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Integer Points on A^4+20006*B^4=C^4+20006*D^4

[2026.06.20]A^4+20006*B^4=C^4+20006*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=20006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
    89^4+20006*68^4=551^4+20006*64^4
    709^4+20006*92^4=877^4+20006*86^4
    5739^4+20006*684^4=8597^4+20006*92^4
    6449^4+20006*2278^4=27113^4+20006*244^4
    6943^4+20006*512^4=7657^4+20006*338^4
    7277^4+20006*956^4=11819^4+20006*98^4
    10347^4+20006*5758^4=66093^4+20006*3476^4
    12991^4+20006*1549^4=19153^4+20006*821^4
    63022^4+20006*7063^4=87662^4+20006*4247^4
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 = 560. 7145/6. 14885/64 2044/29, 21196/221, 264365/1746, 1820/9, 574/5, 3031/26
となる。
{gp2cによる計算]
gp > sss(20006,100000)
1:89^4+20006*68^4=551^4+20006*64^4
2:709^4+20006*92^4=877^4+20006*86^4
3:5739^4+20006*684^4=8597^4+20006*92^4
4:6449^4+20006*2278^4=27113^4+20006*244^4
5:6943^4+20006*512^4=7657^4+20006*338^4
6:7277^4+20006*956^4=11819^4+20006*98^4
7:10347^4+20006*5758^4=66093^4+20006*3476^4
8:12991^4+20006*1549^4=19153^4+20006*821^4
9:63022^4+20006*7063^4=87662^4+20006*4247^4
9 solutions,
time = -361519717 ms.
%1 = 9
{Pari/GPによる計算]
gp > uu(20006,89,68,551,64)
[u - 560, 1; u, -1; 264*u + 155761, 1]
[u - 560, 1; u, -1; 264*u - 155761, 1]
[u, -1; u + 560, 1; 264*u + 155761, 1]
time = 1 ms.
%1 = [560, -155761/264]
gp > uu(20006,709,92,877,86)
[u, -1; 6*u + 7145, 1; 89*u - 22204, 1]
[u, -1; 6*u - 7145, 1; 89*u - 22204, 1]
[u, -1; 6*u + 7145, 1; 89*u + 22204, 1]
time = 1 ms.
%2 = [-7145/6, 22204/89]
gp > uu(20006,5739,684,8597,92)
[u, -1; 64*u + 14885, 1; 3589*u - 320096, 1]
[u, -1; 64*u - 14885, 1; 3589*u - 320096, 1]
[u, -1; 64*u + 14885, 1; 3589*u + 320096, 1]
time = 2 ms.
%3 = [-14885/64, 320096/3589]
gp > uu(20006,6449,2278,27113,244)
[u, -1; 1469*u - 198604, 1; 1746*u + 264365, 1]
[u, -1; 1469*u - 198604, 1; 1746*u - 264365, 1]
[u, -1; 1469*u + 198604, 1; 1746*u + 264365, 1]
time = 1 ms.
%4 = [198604/1469, -264365/1746]
gp > uu(20006,6943,512,7657,338)
[u, -1; 29*u - 2044, 1; 2550*u + 1841981, 1]
[u, -1; 29*u - 2044, 1; 2550*u - 1841981, 1]
[u, -1; 29*u + 2044, 1; 2550*u + 1841981, 1]
time = 1 ms.
%5 = [2044/29, -1841981/2550]
gp > uu(20006,7277,956,11819,98)
[u, -1; 221*u - 21196, 1; 2046*u + 435845, 1]
[u, -1; 221*u - 21196, 1; 2046*u - 435845, 1]
[u, -1; 221*u + 21196, 1; 2046*u + 435845, 1]
time = 1 ms.
%6 = [21196/221, -435845/2046]
gp > uu(20006,10347,5758,66093,3476)
[u, -1; 9*u - 1820, 1; 1170666*u + 248630281, 1]
[u, -1; 9*u - 1820, 1; 1170666*u - 248630281, 1]
[u, -1; 9*u + 1820, 1; 1170666*u + 248630281, 1]
time = 1 ms.
