ( 523/2719 , -162881/2719 )
**u= 26/17 , ht= 71977 ; ( -115897/60427 , -10568999/60427 ), ( -71977/24005 , 6135857/24005 )
**u= 26/49 , ht= 31 ; ( 31/27 , 13169/63 ), ( -8744877/10069849 , -1831006059/10069849 )
**u= 26/89 , ht= 419 ; ( 594169/382883 , 130784599/382883 ), ( 419/63 , 78397/63 )
**u= 26/105 , ht= 910033 ; ( 878323/910033 , 254177411/910033 ), ( -1628993/1040107 , -388419385/1040107 )
**u= 26/113 , ht= 311671 ; ( -625846029/72123881 , 131342749211/72123881 ), ( -311671/5181 , 64987709/5181 )
**u= 26/121 , ht= 184209 ; ( 63757/184209 , 154198237/675433 ), ( -197907/524705 , 120983349/524705 )
**u= 26/137 , ht= 3113 ; ( 373/3113 , 719731/3113 ), ( 1532333/2419545 , 657448073/2419545 )
**u= 26/177 , ht= 1042627 ; ( 1772035/2194711 , -736025229/2194711 ), ( 254977/1042627 , 280068071/1042627 )
**u= 26/185 , ht= 48927 ; ( -9364841/106419 , -832775389/35473 ), ( -48927/15157 , 13663713/15157 )
**u= 32/25 , ht= 275 ; ( 19003/4155 , 11462313/27700 ), ( -189/275 , 589899/5500 )
**u= 32/33 , ht= 2147 ; ( -764041/98183 , 312915481/392732 ), ( 2147/211 , -876381/844 )
**u= 32/97 , ht= 55769 ; ( -128569/33897 , 30867331/45196 ), ( -46857/55769 , 50730153/223076 )
**u= 32/105 , ht= 1387 ; ( 313423/41543 , 229093663/166172 ), ( 1387/883 , -1191411/3532 )
**u= 34/13 , ht= 7927 ; ( 3845/7927 , -544867/7927 ), ( 19633/4683 , 1248311/4683 )
**u= 34/21 , ht= 2501 ; ( 37019/50977 , -4951589/50977 ), ( -2501/2473 , 276433/2473 )
**u= 34/37 , ht= 456021 ; ( 475499/1736361 , -62605507/578787 ), ( 456021/8329 , -47582751/8329 )
**u= 34/69 , ht= 236413 ; ( -472805/429577 , -91008031/429577 ), ( -236413/38345 , 34120811/38345 )
**u= 34/93 , ht= 1972951 ; ( 1972951/266029 , 329269263/266029 ), ( 10503379/1270711 , -1749884577/1270711 )
**u= 34/157 , ht= 382571 ; ( -382571/178471 , 90719531/178471 ), ( -559603/48615 , -17244367/6945 )
**u= 34/165 , ht= 128929 ; ( -128929/11981 , -28526041/11981 ), ( -14577787/1517633 , 3228918363/1517633 )
**u= 34/189 , ht= 11807 ; ( -11807/3691 , -2916751/3691 ), ( 6453883/255481 , -4568733841/766443 )
**u= 34/197 , ht= 7689 ; ( 4525/7689 , -715881/2563 ), ( 2429661/1129855 , -645018993/1129855 )
**u= 36/17 , ht= 904 ; ( 32220/2623 , -13327787/15738 ), ( 819/904 , -502967/5424 )
**u= 36/25 , ht= 1981 ; ( -1043/6668 , -3374875/40008 ), ( -172/1981 , -710161/8490 )
**u= 36/65 , ht= 10844 ; ( 32112/11117 , 27398273/66702 ), ( 10844/1731 , -8853905/10386 )
**u= 36/89 , ht= 77932 ; ( -77932/10089 , -74137847/60534 ), ( -92196/26263 , -90442463/157578 )
**u= 36/97 , ht= 947977 ; ( -947977/344548 , 993451549/2067288 ), ( -1632184/229 , -1607600737/1374 )
**u= 36/113 , ht= 116076 ; ( 116076/29179 , 127235431/175074 ), ( -8416821/1926860 , 9179145463/11561160 )
**u= 36/121 , ht= 3503 ; ( 1300/3503 , 45213301/231198 ), ( 18044/25297 , -376000429/1669602 )
**u= 36/145 , ht= 16444 ; ( -6901/16444 , 21475313/98664 ), ( -837973/196892 , 1036585721/1181352 )
**u= 36/169 , ht= 14744 ; ( -8197/14744 , -285107129/1150032 ), ( -5911/15820 , 285425117/1233960 )
**u= 40 , ht= 409 ; ( 409/373 , -17441/746 ), ( -881/843 , 38629/1686 )
**u= 40/33 , ht= 2381 ; ( -137/2381 , 433283/4762 ), ( 8953/3211 , 1727967/6422 )
**u= 40/49 , ht= 1393 ; ( 549/1393 , 2320205/19502 ), ( -24097/11651 , 5925233/23302 )
**u= 40/57 , ht= 53077 ; ( -136139/22207 , 32933583/44414 ), ( 45401/53077 , 16676229/106154 )
**u= 40/73 , ht= 763 ; ( -551/763 , -254299/1526 ), ( 5843539/4698753 , -2026046717/9397506 )
**u= 40/169 , ht= 61963 ; ( 61963/58039 , -453747167/1509014 ), ( -71397/111751 , -708744081/2905526 )
**u= 49/4 , ht= 227 ; ( -65/227 , -47377/1589 ), ( -97/779 , 156439/5453 )
**u= 49/12 , ht= 1273 ; ( -1273/1127 , -588964/7889 ), ( -1127/1273 , 588964/8911 )
**u= 49/24 , ht= 287 ; ( 287/134 , -620673/3752 ), ( 949/19285 , -9458606/134995 )
**u= 49/34 , ht= 93611 ; ( -321472/2079459 , -272677399/3234714 ), ( 93611/14493 , -55239928/101451 )
**u= 49/36 , ht= 489 ; ( -445/489 , -1190171/10269 ), ( 489/445 , -1190171/9345 )
**u= 49/46 , ht= 9213 ; ( 9213/2221 , -6427726/15547 ), ( -33928/46135 , 77684417/645890 )
**u= 49/48 , ht= 2969 ; ( 73439/38299 , -114770713/536186 ), ( -2969/1541 , 201539/938 )
**u= 49/58 , ht= 699914 ; ( -699914/349 , 1066111657/4886 ), ( -9273071/3423390 , 15057049171/47927460 )
**u= 49/66 , ht= 47897 ; ( 587377/25871 , -477667294/181097 ), ( 47897/9089 , -39608592/63623 )
**u= 49/72 , ht= 620 ; ( -8119/9271 , -10457408/64897 ), ( -197/620 , -2208113/17360 )
**u= 49/76 , ht= 25253 ; ( -25253/21031 , 28651211/147217 ), ( 59939/86653 , 91857856/606571 )
**u= 49/78 , ht= 5339 ; ( -5339/4099 , 5944964/28693 ), ( -6343/21815 , -20065226/152705 )
**u= 49/82 , ht= 683981 ; ( -370580/683981 , 1408950373/9575734 ), ( 15783541/2427996 , -28923142541/33991944 )
**u= 49/88 , ht= 190983 ; ( 352637/53283 , -111524006/124327 ), ( 190983/85625 , -196350864/599375 )
**u= 49/100 , ht= 16899 ; ( 16899/10601 , -19949815/74207 ), ( 16997/10699 , 100425226/374465 )
**u= 49/106 , ht= 31991 ; ( 22705/31991 , -40391168/223937 ), ( 6747717/6806 , 13895175699/95284 )
**u= 49/108 , ht= 41969 ; ( 45175/33127 , -174659822/695667 ), ( 41969/6241 , -132292462/131061 )
**u= 49/120 , ht= 50996 ; ( 278459/794384 , 3688670205/22242752 ), ( -50996/409 , -223472567/11452 )
**u= 49/136 , ht= 635 ; ( 635/397 , -873389/2779 ), ( 397/635 , -873389/4445 )
**u= 49/138 , ht= 12505 ; ( 12505/9607 , 18525594/67249 ), ( -1510397/660067 , 1936448762/4620469 )
**u= 49/142 , ht= 739789 ; ( -739789/80616 , -591213887/376208 ), ( 858867/109675 , -1031823528/767725 )
**u= 49/144 , ht= 20563 ; ( 29137/16941 , -242680867/711522 ), ( 20563/9871 , 54745653/138194 )
**u= 49/148 , ht= 48207 ; ( 48207/30817 , 69608942/215719 ), ( 55411/76575 , 114994703/536025 )
**u= 49/156 , ht= 7649 ; ( -528635/134677 , 681387754/942739 ), ( 7649/1123 , -9656519/7861 )
**u= 49/174 , ht= 2643389 ; ( 4083905/14285351 , 19599589056/99997457 ), ( 2643389/1445147 , -50305548/128051 )
**u= 49/198 , ht= 20747 ; ( 20747/2322 , -176263877/97524 ), ( -238974/32891 , -2036722379/1381422 )
**u= 49/200 , ht= 540592 ; ( 540592/112845 , -5206861823/5266100 ), ( -2321085/135491 , 16441335228/4742185 )
**u= 50/17 , ht= 3801 ; ( 36183/6385 , -10710413/31925 ), ( 2407/3801 , 37475/543 )
**u= 50/89 , ht= 24763 ; ( -5141/24763 , 3374437/24763 ), ( -82381/22283 , 11386615/22283 )
**u= 50/129 , ht= 157421 ; ( 157421/72797 , 27859609/72797 ), ( 72797/157421 , 27859609/157421 )
**u= 50/153 , ht= 131885 ; ( -1653305/383567 , -1484528111/1917835 ), ( -131885/19619 , 116627727/98095 )
**u= 50/169 , ht= 78447 ; ( 174809/59679 , -735824863/1293045 ), ( -78447/14903 , 73405089/74515 )
**u= 52 , ht= 63 ; ( -63/62 , -79/4 ), ( 279/74 , 8463/148 )
**u= 52/17 , ht= 133 ; ( 133/54 , 5471/36 ), ( -54/133 , 16413/266 )
**u= 52/25 , ht= 6382 ; ( 6382/4285 , 5330077/42850 ), ( -412549/15474 , -57259519/30948 )
**u= 52/97 , ht= 10003 ; ( -4450/10003 , 2990747/20006 ), ( -10003/4450 , -2990747/8900 )
**u= 52/121 , ht= 11506 ; ( -11506/3671 , -40532953/80762 ), ( 148829/153750 , -718152301/3382500 )
**u= 52/129 , ht= 92171 ; ( 14126/92171 , 29375299/184342 ), ( 92171/14126 , -29375299/28252 )
**u= 52/177 , ht= 162653 ; ( 4874042/1913009 , -1932133583/3826018 ), ( -162653/38950 , 61717427/77900 )
**u= 58/9 , ht= 14001 ; ( 10339/14001 , -2057543/42003 ), ( 51593/7835 , 2057789/7835 )
**u= 58/25 , ht= 9509 ; ( -27535/26181 , -4157671/43635 ), ( -8055/9509 , 4091001/47545 )
**u= 58/49 , ht= 1253 ; ( 155077/87847 , 114677191/614929 ), ( -247/1253 , -821711/8771 )
**u= 58/97 , ht= 37119 ; ( -10660739/3124033 , -1436703479/3124033 ), ( 37119/14227 , 5141103/14227 )
**u= 58/105 , ht= 188339 ; ( 321781/146921 , 47596917/146921 ), ( -85249/188339 , -27817123/188339 )
**u= 58/129 , ht= 18095 ; ( 18095/4507 , 2781181/4507 ), ( -6066383/101107 , -904888153/101107 )
**u= 64 , ht= 105 ; ( -137/104 , 16721/832 ), ( 32/105 , 10001/840 )
**u= 64/9 , ht= 784 ; ( -305/784 , 252619/6272 ), ( -241/848 , -264203/6784 )
**u= 64/49 , ht= 676 ; ( -137/676 , -482863/5408 ), ( -1859/1392 , -11380469/77952 )
**u= 64/97 , ht= 47172 ; ( -47172/3947 , -46623211/31576 ), ( 12680959/127704 , -12490689863/1021632 )
**u= 64/105 , ht= 14972 ; ( 14972/293 , 15345297/2344 ), ( 38272/72187 , 83726473/577496 )
**u= 64/121 , ht= 3648 ; ( -11008/5089 , 146748289/447832 ), ( -3648/1675 , 48574059/147400 )
**u= 64/129 , ht= 123329 ; ( 478945/170248 , 577348183/1361984 ), ( -123329/36760 , -146172791/294080 )
**u= 64/145 , ht= 65747 ; ( -310504/291391 , -512778929/2331128 ), ( -65747/22732 , 83772923/181856 )
**u= 64/169 , ht= 13016 ; ( 13080/10639 , -284955617/1106456 ), ( 13016/10703 , -284803489/1113112 )
**u= 64/177 , ht= 604088 ; ( -21689/896056 , -1192537657/7168448 ), ( 604088/279647 , -885670689/2237176 )
**u= 65/2 , ht= 1141 ; ( -492/1141 , -44377/2282 ), ( 6163/7051 , -163696/7051 )
**u= 65/6 , ht= 1294 ; ( -1283/1294 , -110735/2588 ), ( -1855147/312701 , 57319572/312701 )
**u= 65/14 , ht= 53176 ; ( -12757/53176 , 5078485/106352 ), ( 1105921/680657 , -60284962/680657 )
**u= 65/32 , ht= 3073 ; ( -4063733/653429 , 288779038/653429 ), ( -3073/729 , -443263/1458 )
**u= 65/44 , ht= 38729 ; ( 38729/3507 , 1066489/1169 ), ( -359583/55469 , -29937795/55469 )
**u= 65/48 , ht= 1291 ; ( -15053/57439 , 10206193/114878 ), ( -1291/943 , 274787/1886 )
**u= 65/68 , ht= 213119 ; ( -129749/413537 , 44333347/413537 ), ( 213119/77153 , 23184172/77153 )
**u= 65/86 , ht= 73677 ; ( -73677/3766 , -16972141/7532 ), ( 180071/268737 , 37211008/268737 )
**u= 65/108 , ht= 212851 ; ( 212851/48877 , -84455672/146631 ), ( -447623/389029 , -229345921/1167087 )
**u= 65/114 , ht= 918617 ; ( -7450688/154879 , 1973940777/309758 ), ( -385999/918617 , -131964568/918617 )
**u= 65/116 , ht= 7901 ; ( 7901/4743 , -410375/1581 ), ( 5887539/1196963 , -802649094/1196963 )
**u= 65/144 , ht= 79229 ; ( -651543/639269 , 815201779/3835614 ), ( -79229/51757 , 14086553/51757 )
**u= 65/152 , ht= 1949 ; ( 41151683/4591254 , 8443014199/6121672 ), ( 1227/1949 , -352203/1949 )
**u= 65/158 , ht= 690211 ; ( 690211/436147 , 127300606/436147 ), ( 20368631/15029337 , 3946776454/15029337 )
**u= 65/168 , ht= 13411 ; ( 9827/13411 , 2673060/13411 ), ( -16892033/1474294 , 10904631885/5897176 )
**u= 65/188 , ht= 23639 ; ( 21477/23639 , -5431975/23639 ), ( -409371/50383 , -70149861/50383 )
**u= 68/21 , ht= 169378 ; ( -2040818/1622233 , 289751953/3244466 ), ( 105377/169378 , -22169767/338756 )
**u= 68/29 , ht= 9257 ; ( 9257/342 , -134419/76 ), ( 219519/56126 , 4228371/16036 )
**u= 68/45 , ht= 5062 ; ( 246977/15202 , 3659855/2764 ), ( -1463/5062 , 857285/10124 )
**u= 68/61 , ht= 413 ; ( -41258/327 , -2605249/218 ), ( 258/413 , -92247/826 )
**u= 68/77 , ht= 16166 ; ( 14395/16166 , -4607063/32332 ), ( 8479/38810 , -8454803/77620 )
**u= 68/93 , ht= 35570 ; ( 673510/239293 , -167183087/478586 ), ( -29437/35570 , -10799583/71140 )
**u= 68/101 , ht= 201549 ; ( -201549/71966 , -52166549/143932 ), ( -1896426/422635 , 473613267/845270 )
**u= 68/165 , ht= 336887 ; ( -336887/38378 , -105638351/76756 ), ( -11288434/805241 , 3525955985/1610482 )
**u= 68/189 , ht= 6649 ; ( -3970/6649 , -7746713/39894 ), ( -3037399/1520686 , -3397979561/9124116 )
**u= 68/197 , ht= 26646 ; ( 26646/737 , 824963/134 ), ( -19871302/486481 , -6766873993/972962 )
**u= 72 , ht= 33 ; ( -217/87 , -14903/522 ), ( -33/7 , 379/6 )
**u= 72/17 , ht= 14815 ; ( 14815/1487 , -4339289/8922 ), ( 25961/30007 , 11568041/180042 )
**u= 72/41 , ht= 40837 ; ( 16867/48437 , 23224987/290622 ), ( -40837/24517 , -21567863/147102 )
**u= 72/49 , ht= 13279 ; ( -13279/1599 , 46341641/67158 ), ( 57879/169 , -200562209/7098 )
**u= 72/121 , ht= 11847 ; ( 11847/503 , 101458337/33198 ), ( 503/11847 , 101458337/781902 )
**u= 72/161 , ht= 1733 ; ( 445/1733 , -1605409/10398 ), ( 1733/445 , 1605409/2670 )
**u= 73/4 , ht= 25 ; ( 313/117 , -2591/39 ), ( 9/25 , 618/25 )
**u= 73/16 , ht= 19 ; ( -511 , 23912 ), ( 19/15 , 2267/30 )
**u= 73/24 , ht= 35 ; ( -35/2 , 8039/8 ), ( -2437/2195 , 188071/2195 )
**u= 73/32 , ht= 4820 ; ( 13283/10341 , 247679/2298 ), ( 3699/4820 , -3218193/38560 )
**u= 73/36 , ht= 729 ; ( 505/729 , 186848/2187 ), ( 253/865 , -63287/865 )
**u= 73/46 , ht= 31 ; ( -1987/1067 , -16276/97 ), ( 31/15 , -2734/15 )
**u= 73/50 , ht= 329 ; ( -329/33 , -45608/55 ), ( 933/2021 , -921126/10105 )
**u= 73/70 , ht= 2897 ; ( -7789/2774 , 1619357/5548 ), ( -2897/1023 , -300874/1023 )
**u= 73/76 , ht= 5455 ; ( 5455/321 , -185858/107 ), ( -43161/1019 , 4405443/1019 )
**u= 73/114 , ht= 176 ; ( 7604/1771 , 1951423/3542 ), ( 109/176 , 51743/352 )
**u= 73/142 , ht= 18433 ; ( 18433/4304 , -5280259/8608 ), ( 32812/27345 , -11914949/54690 )
**u= 73/144 , ht= 18927 ; ( 14575/18927 , -10065908/56781 ), ( 72561/2875 , -61198313/17250 )
**u= 73/148 , ht= 80063 ; ( -80063/44691 , 4352097/14897 ), ( 320973/44735 , -46146609/44735 )
**u= 73/162 , ht= 5992 ; ( -5992/1051 , -1812581/2102 ), ( 14741/3608 , 13565377/21648 )
**u= 73/178 , ht= 9101 ; ( 80899/19133 , -12981662/19133 ), ( -9101/1312 , -2871823/2624 )
**u= 73/184 , ht= 19 ; ( 19/4 , 12331/16 ), ( -99273/1615 , 829671/85 )
**u= 73/192 , ht= 5699 ; ( -3517/5699 , 1086132/5699 ), ( -72947/8221 , -23811703/16442 )
**u= 74/17 , ht= 19143 ; ( 67045/25129 , 3430723/25129 ), ( -19143/3875 , 936591/3875 )
**u= 74/65 , ht= 207697 ; ( -6531/516079 , 48375733/516079 ), ( -143797/207697 , -3382385/29671 )
**u= 74/81 , ht= 11033 ; ( -537887/429869 , -19647821/117237 ), ( -11033/8949 , -13377221/80541 )
**u= 74/161 , ht= 2588223 ; ( -2765341/326513 , 37340401/29683 ), ( 2588223/382549 , -385936089/382549 )
**u= 80/17 , ht= 1538 ; ( 2498/1631 , 550001/6524 ), ( 1538/841 , 323335/3364 )
**u= 80/41 , ht= 20739 ; ( -88066/149607 , 16572337/199476 ), ( -2918/20739 , -5996993/82956 )
**u= 80/113 , ht= 2446 ; ( -13251943/2918206 , -6451091129/11672824 ), ( 2446/1057 , 1266817/4228 )
**u= 80/121 , ht= 7771 ; ( -17782/48939 , 93926311/717772 ), ( 6798/7771 , -55873077/341924 )
**u= 80/129 , ht= 204542 ; ( 1642301/567602 , 882639147/2270408 ), ( 204542/161069 , -132246875/644276 )
**u= 81/4 , ht= 631 ; ( -631/343 , -142844/3087 ), ( -793/505 , -188852/4545 )
**u= 81/14 , ht= 145 ; ( -145/9 , -54316/81 ), ( 1972/2469 , 2364307/44442 )
**u= 81/22 , ht= 11233 ; ( 11233/4633 , -5698204/41697 ), ( -3853/12427 , 6101116/111843 )
**u= 81/32 , ht= 3215 ; ( -3215/1767 , 4150381/31806 ), ( -68349/45584 , 371833741/3282048 )
**u= 81/34 , ht= 4871 ; ( -58744/58267 , -96495257/1048806 ), ( -3527/4871 , 3506666/43839 )
**u= 81/38 , ht= 1245 ; ( 1245/181 , -775492/1629 ), ( 12423/16033 , 12503368/144297 )
**u= 81/40 , ht= 9833 ; ( -59317/19613 , -39511639/176517 ), ( 5233/9833 , -7044725/88497 )
**u= 81/44 , ht= 553 ; ( -553/201 , 390292/1809 ), ( 10443/133679 , -12705683/171873 )
**u= 81/52 , ht= 3247 ; ( -293255/5381 , 211502539/48429 ), ( 1693/3247 , 2640649/29223 )
**u= 81/56 , ht= 13581 ; ( 17575/6144 , -55730203/221184 ), ( 9332/13581 , 49326047/488916 )
**u= 81/64 , ht= 12191 ; ( 52193/36239 , -101668199/652302 ), ( -3743/12191 , 10202306/109719 )
**u= 81/74 , ht= 177707 ; ( -309531/157853 , 298904296/1420677 ), ( -177707/99645 , 175272988/896805 )
**u= 81/76 , ht= 1675 ; ( 1675/73 , 1461649/657 ), ( -73/1675 , -1461649/15075 )
**u= 81/82 , ht= 45659 ; ( -25435/45659 , -47331406/410931 ), ( 53569/36964 , 117879719/665352 )
**u= 81/92 , ht= 109485 ; ( 186045/126751 , -215935811/1140759 ), ( 55343/109485 , -117672827/985365 )
**u= 81/94 , ht= 6217 ; ( -6217/2384 , -12911621/42912 ), ( -2384/6217 , 12911621/111906 )
**u= 81/106 , ht= 124255 ; ( -263470/80659 , -567393179/1451862 ), ( 85247/124255 , 155147966/1118295 )
**u= 81/110 , ht= 229 ; ( -229/189 , 311428/1701 ), ( 832804/116979 , -1764164755/2105622 )
**u= 81/122 , ht= 142151 ; ( 24726/142151 , 318756149/2558718 ), ( 142151/24726 , 318756149/445068 )
**u= 81/124 , ht= 157915 ; ( -157915/139127 , -234369893/1252143 ), ( 2734607/16465 , 3045369412/148185 )
**u= 81/140 , ht= 281143 ; ( -281143/226707 , -427352365/2040363 ), ( 747813/5424443 , -6479263072/48819987 )
**u= 81/142 , ht= 220913 ; ( 461909/243476 , -1244483261/4382568 ), ( -117608/220913 , -29701/198 )
**u= 81/148 , ht= 649391 ; ( 649391/430721 , 948042154/3876489 ), ( -766129/141185 , -947784218/1270665 )
**u= 81/166 , ht= 10453 ; ( -10453/1499 , 13606126/13491 ), ( -6947662/407549 , -17934650569/7335882 )
**u= 81/176 , ht= 18329 ; ( -7785/18329 , -26420027/164961 ), ( 3817423/1748829 , -11141649979/31478922 )
**u= 81/188 , ht= 98131 ; ( 90069/167431 , -260692931/1506879 ), ( 98131/37689 , -144140591/339201 )
**u= 82/5 , ht= 2649 ; ( -61/2649 , -21957/883 ), ( -2967/2653 , 98361/2653 )
**u= 82/37 , ht= 11479 ; ( -153033/221 , 10278929/221 ), ( 11479/6077 , 872569/6077 )
**u= 82/77 , ht= 675589 ; ( -675589/669447 , -10240967/74383 ), ( 2139849/1031357 , 230202171/1031357 )
**u= 82/117 , ht= 32705 ; ( -2274245/208833 , -818435123/626499 ), ( 12581/32705 , 4185863/32705 )
**u= 82/133 , ht= 5166643 ; ( 2949001/5166643 , -757683689/5166643 ), ( -30637213/603689 , -3902817841/603689 )
**u= 82/141 , ht= 6517 ; ( 1535/6517 , -878011/6517 ), ( 145876553/31820525 , -19579753101/31820525 )
**u= 82/149 , ht= 3885785 ; ( 3885785/1628733 , -189325531/542911 ), ( -4900389/44479 , -660632073/44479 )
**u= 85/6 , ht= 1021 ; ( -266243/106699 , -7593240/106699 ), ( -557/1021 , -30818/1021 )
**u= 85/16 , ht= 95449 ; ( 95449/55767 , -1598249/18589 ), ( 4646613/108329 , -403610613/216658 )
**u= 85/22 , ht= 70391 ; ( 71297/91889 , -5917826/91889 ), ( -70391/2033 , -3584128/2033 )
**u= 85/24 , ht= 10537 ; ( -20503/21076 , 6250027/84304 ), ( -10537/7441 , 97935/1063 )
**u= 85/28 , ht= 9101 ; ( -476121/208003 , 29817425/208003 ), ( -9101/3373 , 557192/3373 )
**u= 85/36 , ht= 2083 ; ( -49/2083 , -135623/2083 ), ( 177203/108191 , -13513287/108191 )
**u= 85/52 , ht= 47347 ; ( -47347/36711 , -1562063/12237 ), ( -72617/26781 , -6054325/26781 )
**u= 85/54 , ht= 24308 ; ( 10549/24308 , 4224503/48616 ), ( 529913/241121 , -46407992/241121 )
**u= 85/66 , ht= 326123 ; ( -816889/579737 , 88270562/579737 ), ( -326123/74134 , -58945743/148268 )
**u= 85/84 , ht= 25187 ; ( 25187/589 , -2504586/589 ), ( -354539/49333 , 35586952/49333 )
**u= 85/94 , ht= 217011 ; ( 217011/18013 , -22900246/18013 ), ( -198086/691483 , -151288639/1382966 )
**u= 85/112 , ht= 288621 ; ( 12459/309833 , -71192789/619666 ), ( -180883/288621 , -39100864/288621 )
**u= 85/124 , ht= 78851 ; ( -4597483/746601 , -187529437/248867 ), ( -71907/78851 , -12889860/78851 )
**u= 85/126 , ht= 193673 ; ( 193673/112606 , -54554531/225212 ), ( -4418917/159531 , -1615185410/478593 )
**u= 85/132 , ht= 148387 ; ( 1145057/863699 , -178742985/863699 ), ( -92249/148387 , -21774617/148387 )
**u= 85/138 , ht= 2679703 ; ( -21242392/3859831 , -5502182125/7719662 ), ( -2679703/1580564 , -792864219/3161128 )
**u= 85/142 , ht= 280541 ; ( 280541/103893 , 12889512/34631 ), ( 958938/325831 , 261821673/651662 )
**u= 85/154 , ht= 35437 ; ( -2675669/7674517 , -1094047030/7674517 ), ( 35437/1579 , 4774892/1579 )
**u= 85/156 , ht= 8369609 ; ( -11849741/4777363 , -1730951640/4777363 ), ( -8369609/690247 , -1137770858/690247 )
**u= 85/174 , ht= 224666 ; ( 16823/224666 , -64471969/449332 ), ( 912718/186181 , 266568259/372362 )
**u= 85/178 , ht= 894969 ; ( 8727326/1004913 , -282519647/223314 ), ( 894969/812822 , -349924257/1625644 )
**u= 85/182 , ht= 477101 ; ( 477101/244323 , 26146138/81441 ), ( 341733459/46627843 , 50471154954/46627843 )
**u= 85/196 , ht= 20797 ; ( -20797/11859 , 8483022/27671 ), ( -5035263/5308229 , 7777527957/37157603 )
**u= 90/121 , ht= 41891 ; ( -3377/41891 , -160820101/1382403 ), ( 811147/414799 , 3486215809/13688367 )
**u= 90/169 , ht= 62903 ; ( -62903/21899 , -355974659/854061 ), ( 150707/61249 , 869438155/2388711 )
**u= 90/193 , ht= 127753 ; ( 6145717/122959 , 2700582641/368877 ), ( -127753/24331 , 57136109/72993 )
**u= 98/9 , ht= 505 ; ( -505/143 , -111063/1001 ), ( -1553/905 , -381873/6335 )
**u= 98/17 , ht= 1787 ; ( 68439/44033 , -2156303/28021 ), ( 1787/189 , 524353/1323 )
**u= 98/41 , ht= 7109 ; ( -15485/17781 , -3558799/41489 ), ( 6699/7109 , -4422849/49763 )
**u= 98/65 , ht= 26141 ; ( -2589/26141 , 14977157/182987 ), ( 82673/15073 , 47912735/105511 )
**u= 98/81 , ht= 55717 ; ( -103531/126221 , -311691907/2650641 ), ( -55717/9893 , 36015723/69251 )
**u= 98/129 , ht= 56987 ; ( 29357/56987 , -51486321/398909 ), ( 825347/132445 , -671374933/927115 )
**u= 98/185 , ht= 1297 ; ( 1709749/276781 , 1665863501/1937467 ), ( -257/1297 , -1271723/9079 )
**u= 98/193 , ht= 928955 ; ( 928955/133317 , -307314481/311073 ), ( 6606891/332075 , -6498800061/2324525 )
**u= 100/33 , ht= 15692 ; ( -2887/15692 , 9168083/156920 ), ( -38548/8213 , -22647047/82130 )
**u= 100/41 , ht= 21388 ; ( -21388/19215 , 6136909/64050 ), ( 46425/12272 , 30753063/122720 )
**u= 100/49 , ht= 384 ; ( -384/235 , -2205991/16450 ), ( -795/1408 , 7922967/98560 )
**u= 100/129 , ht= 131740 ; ( -131740/80551 , -175388923/805510 ), ( -219899/15268 , -50074957/30536 )
**u= 100/177 , ht= 28043 ; ( -463547/644296 , 211206539/1288592 ), ( -28043/6472 , -38291679/64720 )
**u= 101/2 , ht= 5833 ; ( 74641/3919 , 982898/3919 ), ( -5833/1967 , -12988/281 )
**u= 101/8 , ht= 10 ; ( -10 , -1127/4 ), ( 10207/11858 , 1760393/47432 )
**u= 101/10 , ht= 34868 ; ( -718409/236481 , -7916956/78827 ), ( 26187/34868 , 2742423/69736 )
**u= 101/14 , ht= 10665 ; ( -123997/10814 , 9256661/21628 ), ( -5953/10665 , -454448/10665 )
**u= 101/30 , ht= 7607 ; ( 162607/172882 , -25871765/345764 ), ( 7607/17 , 414724/17 )
**u= 101/32 , ht= 2239 ; ( 136826/54149 , 66253811/433192 ), ( -2239/1345 , 147044/1345 )
**u= 101/36 , ht= 42451 ; ( 249735/47191 , 45514726/141573 ), ( -42451/24435 , -8774164/73305 )
**u= 101/38 , ht= 49636 ; ( -220355/83797 , 14459324/83797 ), ( 49636/10651 , -6229093/21302 )
**u= 101/42 , ht= 156287 ; ( 33824081/6686999 , 2223203192/6686999 ), ( 156287/19388 , 20314947/38776 )
**u= 101/44 , ht= 222813 ; ( 206071/222813 , -6677687/74271 ), ( -999909/25643 , 66030978/25643 )
**u= 101/48 , ht= 56737 ; ( -244567/135587 , -38555009/271174 ), ( 56737/52573 , -10665349/105146 )
**u= 101/50 , ht= 91927 ; ( 608139/127339 , -43714574/127339 ), ( 91927/48650 , 73186723/486500 )
**u= 101/60 , ht= 167663 ; ( -167663/2293 , -12923607/2293 ), ( 580117/1624517 , -132953410/1624517 )
**u= 101/66 , ht= 1258030 ; ( 49512577/703399 , 4002828024/703399 ), ( 70657/1258030 , 203710133/2516060 )
**u= 101/74 , ht= 358711 ; ( -100792/358711 , -63792539/717422 ), ( 2040083/951765 , -192707852/951765 )
**u= 101/80 , ht= 109161 ; ( -14441/109161 , 1088968/12129 ), ( 109161/14441 , 9800712/14441 )
**u= 101/84 , ht= 779405 ; ( 779405/176327 , -72876667/176327 ), ( 34455269/1210967 , 3144416273/1210967 )
**u= 101/96 , ht= 50327 ; ( 73801/112612 , -105018781/900896 ), ( 50327/48265 , 971227/6895 )
**u= 101/104 , ht= 3116863 ; ( 60377305/51260327 , -8037390448/51260327 ), ( -896809/3116863 , -329123083/3116863 )
**u= 101/108 , ht= 124669 ; ( 619733/37643 , -192614524/112929 ), ( 124669/23075 , -13111572/23075 )
**u= 101/122 , ht= 14863 ; ( -14863/143 , 148514/13 ), ( 1340449/1928724 , -516310751/3857448 )
**u= 101/126 , ht= 287863 ; ( -114361/287863 , -34598658/287863 ), ( -611602/127699 , 7346221/13442 )
**u= 101/128 , ht= 704323 ; ( -2463233/2610593 , -808165039/5221186 ), ( 704323/4035 , 294781/15 )
**u= 101/154 , ht= 300033 ; ( 562951/482247 , -30512218/160749 ), ( 300033/27145 , 37201908/27145 )
**u= 101/164 , ht= 830141 ; ( -362519759/41347133 , -46496168629/41347133 ), ( -830141/498013 , 123364352/498013 )
**u= 101/172 , ht= 1428097 ; ( -15548599/5141481 , -712402316/1713827 ), ( 1138143/1428097 , -238320444/1428097 )
**u= 101/174 , ht= 2370281 ; ( -19745905/1112606 , 5191912703/2225212 ), ( 613142/2370281 , -642727913/4740562 )
**u= 101/182 , ht= 566373 ; ( -1048769625/252845144 , 289648941677/505690288 ), ( -379208/566373 , -183001367/1132746 )
**u= 101/186 , ht= 856343 ; ( -36874436/30936173 , 13064370141/61872346 ), ( -856343/155900 , 236253657/311800 )
**u= 101/192 , ht= 191519 ; ( -1903825/652609 , 554997213/1305218 ), ( -191519/80671 , -28654442/80671 )
**u= 104/25 , ht= 3089 ; ( 3089/1587 , 567449/5290 ), ( 16197/8159 , -8894607/81590 )
**u= 104/81 , ht= 3915 ; ( -8607/10499 , -21567379/188982 ), ( -2311/3915 , -7222043/70470 )
**u= 104/113 , ht= 153735 ; ( 153735/117349 , 40321627/234698 ), ( 1186937/270507 , -253808227/541014 )
**u= 104/137 , ht= 87911 ; ( -87911/40509 , -7406689/27006 ), ( 2790523/2037023 , -793113589/4074046 )
**u= 104/177 , ht= 2147 ; ( -1950759485495/20896809677 , 509033739764007/41793619354 ), ( 23/2147 , 560241/4294 )
**u= 106 , ht= 209 ; ( -845/567 , 2923/189 ), ( -37/209 , 1417/209 )
**u= 106/49 , ht= 6577 ; ( -24573/58711 , -30295049/410977 ), ( 6577/1051 , 3170417/7357 )
**u= 106/57 , ht= 41555 ; ( 41555/32473 , 3867427/32473 ), ( -18769/102451 , 7637577/102451 )
**u= 106/105 , ht= 16627 ; ( 16627/12217 , -2053611/12217 ), ( 4523983/306133 , -451322955/306133 )
**u= 106/121 , ht= 15277 ; ( -15277/11641 , -22573871/128051 ), ( -131603/144199 , -229450103/1586189 )
**u= 106/129 , ht= 495959 ; ( -293435/495959 , -63575261/495959 ), ( -4211917/1935191 , 511373481/1935191 )
**u= 106/169 , ht= 43661 ; ( 875005/38311 , 1437731017/498043 ), ( -43661/22855 , -80898217/297115 )
**u= 106/177 , ht= 619343 ; ( -2127371/2138801 , -389834577/2138801 ), ( 619343/266405 , 87126937/266405 )
**u= 113/2 , ht= 4071 ; ( 2717/4821 , 25536/1607 ), ( 4071/2983 , -3618/157 )
**u= 113/12 , ht= 65075 ; ( -65075/30253 , 2335368/30253 ), ( 806443/347131 , -28656214/347131 )
**u= 113/36 , ht= 35623 ; ( 72597/21557 , 12821332/64671 ), ( 35623/18387 , 6789467/55161 )
**u= 113/38 , ht= 32815 ; ( 32815/4178 , 3835973/8356 ), ( 102114/255827 , -31943553/511654 )
**u= 113/42 , ht= 11539 ; ( 1613/11539 , -710482/11539 ), ( -94697/31030 , -12152999/62060 )
**u= 113/52 , ht= 359101 ; ( 359101/483 , 8119488/161 ), ( 2000889/24265 , 135765786/24265 )
**u= 113/54 , ht= 3400043 ; ( 4643891/596381 , 970931762/1789143 ), ( -3400043/21115 , 705246502/63345 )
**u= 113/56 , ht= 91995 ; ( -91995/41728 , -28444553/166912 ), ( -475752/2495 , 133988427/9980 )
**u= 113/66 , ht= 1728233 ; ( 1728233/239273 , -133336506/239273 ), ( -1852151/1657399 , -189959958/1657399 )
**u= 113/70 , ht= 224673 ; ( -359867/4638 , 18883829/3092 ), ( -224673/143753 , -20994672/143753 )
**u= 113/72 , ht= 1993 ; ( -1993/841 , 172672/841 ), ( -841/1993 , -172672/1993 )
**u= 113/80 , ht= 1200377 ; ( -1200377/618553 , -113623802/618553 ), ( -1732103/46423 , -145807127/46423 )
**u= 113/86 , ht= 195833 ; ( 20393/195833 , -17178214/195833 ), ( -265126/15965 , -46346677/31930 )
**u= 113/90 , ht= 687891 ; ( 687891/206221 , 192272942/618663 ), ( -2725663/584247 , 746392768/1752741 )
**u= 113/92 , ht= 1089153 ; ( 893999/2799963 , 88409903/933321 ), ( -1089153/492925 , -107879313/492925 )
**u= 113/98 , ht= 10609 ; ( -66260/839 , 86396347/11746 ), ( -4777/10609 , 1083548/10609 )
**u= 113/108 , ht= 194533 ; ( 1501129/283753 , -149356732/283753 ), ( -194533/132545 , 3287757/18935 )
**u= 113/110 , ht= 567933 ; ( -807197/391088 , -176998381/782176 ), ( -552043/567933 , 78147652/567933 )
**u= 113/126 , ht= 37116 ; ( 1328311/1305316 , -393326701/2610632 ), ( 37116/31855 , 30990719/191130 )
**u= 113/132 , ht= 2288779 ; ( 2483065/2807191 , -405084594/2807191 ), ( -2288779/1996007 , -328243493/1996007 )
**u= 113/160 , ht= 2460286 ; ( 2460286/1278441 , 879829193/3409176 ), ( 3972471/738134 , -3846528981/5905072 )
**u= 113/162 , ht= 45829 ; ( 163940/3497 , -117806617/20982 ), ( 1413/45829 , -49411034/412461 )
**u= 113/164 , ht= 1916971 ; ( -132569/1916971 , -231505381/1916971 ), ( -14117971/3499729 , 1752391339/3499729 )
**u= 113/168 , ht= 1957033 ; ( 1957033/388849 , 243298031/388849 ), ( -22001867/791134 , 10738463287/3164536 )
**u= 113/174 , ht= 398627 ; ( 96237329/8856791 , -11992993336/8856791 ), ( 130957/398627 , -52068542/398627 )
**u= 113/176 , ht= 587483 ; ( -1285759/79781 , 321557321/159562 ), ( 587483/43445 , 73522937/43445 )
**u= 113/186 , ht= 8365294 ; ( -8365294/379031 , 2148777869/758062 ), ( -8529307/403963 , -156511122/57709 )
**u= 116/81 , ht= 86142 ; ( -86142/61769 , -159442327/1111842 ), ( 61769/86142 , 159442327/1550556 )
**u= 116/89 , ht= 804626 ; ( 8417070/5235587 , 1736578799/10471174 ), ( 679403/804626 , 184494167/1609252 )
**u= 116/153 , ht= 234578 ; ( 257446/189939 , -220466669/1139634 ), ( -234578/107003 , 59225199/214006 )
**u= 116/161 , ht= 371858 ; ( 108075542/17664221 , -25803725437/35328442 ), ( 328549/371858 , -116922773/743716 )
**u= 121/16 , ht= 337 ; ( 175/337 , 152024/3707 ), ( -1743/4891 , 4149441/107602 )
**u= 121/28 , ht= 4077 ; ( 4077/3853 , 2968388/42383 ), ( -40433/14867 , 22804847/163537 )
**u= 121/30 , ht= 3644 ; ( 1799/3644 , -4453083/80168 ), ( 11011/2291 , 6162738/25201 )
**u= 121/34 , ht= 1693 ; ( -76777/52841 , -54343042/581251 ), ( 1693/1115 , -1182248/12265 )
**u= 121/40 , ht= 321 ; ( 2006/2671 , -8451581/117524 ), ( -319/321 , -286232/3531 )
**u= 121/42 , ht= 13561 ; ( 12889/13561 , 12125558/149171 ), ( 12241/15706 , -25809951/345532 )
**u= 121/46 , ht= 15116 ; ( -5380813/94981 , -3649564936/1044791 ), ( -15116/7953 , 23172553/174966 )
**u= 121/48 , ht= 1295 ; ( -75551/93679 , -83386709/1030469 ), ( -949/1295 , -317807/4070 )
**u= 121/52 , ht= 4257 ; ( -4257/2477 , 3551551/27247 ), ( 489/16115 , 11625051/177265 )
**u= 121/54 , ht= 2171 ; ( -2171/1597 , 5941568/52701 ), ( 120362/18059 , 178905723/397298 )
**u= 121/76 , ht= 24229 ; ( -5183/24229 , 21602561/266519 ), ( 44239/33105 , -48172792/364155 )
**u= 121/82 , ht= 16221 ; ( -8016/16789 , -33696989/369358 ), ( -11741/16221 , -18133358/178431 )
**u= 121/90 , ht= 17747 ; ( 72302/176027 , -361093505/3872594 ), ( 15067/17747 , 22086536/195217 )
**u= 121/94 , ht= 46331 ; ( -46331/2173 , -44969344/23903 ), ( 114724/146825 , -361323323/3230150 )
**u= 121/102 , ht= 346 ; ( 125/346 , -743153/7612 ), ( -292838/38543 , 596655333/847946 )
**u= 121/112 , ht= 5445 ; ( -5445/1541 , -5988926/16951 ), ( 1541/5445 , -5988926/59895 )
**u= 121/114 , ht= 19444 ; ( -39928/14621 , 90802179/321662 ), ( 15967/19444 , -53728967/427768 )
**u= 121/118 , ht= 7553 ; ( 64247/55102 , -183893321/1212244 ), ( -7553/6135 , -10570804/67485 )
**u= 121/126 , ht= 1648 ; ( -501/1648 , -11601613/108768 ), ( 14157076/4436127 , -99925993679/292784382 )
**u= 121/136 , ht= 198212 ; ( 198212/82459 , 1001466899/3628196 ), ( 1234721/668740 , 6550609097/29424560 )
**u= 121/150 , ht= 13223 ; ( 13223/7034 , -183443413/773740 ), ( 31889/67967 , 459763574/3738185 )
**u= 121/156 , ht= 236335 ; ( -236335/63887 , -305788716/702757 ), ( -765821/141917 , 972857596/1561087 )
**u= 121/164 , ht= 628029 ; ( 529793/628029 , 3771593/24761 ), ( -25873623/4166665 , -33563413188/45833315 )
**u= 121/166 , ht= 80687 ; ( -5799/80687 , 104232652/887557 ), ( -80687/5799 , -104232652/63789 )
**u= 121/172 , ht= 295825 ; ( -295825/270571 , 525801599/2976281 ), ( 568459/536811 , -1025457328/5904921 )
**u= 121/178 , ht= 326297 ; ( -326297/189593 , -503509258/2085523 ), ( 2529053/568027 , -3458445488/6248297 )
**u= 121/180 , ht= 6569 ; ( 35103/491713 , 1984264808/16226529 ), ( 6569/269 , -8821165/2959 )
**u= 121/184 , ht= 29401 ; ( 3640683/4133389 , -7471972507/45467279 ), ( 27193/29401 , 54327026/323411 )
**u= 121/186 , ht= 93335 ; ( 93335/32513 , -134799696/357643 ), ( -1058921/523102 , -3221737843/11508244 )
**u= 121/196 , ht= 11875 ; ( -11875/5373 , -6082792/19701 ), ( 5373/11875 , -18248376/130625 )
**u= 122/9 , ht= 210967 ; ( -15264891/898241 , 1240333301/2694723 ), ( -210967/70347 , -18187903/211041 )
**u= 122/33 , ht= 382693 ; ( 1898939/74153 , -98808129/74153 ), ( 382693/17311 , 2847363/2473 )
**u= 122/57 , ht= 1983031 ; ( -36573305/18515827 , 2801957929/18515827 ), ( -1983031/655613 , -142782161/655613 )
**u= 122/65 , ht= 155827 ; ( 851963/447 , -20728113/149 ), ( 59783/155827 , 1740331/22261 )
**u= 122/97 , ht= 637413 ; ( 182431/637413 , 19707801/212471 ), ( 59427219/55574767 , 7255422369/55574767 )
**u= 122/129 , ht= 23215 ; ( -23215/5509 , 2453539/5509 ), ( 5979509/3730265 , -724745289/3730265 )
**u= 122/169 , ht= 27357 ; ( -42493/33753 , -27678511/146263 ), ( 27357/6175 , -42913401/80275 )
**u= 122/177 , ht= 35588477 ; ( 35588477/145289 , -4286830341/145289 ), ( 23743689983/1010291069 ,
2862691651423/1010291069 )
**u= 122/185 , ht= 288533 ; ( 169389/403969 , 53944837/403969 ), ( -171377/288533 , -41327183/288533 )
**u= 128/41 , ht= 6035 ; ( -6035/4789 , -3488237/38312 ), ( -4789/6035 , 3488237/48280 )
**u= 128/153 , ht= 13591 ; ( -2593/13591 , 12102487/108728 ), ( 291161/101135 , -269604471/809080 )
**u= 128/161 , ht= 9357 ; ( -6053/9357 , -3333101/24952 ), ( 1055213/764493 , -1169178763/6115944 )
**u= 128/177 , ht= 274847 ; ( -1753381/16978883 , 16058741063/135831064 ), ( -274847/145351 , 292507787/1162808 )
**u= 130 , ht= 293 ; ( -2637/491 , -10825/491 ), ( 293/61 , 3497/61 )
**u= 130/97 , ht= 117243 ; ( -117243/41971 , -10757009/41971 ), ( -1025487/77791 , 88844655/77791 )
**u= 130/129 , ht= 107749 ; ( 717581/2595563 , -268274473/2595563 ), ( -49373/107749 , -11806953/107749 )
**u= 130/193 , ht= 609989 ; ( -609989/259283 , -80763089/259283 ), ( -1756643/1158429 , -256402523/1158429 )
**u= 136/93 , ht= 4031 ; ( 1861/4031 , -734357/8062 ), ( 3009787/302495 , 500339333/604990 )
**u= 136/141 , ht= 956183 ; ( -9837649/1270613 , 2020067581/2541226 ), ( -956183/228875 , -200234943/457750 )
**u= 136/157 , ht= 10589 ; ( -378851/997431 , 76429977/664954 ), ( -10589/3681 , 2409161/7362 )
**u= 137/14 , ht= 361 ; ( -185/361 , 12986/361 ), ( 368/515 , 40439/1030 )
**u= 137/28 , ht= 605 ; ( -605/381 , 10772/127 ), ( -381/605 , -32316/605 )
**u= 137/36 , ht= 705 ; ( 2055/743 , -335972/2229 ), ( 49/705 , 108646/2115 )
**u= 137/38 , ht= 125 ; ( -125/46 , -14027/92 ), ( -179/277 , 17368/277 )
**u= 137/50 , ht= 181 ; ( -44/181 , -22511/362 ), ( 329/183 , -113734/915 )
**u= 137/60 , ht= 91 ; ( -91/41 , -6605/41 ), ( -2887/2777 , 265112/2777 )
**u= 137/98 , ht= 80 ; ( 80/31 , -14513/62 ), ( -2217/2839 , 2132658/19873 )
**u= 137/126 , ht= 1501 ; ( -1019/1501 , 173996/1501 ), ( 137097/8423 , -39520894/25269 )
**u= 137/136 , ht= 3867 ; ( 3613/3867 , -175771/1289 ), ( -40197/7165 , -4068429/7165 )
**u= 137/138 , ht= 1039 ; ( -485/3517 , 356348/3517 ), ( -617/1039 , -121284/1039 )
**u= 137/146 , ht= 1 ; ( 1 , -146 ), ( -473/1057 , -119548/1057 )
**u= 137/152 , ht= 15087 ; ( -8399/15087 , 606306/5029 ), ( -29139/16333 , 3518766/16333 )
**u= 137/174 , ht= 259243 ; ( -62855/262351 , 30404964/262351 ), ( 259243/16165 , 29274698/16165 )
**u= 137/176 , ht= 679 ; ( -69/679 , 154723/1358 ), ( -352753/43313 , -40285078/43313 )
**u= 137/200 , ht= 6461 ; ( -6461/1861 , 812417/1861 ), ( 9191/8313 , -7487122/41565 )
**u= 144 , ht= 32 ; ( 1/32 , -4567/384 ), ( 104/137 , 14609/1644 )
**u= 144/17 , ht= 3421 ; ( -10436/8481 , 503893/9252 ), ( 3421/1704 , 1577587/20448 )
**u= 144/41 , ht= 940 ; ( 266636/74223 , -177183853/890676 ), ( -249/940 , 622517/11280 )
**u= 144/49 , ht= 4756 ; ( 113/4756 , -23317169/399504 ), ( -9047/4396 , 49295593/369264 )
**u= 144/113 , ht= 241197 ; ( -37348/241197 , -259474507/2894364 ), ( 4139003/354972 , 4416369203/4259664 )
**u= 144/161 , ht= 40069 ; ( 55865/32244 , -81847783/386928 ), ( 40069/8352 , 51938141/100224 )
**u= 144/169 , ht= 16727 ; ( -16727/2564 , 285997409/399984 ), ( 2276/16727 , -285300449/2609412 )
**u= 144/185 , ht= 43164 ; ( 37817776/7168641 , 52355591587/86023692 ), ( -8821/43164 , -59924951/517968 )
**u= 145/4 , ht= 57 ; ( 57/41 , -1154/41 ), ( -17961/7043 , 328077/7043 )
**u= 145/16 , ht= 3793 ; ( 1349/3793 , -267895/7586 ), ( 88463/23131 , -6085921/46262 )
**u= 145/18 , ht= 7993 ; ( -850536/79487 , 180307099/476922 ), ( 2449/7993 , -294160/7993 )
**u= 145/24 , ht= 11057 ; ( -11057/3761 , -474832/3761 ), ( -200021/27733 , -8223084/27733 )
**u= 145/42 , ht= 732182 ; ( -3336898/3258509 , 502049517/6517018 ), ( 624221/732182 , 103565939/1464364 )
**u= 145/46 , ht= 81181 ; ( 81181/58112 , -11246099/116224 ), ( 437429/104803 , 25341980/104803 )
**u= 145/68 , ht= 687699 ; ( -586519/2310183 , -18138590/256687 ), ( 687699/594097 , 62238357/594097 )
**u= 145/72 , ht= 125148 ; ( 125148/58511 , -116817905/702132 ), ( 793711/156567 , -171047833/469701 )
**u= 145/82 , ht= 32533 ; ( 26301/32533 , -3146212/32533 ), ( 32533/26301 , 3146212/26301 )
**u= 145/86 , ht= 283263 ; ( -203369/283263 , 8952344/94421 ), ( 23410509/219883 , 1803216216/219883 )
**u= 145/88 , ht= 36122 ; ( 93661/9533 , -7334102/9533 ), ( -36122/7251 , -11481967/29004 )
**u= 145/96 , ht= 1196557 ; ( 1196557/69541 , 195049269/139082 ), ( -3015299/343387 , -493919213/686774 )
**u= 145/108 , ht= 373349 ; ( -373349/7 , -32221445/7 ), ( 1468781/19127 , -380350747/57381 )
**u= 145/114 , ht= 38068 ; ( -321452/105839 , 60016665/211678 ), ( -1661/38068 , -6757339/76136 )
**u= 145/124 , ht= 268313 ; ( -268313/88249 , 26120588/88249 ), ( -2217793/73789 , 205222333/73789 )
**u= 145/126 , ht= 18867 ; ( 18867/16616 , -14062091/99696 ), ( -16616/18867 , 14062091/113202 )
**u= 145/142 , ht= 611416 ; ( 931437/360029 , -98824294/360029 ), ( 611416/96027 , -122504147/192054 )
**u= 145/184 , ht= 249 ; ( 249/88 , 119003/352 ), ( -5581379/3247677 , 727467848/3247677 )
**u= 145/192 , ht= 2131 ; ( -81879053/8514529 , 37892271175/34058116 ), ( 2131/23 , -980987/92 )
**u= 145/198 , ht= 162241 ; ( -777251/19578 , 545148427/117468 ), ( -39202/162241 , 39010131/324482 )
**u= 146 , ht= 11 ; ( 11/3 , 9 ), ( 9/47 , -39/47 )
**u= 146/5 , ht= 7 ; ( -3/7 , 143/7 ), ( -7/3 , -143/3 )
**u= 146/21 , ht= 593 ; ( 6781/3127 , 282993/3127 ), ( 593/5 , 22519/5 )
**u= 146/29 , ht= 491 ; ( -6309/1727 , 26489/157 ), ( 143/491 , 22781/491 )
**u= 146/41 , ht= 1265 ; ( -805/1573 , 93661/1573 ), ( 199/1265 , -67843/1265 )
**u= 146/81 , ht= 497 ; ( -497/151 , -38689/151 ), ( 32027/16989 , 3472229/21843 )
**u= 146/93 , ht= 2677 ; ( -2677/113 , 213843/113 ), ( 5353/13165 , 1134261/13165 )
**u= 146/125 , ht= 427 ; ( 427/177 , 14257/59 ), ( 2283/577 , 1089519/2885 )
**u= 146/137 , ht= 346159 ; ( 346159/240121 , -40811357/240121 ), ( -100580749/100655811 , 13784705027/100655811 )
**u= 146/189 , ht= 10463 ; ( -394367/71371 , -136801241/214113 ), ( 10463/3659 , -1261221/3659 )
**u= 148 , ht= 29 ; ( -630/709 , 16567/1418 ), ( 26/29 , 601/58 )
**u= 148/73 , ht= 52066 ; ( -42323/52066 , 9425387/104132 ), ( -7903519/11018738 , 1904728577/22037476 )
**u= 148/81 , ht= 4226 ; ( -581/4226 , 631239/8452 ), ( -286233/137942 , 60450421/354708 )
**u= 148/105 , ht= 5855474 ; ( -10736197/1367258 , -1823217811/2734516 ), ( 5855474/186209 , 987003223/372418 )
**u= 148/121 , ht= 38939 ; ( -38939/13706 , -82118521/301532 ), ( -472829/152846 , -988565801/3362612 )
**u= 148/153 , ht= 368714 ; ( -9360751/7726918 , 2468360831/15453836 ), ( -368714/11117 , -75017683/22234 )
**u= 148/169 , ht= 14243 ; ( -14243/7042 , 44146127/183092 ), ( -10324241/261914 , 28695494371/6809764 )
**u= 148/177 , ht= 1216742 ; ( -15690070/147359 , 3431973533/294718 ), ( -1216742/46861 , 266339393/93722 )
**u= 149/2 , ht= 2373 ; ( -8393/2463 , 29576/821 ), ( -2373/1453 , -34212/1453 )
**u= 149/12 , ht= 9661 ; ( 15935/24341 , -826947/24341 ), ( 4159/9661 , 297719/9661 )
**u= 149/16 , ht= 52411 ; ( -52411/33663 , 1359651/22442 ), ( -67953/8591 , -2249316/8591 )
**u= 149/18 , ht= 1769 ; ( -1769/1399 , -78362/1399 ), ( 71181/44269 , 8747494/132807 )
**u= 149/20 , ht= 18893 ; ( -278507/252603 , 4591442/84201 ), ( 18893/10707 , 796220/10707 )
**u= 149/30 , ht= 10702 ; ( 1687/10702 , 972901/21404 ), ( 597191/86911 , 27096308/86911 )
**u= 149/34 , ht= 237370 ; ( -18823/338151 , 5395558/112717 ), ( -184587/237370 , 4103679/67820 )
**u= 149/38 , ht= 124775 ; ( -724201/799536 , -36322083/533024 ), ( -93824/124775 , 685519/10850 )
**u= 149/48 , ht= 33629 ; ( -653477/5077 , 37084121/5077 ), ( 9791/33629 , 3975329/67258 )
**u= 149/54 , ht= 78265 ; ( 4834339/726667 , -294262692/726667 ), ( -76316/78265 , 39486647/469590 )
**u= 149/70 , ht= 4728 ; ( 1313/4728 , 224253/3152 ), ( 46317/66677 , 5564664/66677 )
**u= 149/74 , ht= 173503 ; ( -48904/173503 , 2310079/31546 ), ( 818453/31019 , 57728962/31019 )
**u= 149/90 , ht= 12484 ; ( -147253/59467 , -12342382/59467 ), ( -3941/12484 , -2034875/24968 )
**u= 149/104 , ht= 97508 ; ( -47108809/2753393 , -3942446203/2753393 ), ( 24929/97508 , -33633701/390032 )
**u= 149/114 , ht= 1087774 ; ( 1662530/2004131 , 455561269/4008262 ), ( 723373/1087774 , -228538629/2175548 )
**u= 149/126 , ht= 957451 ; ( -614147/957451 , 104608912/957451 ), ( -1218037/1151141 , 154124066/1151141 )
**u= 149/132 , ht= 206357 ; ( -826337/33677 , -77842957/33677 ), ( -206357/104503 , 3110421/14929 )
**u= 149/138 , ht= 309712 ; ( 3980423/2444207 , 449540704/2444207 ), ( 78907/309712 , 61517913/619424 )
**u= 149/140 , ht= 282427 ; ( -29784849/4622111 , 2921753794/4622111 ), ( -282427/50997 , 27821294/50997 )
**u= 149/150 , ht= 149359 ; ( -5961719/275743 , 2994119224/1378715 ), ( -50807/149359 , -79148644/746795 )
**u= 149/156 , ht= 8350267 ; ( -106464589/6587363 , 10914804534/6587363 ), ( -8350267/353675 , 855244538/353675 )
**u= 149/162 , ht= 29023 ; ( 717850/1330503 , 2837649157/23949054 ), ( 29023/8575 , -1352486/3675 )
**u= 149/164 , ht= 130505 ; ( 130505/6763 , 13710431/6763 ), ( -7237331/2526655 , -804283753/2526655 )
**u= 149/186 , ht= 178600 ; ( 634265/1027097 , -134881002/1027097 ), ( 142733/178600 , 51090781/357200 )
**u= 149/192 , ht= 31649131 ; ( 89935781/8993609 , 41041761139/35974436 ), ( -31649131/3439979 , 3614079144/3439979 )
**u= 149/200 , ht= 132087 ; ( 132087/10513 , -15352141/10513 ), ( -600199/220665 , -370461992/1103325 )
**u= 160 , ht= 151 ; ( -151/117 , -1661/156 ), ( -67/151 , -13/4 )
**u= 160/9 , ht= 379 ; ( -2149/767 , -215151/3068 ), ( 57/379 , 108257/4548 )
**u= 160/33 , ht= 2017 ; ( 2017/1379 , 443783/5516 ), ( 1379/2017 , 443783/8068 )
**u= 160/57 , ht= 145573 ; ( -42071/145573 , -36185227/582292 ), ( 2419709/103343 , 578366619/413372 )
**u= 160/73 , ht= 24829 ; ( 1406839/636723 , 46356975/282988 ), ( -22767/24829 , 1300299/14188 )
**u= 160/81 , ht= 8831 ; ( -59453/12799 , 17307601/51196 ), ( 1843/8831 , 7702187/105972 )
**u= 160/97 , ht= 16877 ; ( -810207/319261 , 271221767/1277044 ), ( 9311/16877 , -6003265/67508 )
**u= 160/169 , ht= 53101 ; ( -33767/53101 , -336322643/2761252 ), ( 212141/40777 , -1154570309/2120404 )
**u= 162 , ht= 203 ; ( -345/137 , 5591/1233 ), ( 123/203 , 10517/1827 )
**u= 162/17 , ht= 2603 ; ( 2603/1141 , -827371/10269 ), ( 4429/8917 , -2898911/80253 )
**u= 162/41 , ht= 137587 ; ( 137587/37547 , -64554667/337923 ), ( 145927/162655 , 98940271/1463895 )
**u= 162/49 , ht= 8207 ; ( -1785/8207 , -29109299/517041 ), ( -7781/13005 , 52505233/819315 )
**u= 162/65 , ht= 130601 ; ( 190793/167783 , -144848279/1510047 ), ( -130601/36319 , -77293501/326871 )
**u= 162/89 , ht= 7496417 ; ( -32488295/13914881 , -23576416921/125233929 ), ( 7496417/329425 , -5006223253/2964825 )
**u= 162/97 , ht= 339177 ; ( -377149/198837 , 296924177/1789533 ), ( 339177/80575 , 242810461/725175 )
**u= 164/21 , ht= 1642 ; ( -200066/380647 , -30803837/761294 ), ( -1642/515 , 123319/1030 )
**u= 164/29 , ht= 16262 ; ( 16262/5851 , -1452677/11702 ), ( 132506/12533 , 11203123/25066 )
**u= 164/37 , ht= 16189 ; ( 4837655/1806102 , 163462559/1204068 ), ( 54/16189 , -1537233/32378 )
**u= 164/53 , ht= 23462 ; ( -23462/15777 , -1071479/10518 ), ( 15777/23462 , -3214437/46924 )
**u= 164/69 , ht= 30047 ; ( 1632955/741958 , -232671113/1483916 ), ( -17494/30047 , 4510373/60094 )
**u= 164/77 , ht= 1121929 ; ( -130630/1121929 , 154814801/2243858 ), ( 2588407/476990 , -360749183/953980 )
**u= 164/101 , ht= 13982 ; ( -13982/4501 , 2305409/9002 ), ( 548134/33337 , -86200019/66674 )
**u= 164/125 , ht= 29454 ; ( -27526/173869 , 153698671/1738690 ), ( -29454/17609 , -29961621/176090 )
**u= 164/189 , ht= 28181 ; ( 1090/69137 , 14846817/138274 ), ( 2498/28181 , 18223417/169086 )
**u= 169/6 , ht= 61 ; ( 37/61 , -17636/793 ), ( 1571/347 , 401136/4511 )
**u= 169/10 , ht= 1517 ; ( 5386/5301 , 1593073/45942 ), ( -1517/62 , 967093/1612 )
**u= 169/28 , ht= 14013 ; ( 937/14013 , -826513/20241 ), ( 252963/156797 , 157555512/2038361 )
**u= 169/30 , ht= 14684 ; ( 30209/25871 , 21783006/336323 ), ( -14684/4771 , -16924969/124046 )
**u= 169/36 , ht= 5757 ; ( -6095/4179 , 13300141/162981 ), ( -5757/3841 , 541763/6513 )
**u= 169/54 , ht= 1103 ; ( -1055/1103 , -3365026/43017 ), ( -16358/10249 , -85124171/799422 )
**u= 169/72 , ht= 2943 ; ( 2943/2857 , 10441627/111423 ), ( 78491/14011 , -67666722/182143 )
**u= 169/94 , ht= 78412 ; ( 117969/105608 , 23618009/211216 ), ( -78412/75325 , 210847271/1958450 )
**u= 169/96 , ht= 39547 ; ( 1939/39547 , -77598931/1028222 ), ( -4468574/1183345 , -36237910809/123067880 )
**u= 169/98 , ht= 10041 ; ( 30047/5217 , 70442018/158249 ), ( -10041/2324 , -142856619/422968 )
**u= 169/100 , ht= 609 ; ( 20149/24549 , -814386/8183 ), ( -609/383 , 3597408/24895 )
**u= 169/106 , ht= 9779 ; ( -9779/6757 , -12238108/87841 ), ( -14357/5730 , -31833467/148980 )
**u= 169/108 , ht= 8021 ; ( 24505/2377 , -76756838/92703 ), ( -8021/2671 , -8786559/34723 )
**u= 169/116 , ht= 52871 ; ( 144109/32733 , 53054666/141843 ), ( 29787/52871 , 9337239/98189 )
**u= 169/142 , ht= 206461 ; ( -7638625/112303 , -700276066/112303 ), ( -54251/206461 , 254383388/2683993 )
**u= 169/148 , ht= 32711 ; ( 271117/175213 , -392725408/2277769 ), ( -32711/1255 , -39827212/16315 )
**u= 169/154 , ht= 20299 ; ( -21990/49367 , 134140949/1283542 ), ( 15987/20299 , -32066244/263887 )
**u= 169/170 , ht= 89078 ; ( -63893/129717 , 4834514/43239 ), ( 16947/89078 , 236459805/2316028 )
**u= 169/180 , ht= 35329 ; ( 35329/499 , -47404881/6487 ), ( -396337/996223 , 1438512618/12950899 )
**u= 169/190 , ht= 1727 ; ( 1727/207 , -266402/299 ), ( -92804/440469 , 1240988407/11452194 )
**u= 169/192 , ht= 24821 ; ( 70367/67163 , -539178039/3492476 ), ( -24821/8395 , 36309281/109135 )
**u= 170/13 , ht= 2383 ; ( -5981/19317 , 187065/6439 ), ( -1161/2383 , -73113/2383 )
**u= 170/77 , ht= 400513 ; ( -400513/168561 , 9747929/56187 ), ( 912977/293809 , 64557139/293809 )
**u= 170/93 , ht= 70619 ; ( 30097/166249 , -12497823/166249 ), ( -31123/70619 , -5707949/70619 )
**u= 170/101 , ht= 37021 ; ( -665381/362597 , -58408501/362597 ), ( 37021/22477 , -3338581/22477 )
**u= 170/149 , ht= 151841 ; ( 3173149/2057437 , -354064037/2057437 ), ( 57057/151841 , 15186231/151841 )
**u= 178/21 , ht= 737 ; ( 485/737 , -2757/67 ), ( 1549361/243301 , -53964097/243301 )
**u= 178/37 , ht= 12993 ; ( 17381/14513 , -1032311/14513 ), ( 12993/5285 , 91401/755 )
**u= 178/53 , ht= 296549 ; ( 296549/7353 , 5394311/2451 ), ( -128319/410267 , 23452119/410267 )
**u= 178/85 , ht= 181277 ; ( -99765819/10548439 , -6932190077/10548439 ), ( 181277/48337 , -12966427/48337 )
**u= 178/109 , ht= 9687 ; ( 9687/7507 , -959053/7507 ), ( 440417/516717 , -53131697/516717 )
**u= 178/141 , ht= 10463 ; ( -10463/347 , 931749/347 ), ( -630851/494329 , -71335533/494329 )
**u= 178/165 , ht= 7245683 ; ( 31631513/1500203 , -3048940627/1500203 ), ( 1181273/7245683 , 706833727/7245683 )
**u= 180/17 , ht= 2417 ; ( 2417/1414 , -515653/8484 ), ( -734209/143722 , -138315389/862332 )
**u= 180/113 , ht= 175813 ; ( -175813/2614 , -83588663/15684 ), ( 206746109/2317922 , 98303302663/13907532 )
**u= 180/121 , ht= 6679 ; ( -6679/402 , -36207197/26532 ), ( -35041/13638 , 203490445/900108 )
**u= 180/193 , ht= 782079 ; ( 1330233/705106 , 935419765/4230636 ), ( 782079/14518 , 486015637/87108 )
**u= 185/2 , ht= 261 ; ( -32/289 , -7199/578 ), ( 261/197 , -3594/197 )
**u= 185/6 , ht= 19511 ; ( 2033/19511 , 361416/19511 ), ( -707231/71377 , 13096578/71377 )
**u= 185/14 , ht= 1487 ; ( 1487/809 , -46460/809 ), ( -351727/33959 , 9762772/33959 )
**u= 185/24 , ht= 168694 ; ( -825887/42721 , -29744028/42721 ), ( -168694/116357 , 29541847/465428 )
**u= 185/36 , ht= 13961 ; ( -4183/13961 , 643296/13961 ), ( 13961/4183 , 643296/4183 )
**u= 185/44 , ht= 494899 ; ( 546073837/30669629 , 26662854838/30669629 ), ( -241133/494899 , -26842036/494899 )
**u= 185/48 , ht= 221027 ; ( -550601/11827 , -56087031/23654 ), ( -97651/221027 , 647677/11633 )
**u= 185/56 , ht= 1451 ; ( 803/1451 , -91259/1451 ), ( -828893/219 , -45619109/219 )
**u= 185/66 , ht= 4524352 ; ( -4524352/719491 , 547187693/1438982 ), ( 8406971/620357 , -503629198/620357 )
**u= 185/78 , ht= 64811 ; ( -64811/9172 , -8499809/18344 ), ( 6396692/1378679 , 849931899/2757358 )
**u= 185/84 , ht= 1311911 ; ( 1809151/204653 , 122676123/204653 ), ( 1074233/1311911 , 114259048/1311911 )
**u= 185/88 , ht= 555141 ; ( -2412161/600057 , 57142363/200019 ), ( -555141/185893 , 40382802/185893 )
**u= 185/104 , ht= 167526 ; ( -167526/22547 , -50694377/90188 ), ( -346898339/1661067 , 26013217153/1661067 )
**u= 185/106 , ht= 201657 ; ( 31789/509637 , 12885596/169879 ), ( -201657/114806 , 35132757/229612 )
**u= 185/114 , ht= 10483153 ; ( -10792429/1753157 , 858294036/1753157 ), ( -10483153/5837999 , -942003458/5837999 )
**u= 185/116 , ht= 8011 ; ( 8011/4517 , 728254/4517 ), ( -290179/9507 , 22992772/9507 )
**u= 185/134 , ht= 1615729 ; ( 1615729/1534793 , -189669704/1534793 ), ( -8178494/2284427 , -1445515523/4568854 )
**u= 185/158 , ht= 1260739 ; ( -9721547/3361564 , 1901267149/6723128 ), ( -791128/1260739 , -275111491/2521478 )
**u= 185/168 , ht= 147239 ; ( 5932834/1974313 , 2383463465/7897252 ), ( -147239/91153 , -16503327/91153 )
**u= 185/174 , ht= 237197 ; ( 237197/195901 , -29836278/195901 ), ( -195901/237197 , 29836278/237197 )
**u= 185/176 , ht= 61909 ; ( 617091/1173647 , 258683731/2347294 ), ( -61909/46927 , -15155117/93854 )
**u= 185/182 , ht= 367624183 ; ( 367624183/20998254 , -24348965511/13998836 ), ( 1214703102/37019699 ,
-241092927207/74039398 )
**u= 185/188 , ht= 260713 ; ( 5108237/1084849 , 526448099/1084849 ), ( 108809/260713 , -28479860/260713 )
**u= 185/198 , ht= 259899 ; ( -544041/180518 , 355815043/1083108 ), ( 190357/259899 , -99988610/779697 )
**u= 194/5 , ht= 5229 ; ( -7177/6283 , -152377/6283 ), ( -2041/5229 , -87589/5229 )
**u= 194/21 , ht= 6367 ; ( 3595/6367 , 240843/6367 ), ( -799399/168197 , -26930647/168197 )
**u= 194/77 , ht= 2731 ; ( -2731/1009 , -183409/1009 ), ( 1223989/4001 , 77128511/4001 )
**u= 194/93 , ht= 8915 ; ( 2363609/1236821 , -184699093/1236821 ), ( 2033/8915 , 633067/8915 )
**u= 194/165 , ht= 1476901 ; ( -1476901/607049 , -147265335/607049 ), ( -17540779/1208039 , 1621638639/1208039 )
**u= 194/173 , ht= 3597921 ; ( 58048749/327991 , -5481886583/327991 ), ( 3597921/2962931 , -440165877/2962931 )
**u= 196 , ht= 39 ; ( -8/39 , 2207/182 ), ( -303/500 , -19491/7000 )
**u= 196/9 , ht= 863 ; ( -4528/1459 , 1413443/20426 ), ( 863/620 , 320007/8680 )
**u= 196/17 , ht= 5439 ; ( -1376/5439 , 773359/25382 ), ( 108408/13327 , -45181461/186578 )
**u= 196/33 , ht= 18208 ; ( 14413/18208 , -13343447/254912 ), ( -25244/8339 , 15283889/116746 )
**u= 196/57 , ht= 1676279 ; ( 1329320/1676279 , 1615395921/23467906 ), ( 5952896/4444049 , -5609345873/62216686 )
**u= 196/73 , ht= 114232 ; ( -115840/157 , 98960941/2198 ), ( 114232/89803 , -124160093/1257242 )
**u= 196/81 , ht= 6920 ; ( 6920/4017 , -64810021/506142 ), ( 4017/6920 , -64810021/871920 )
**u= 196/97 , ht= 37921 ; ( -12616/37921 , -39365707/530894 ), ( -69584/53315 , 86342521/746410 )
**u= 196/145 , ht= 289476 ; ( -416668/127567 , -524729221/1785938 ), ( -289476/154021 , -394874463/2156294 )
**u= 196/177 , ht= 12820 ; ( -1672/36383 , 48459421/509362 ), ( -12137/12820 , 23488083/179480 )
**u= 197/8 , ht= 2203 ; ( -21815/6167 , 450922/6167 ), ( -2203/1686 , -224509/6744 )
**u= 197/12 , ht= 25573 ; ( -16081/25573 , -747787/25573 ), ( -5021/81523 , -2002434/81523 )
**u= 197/14 , ht= 125269 ; ( 25091/125269 , 3421468/125269 ), ( 790689/476759 , -24671628/476759 )
**u= 197/20 , ht= 13723 ; ( 7927/13723 , 505595/13723 ), ( 4061257/1987287 , 144285452/1987287 )
**u= 197/22 , ht= 476208 ; ( -2783501/160308 , 20664809/35624 ), ( -476208/258229 , -36247329/516458 )
**u= 197/26 , ht= 186245 ; ( -186245/70661 , -7229282/70661 ), ( 1518911/97335 , -55375132/97335 )
**u= 197/30 , ht= 94097 ; ( 94097/27167 , -3818686/27167 ), ( 31394/108341 , 8795977/216682 )
**u= 197/32 , ht= 33681 ; ( -543879/107897 , 22328908/107897 ), ( -33681/1295 , -194259/185 )
**u= 197/36 , ht= 62239 ; ( -69925/106043 , -5431844/106043 ), ( 62239/51205 , 3445987/51205 )
**u= 197/42 , ht= 1987060 ; ( -1987060/406093 , -187205943/812186 ), ( -32433649/361568 , -2997123867/723136 )
**u= 197/44 , ht= 813253 ; ( -235685/813253 , -40034692/813253 ), ( 8393041/5531215 , -475162324/5531215 )
**u= 197/50 , ht= 50797 ; ( 131213/230139 , 22249352/383565 ), ( 997/50797 , 2558744/50797 )
**u= 197/58 , ht= 11829 ; ( 5487520/16234959 , -620087629/10823306 ), ( -11829/4405 , -685086/4405 )
**u= 197/66 , ht= 549701 ; ( -249214/549701 , 69882953/1099402 ), ( -3634591/320951 , 211250624/320951 )
**u= 197/78 , ht= 24926 ; ( 18595/24926 , 3913963/49852 ), ( 3638459/80437 , 229049138/80437 )
**u= 197/80 , ht= 180883 ; ( 159303/306133 , 43989383/612266 ), ( -180883/100397 , -13185034/100397 )
**u= 197/86 , ht= 100567 ; ( 100567/93793 , 9086348/93793 ), ( 1766748/86423 , -233785779/172846 )
**u= 197/92 , ht= 649009 ; ( 1340235/420373 , -95984024/420373 ), ( 649009/52113 , -44501836/52113 )
**u= 197/94 , ht= 938885 ; ( 12179135/2214618 , 570028857/1476412 ), ( -606522/938885 , -154421991/1877770 )
**u= 197/96 , ht= 80417 ; ( 8257/80417 , -5644093/80417 ), ( -80417/8257 , 5644093/8257 )
**u= 197/110 , ht= 10012162 ; ( 165551847/77787247 , -13668298540/77787247 ), ( -10012162/5866503 , -1734403127/11733006
)
**u= 197/112 , ht= 5664027 ; ( 134903227/3872949 , 3391907987/1290983 ), ( -5664027/2253151 , -919342941/4506302 )
**u= 197/122 , ht= 137409 ; ( 215340/966413 , -155850883/1932826 ), ( 137409/16372 , 21782223/32744 )
**u= 197/134 , ht= 418345 ; ( -418345/373729 , 46267604/373729 ), ( 276817/912647 , -78656624/912647 )
**u= 197/146 , ht= 2602738 ; ( -2602738/870967 , 472561441/1741934 ), ( 10388221/5715126 , -2041568597/11430252 )
**u= 197/150 , ht= 60467 ; ( 5267075/1546076 , 4790000443/15460760 ), ( 54992/60467 , -14264711/120934 )
**u= 197/152 , ht= 13389655 ; ( 30895201/3439612 , -10922412077/13758448 ), ( 4292056/13389655 , 4940420797/53558620 )
**u= 197/156 , ht= 16801 ; ( -8081/16801 , 1659154/16801 ), ( -6227131/5488859 , 738716164/5488859 )
**u= 197/166 , ht= 92953 ; ( -92953/35697 , -3046816/11899 ), ( -4070172/4243025 , 1079490783/8486050 )
**u= 197/168 , ht= 529636 ; ( 30606860/4614173 , 11433723763/18456692 ), ( -355367/529636 , 235606921/2118544 )
**u= 197/174 , ht= 795043 ; ( -46227637/810218 , -8690558773/1620436 ), ( 795043/114355 , 75494238/114355 )
**u= 197/180 , ht= 7789 ; ( -7789/499 , -746076/499 ), ( 499/7789 , -746076/7789 )
**u= 197/182 , ht= 706973 ; ( 799111/1312687 , -147721888/1312687 ), ( 706973/445075 , -80302154/445075 )
**u= 197/198 , ht= 3671779 ; ( 3671779/3248213 , 491498856/3248213 ), ( 9739556/9637597 , -2747465749/19275194 )
**u= 200/33 , ht= 13609 ; ( 13609/625 , 5529043/6250 ), ( 141227/5861 , -11494289/11722 )
**u= 200/129 , ht= 150931 ; ( -371611/235549 , -353360913/2355490 ), ( -150931/1117 , 24246441/2234 )
**u= 200/153 , ht= 26849 ; ( 44137/186009 , 100334117/1116054 ), ( 26849/16207 , -288757/1706 )
685
>
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。

[2026.06.13追記]u=274のときの整数解を追加した。
[2026.06.14追記]u=289/4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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