%7 = [1820/9, -248630281/1170666]
gp > uu(20006,12991,1549,19153,821)
[u, -1; 5*u - 574, 1; 172536*u + 53560349, 1]
[u, -1; 5*u - 574, 1; 172536*u - 53560349, 1]
[u, -1; 5*u + 574, 1; 172536*u + 53560349, 1]
time = 1 ms.
%8 = [574/5, -53560349/172536]
gp > uu(20006,63022,7063,87662,4247)
[u, -1; 26*u - 3031, 1; 306240*u + 112080757, 1]
[u, -1; 26*u - 3031, 1; 306240*u - 112080757, 1]
[u, -1; 26*u + 3031, 1; 306240*u + 112080757, 1]
time = 1 ms.
%9 = [3031/26, -112080757/306240]

■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687個のuが抽出される。
[MAGMAによる計算]
> PP(20006,1,200);
**u= 1/6 , ht= 253 ; ( -2131/374 , -1060087/748 ), ( -253/67 , 64118/67 )
**u= 1/10 , ht= 623 ; ( -393/623 , 232968/623 ), ( 5443/3243 , 667960/1081 )
**u= 1/24 , ht= 1822 ; ( -9367/23 , -4589576/23 ), ( -1822/727 , 3844681/2908 )
**u= 1/30 , ht= 5141 ; ( 5141/3896 , -7067207/7792 ), ( 1384/23179 , 25440359/46358 )
**u= 1/36 , ht= 1337 ; ( -1337/535 , -172834/107 ), ( -15765/18727 , 44069335/56181 )
**u= 1/46 , ht= 42735 ; ( 2464077/513572 , -3414775419/1027144 ), ( 30223/42735 , -1075926/1295 )
**u= 1/48 , ht= 547 ; ( -128045/168289 , 293056035/336578 ), ( 547/365 , -91133/73 )
**u= 1/50 , ht= 16061 ; ( 35380/323 , 250222327/3230 ), ( 16061/736 , 22740931/1472 )
**u= 1/54 , ht= 248167 ; ( 28016687/4166962 , 41635132257/8333924 ), ( 45863/248167 , -185480634/248167 )
**u= 1/84 , ht= 5249 ; ( 5249/83 , -4812111/83 ), ( -9829/25913 , -25404557/25913 )
**u= 1/100 , ht= 19971 ; ( 1131/19973 , -100039986/99865 ), ( 1133/19971 , -33343522/33285 )
**u= 1/110 , ht= 20771 ; ( 59293/2813 , 2147114/97 ), ( -13456/20771 , -51921107/41542 )
**u= 1/116 , ht= 17943 ; ( 17943/11563 , 22993857/11563 ), ( -11563/17943 , -7664619/5981 )
**u= 1/126 , ht= 11216 ; ( -45579/108428 , 792276443/650568 ), ( 11216/10393 , -34333243/20786 )
**u= 1/144 , ht= 5677 ; ( 4897/28383 , -103704008/85149 ), ( 979/5677 , -6913992/5677 )
**u= 1/154 , ht= 35339 ; ( 86274/102829 , -333193719/205658 ), ( -35339/906 , 29250357/604 )
**u= 1/170 , ht= 171323 ; ( -128523/171323 , -279288270/171323 ), ( 57641/228486 , -204859293/152324 )
**u= 1/186 , ht= 335425 ; ( -41758261/3576325 , -11433545602/715265 ), ( -335425/112153 , 482425020/112153 )
**u= 2/123 , ht= 141067 ; ( -141067/52043 , 117933541/52043 ), ( -270449/388585 , 371332797/388585 )
**u= 2/171 , ht= 14645 ; ( -14645/61 , 13543825/61 ), ( -177721/98615 , -37592747/19723 )
**u= 4/15 , ht= 20038 ; ( 643/20038 , 7765857/40076 ), ( 96791/47266 , -41724235/94532 )
**u= 4/159 , ht= 463 ; ( 149399/3278 , 188458571/6556 ), ( -226/463 , -649757/926 )
**u= 4/167 , ht= 133585 ; ( -133585/64614 , -191792335/129228 ), ( 137613/117730 , 46813919/47092 )
**u= 4/191 , ht= 101195 ; ( 16845145/859018 , 23314200283/1718036 ), ( 8942/101195 , -140420273/202390 )
**u= 5/6 , ht= 568 ; ( -13463/1304 , 2963793/2608 ), ( -568/19 , -124533/38 )
**u= 5/16 , ht= 1517 ; ( 1517/151 , -545501/302 ), ( 151/1517 , -545501/3034 )
**u= 5/18 , ht= 563 ; ( -7426/7623 , -12117067/45738 ), ( 419/563 , 133178/563 )
**u= 5/24 , ht= 133 ; ( -133/101 , -36594/101 ), ( -101/133 , 1926/7 )
**u= 5/36 , ht= 673 ; ( 61027/16409 , -16959411/16409 ), ( -673/591 , -721102/1773 )
**u= 5/58 , ht= 4997 ; ( -1789/4997 , -1807972/4997 ), ( -1503527/93924 , 342105397/62616 )
**u= 5/68 , ht= 80967 ; ( -80967/49321 , 34968042/49321 ), ( -518473/54951 , -64101054/18317 )
**u= 5/72 , ht= 13209 ; ( -73920811/8230973 , -28228564745/8230973 ), ( -13209/2407 , 15287308/7221 )
**u= 5/76 , ht= 10107 ; ( 177057/17419 , 69373179/17419 ), ( 9271/10107 , -594213/1123 )
**u= 5/82 , ht= 128347 ; ( 32981/128347 , 53673262/128347 ), ( 1034259/13318 , -837881123/26636 )
**u= 5/84 , ht= 15371 ; ( -27359/73987 , 32337425/73987 ), ( 15371/2197 , 6365220/2197 )
**u= 5/86 , ht= 90076 ; ( -100629/34237 , -44089704/34237 ), ( 90076/86697 , 295483/494 )
**u= 5/88 , ht= 18484 ; ( -69101/58213 , -37910981/58213 ), ( -18484/2567 , 31320281/10268 )
**u= 5/94 , ht= 20939 ; ( 20939/1763 , -9112424/1763 ), ( -763582/193479 , -227730279/128986 )
**u= 5/136 , ht= 323243 ; ( -1696829/445987 , 915152074/445987 ), ( 323243/155606 , -748510021/622424 )
**u= 5/172 , ht= 99143 ; ( 150679/504483 , -308849329/504483 ), ( -99143/58409 , -67500014/58409 )
**u= 5/174 , ht= 146153 ; ( 146153/18959 , -86953304/18959 ), ( 464162/805289 , 1096795647/1610578 )
**u= 5/192 , ht= 737 ; ( -131/737 , -1855721/2948 ), ( 319342301/56296877 , -401941639363/112593754 )
**u= 7/32 , ht= 121 ; ( -283531/13836 , -485621167/110688 ), ( -7/121 , -25918/121 )
**u= 7/60 , ht= 55933 ; ( -55933/5267 , 16450410/5267 ), ( 424211/80381 , 126425174/80381 )
**u= 7/72 , ht= 419 ; ( -903/3601 , 3572477/10803 ), ( 104/419 , -553909/1676 )
**u= 7/78 , ht= 781 ; ( -116/781 , -527209/1562 ), ( 33211/35 , -11087808/35 )
**u= 7/86 , ht= 35390 ; ( -238085/163437 , 101236760/163437 ), ( 3267/35390 , -4983661/14156 )
**u= 7/88 , ht= 6433 ; ( 11205/26693 , 10265904/26693 ), ( 6433/2682 , -1098471/1192 )
**u= 7/114 , ht= 25255 ; ( 25255/13817 , 11619130/13817 ), ( 3564428/485725 , -580782819/194290 )
**u= 7/158 , ht= 39081 ; ( 12932/39081 , 39120401/78162 ), ( 39081/12932 , 39120401/25864 )
**u= 7/198 , ht= 29486 ; ( 29486/28595 , 8739345/11438 ), ( 90525/112162 , -460014385/672972 )
**u= 8/123 , ht= 8713 ; ( -3080789/473419 , -2444740053/946838 ), ( -8713/1865 , 6988729/3730 )
**u= 8/187 , ht= 4315 ; ( 7901/4645 , -1772741/1858 ), ( -4315/173 , -4176385/346 )
**u= 9/40 , ht= 12409 ; ( -116069/4581 , 73476560/13743 ), ( 249/12409 , 7850888/37227 )
**u= 9/44 , ht= 8147 ; ( 11431/8791 , 9566878/26373 ), ( -8147/4019 , 6026804/12057 )
**u= 9/50 , ht= 1192 ; ( 1820/1831 , -18257821/54930 ), ( 365/1192 , 8816329/35760 )
**u= 9/64 , ht= 2557 ; ( 3147/737 , -10344419/8844 ), ( -2557/189 , -2051489/567 )
**u= 9/80 , ht= 38893 ; ( -104897/82093 , 238313819/492558 ), ( 8503/38893 , -71227883/233358 )
**u= 9/134 , ht= 72046 ; ( 90035/507646 , 1193806271/3045876 ), ( -72046/7483 , 167720999/44898 )
**u= 9/140 , ht= 73723 ; ( 73723/5543 , 87489553/16629 ), ( 415061/12911 , 491418085/38733 )
**u= 9/166 , ht= 154167 ; ( -2146811/111702 , -5540258641/670212 ), ( -72506/154167 , 439068373/925002 )
**u= 9/194 , ht= 136723 ; ( 298297/77719 , -429414358/233157 ), ( -136723/103352 , 477511207/620112 )
**u= 10/159 , ht= 69887 ; ( 61831/69887 , -37213849/69887 ), ( 254047/32719 , 102152763/32719 )
**u= 13/2 , ht= 3 ; ( 2/3 , 283/6 ), ( 3/2 , 283/4 )
**u= 13/22 , ht= 45752 ; ( -178791/66212 , 49611873/132424 ), ( -45752/19137 , 4301669/12758 )
**u= 13/58 , ht= 228889 ; ( -841884/41021 , 356126337/82042 ), ( -228889/22032 , -32385031/14688 )
**u= 13/68 , ht= 15635 ; ( -1372599/444565 , 1118745/1507 ), ( -15635/5571 , 421850/619 )
**u= 13/120 , ht= 353746 ; ( -8917009/387766 , 10848592525/1551064 ), ( -353746/3989 , 429994943/15956 )
**u= 13/126 , ht= 86755 ; ( -86755/27941 , 28379460/27941 ), ( -746734/524155 , 113629681/209662 )
**u= 13/128 , ht= 8169 ; ( 225467/71917 , 297085283/287668 ), ( 8169/5786 , -4569641/8416 )
**u= 13/152 , ht= 10997 ; ( -10997/1581 , -3799547/1581 ), ( 1156319/10379071 , -3571513358/10379071 )
**u= 13/162 , ht= 130794 ; ( 130794/99247 , -1043420903/1786446 ), ( -150703/145370 , 443564179/872220 )
**u= 13/168 , ht= 6767 ; ( -77108/80965 , 32159555/64772 ), ( 1465/6767 , 2489375/6767 )
**u= 13/170 , ht= 764572 ; ( -764572/169323 , -566452261/338646 ), ( -984163/75787 , -357000560/75787 )
**u= 13/180 , ht= 131071 ; ( -131071/14431 , -49074144/14431 ), ( -36581/151429 , -57976876/151429 )
**u= 13/188 , ht= 66099 ; ( -14647/66099 , -25749949/66099 ), ( -130131/85129 , -59143871/85129 )
**u= 14/141 , ht= 39871 ; ( 39871/14743 , -13492611/14743 ), ( 1275101/92675 , -405788203/92675 )
**u= 14/153 , ht= 8161 ; ( -11015/5067 , 12026345/15201 ), ( 8161/8125 , -761513/1625 )
**u= 16/7 , ht= 45 ; ( 45/38 , -15585/152 ), ( 38/45 , -1039/12 )
**u= 16/135 , ht= 10711 ; ( -4814/10711 , 13646285/42844 ), ( 56458/39613 , -240439603/475356 )
**u= 16/143 , ht= 104990 ; ( -104990/48163 , 138150635/192652 ), ( -1053477/499690 , 278903177/399752 )
**u= 16/159 , ht= 35197 ; ( -35197/28562 , 57164851/114248 ), ( 334262/235205 , -515454331/940820 )
**u= 16/191 , ht= 3869 ; ( 2334321/270566 , -3248180751/1082264 ), ( 3869/1386 , 1893553/1848 )
**u= 18/35 , ht= 33507 ; ( -33507/27883 , -18238163/83649 ), ( -464907/13403 , 194595823/40209 )
**u= 18/43 , ht= 96001 ; ( -96001/83137 , 58894361/249411 ), ( -447197/629867 , -358235119/1889601 )
**u= 18/187 , ht= 218509 ; ( -218509/42389 , 215259359/127167 ), ( -2807077/5232053 , 5742162163/15696159 )
**u= 20/3 , ht= 1661 ; ( -1358/2959 , -252409/5918 ), ( -1661/178 , 129557/356 )
**u= 20/11 , ht= 78 ; ( -78 , 11571/2 ), ( -21946/7347 , -1144475/4898 )
**u= 20/51 , ht= 106619 ; ( 42002/106619 , -36603973/213238 ), ( -16456846/194603 , -5257071165/389206 )
**u= 20/67 , ht= 3478 ; ( -13451/3098 , 5053531/6196 ), ( 61/3478 , 1273541/6956 )
**u= 25/12 , ht= 31 ; ( 17/31 , 2450/31 ), ( 31/17 , 2450/17 )
**u= 25/14 , ht= 733 ; ( 733/131 , 55726/131 ), ( 131/733 , 55726/733 )
**u= 25/32 , ht= 14944 ; ( 35932/2699 , -32617951/21592 ), ( -11105/14944 , -84269447/597760 )
**u= 25/36 , ht= 6553 ; ( -7201 , 4321196/5 ), ( -6553/233 , -3934922/1165 )
**u= 25/56 , ht= 18799 ; ( -10143243/415627 , -7597979352/2078135 ), ( 18799/9657 , -12121/37 )
**u= 25/62 , ht= 4763 ; ( -4763/2491 , 846590/2491 ), ( -1008/36719 , 11571073/73438 )
**u= 25/74 , ht= 2573 ; ( 1386/2573 , 1005783/5146 ), ( 70443/25498 , 128909483/254980 )
**u= 25/92 , ht= 2303 ; ( -2303/779 , 466451/779 ), ( 203136025/28531893 , -5045734263/3657935 )
**u= 25/102 , ht= 5231 ; ( 75131/12758 , -30790499/25516 ), ( 5231/3055 , -6118954/15275 )
**u= 25/106 , ht= 357457 ; ( -1665711/1686169 , 2440616976/8430845 ), ( 357457/267201 , 10212210/29689 )
**u= 25/108 , ht= 27697 ; ( -21821/27697 , -7329791/27697 ), ( 45317/4981 , 28431362/14943 )
**u= 25/134 , ht= 166917 ; ( 72581/166917 , 42145754/166917 ), ( -569087/108953 , 670831516/544765 )
**u= 25/144 , ht= 25391 ; ( -142367/22019 , 69158981/44038 ), ( -13551/25391 , 103626254/380865 )
**u= 25/146 , ht= 7847 ; ( 1132/7847 , 19162321/78470 ), ( -65297/40407 , -30932484/67345 )
**u= 26/15 , ht= 8207 ; ( 8207/2737 , 657177/2737 ), ( -1093897/610433 , -95166867/610433 )
**u= 26/79 , ht= 151343 ; ( 730805/93117 , -128436799/93117 ), ( 151343/94937 , 31146451/94937 )
**u= 26/111 , ht= 27961 ; ( 3689/27961 , -5828281/27961 ), ( 148615/136063 , -41639035/136063 )
**u= 26/135 , ht= 541861 ; ( 17881919/648559 , -4077974659/648559 ), ( -541861/151789 , 34975685/41397 )
**u= 28/57 , ht= 14887 ; ( 2518/14887 , 4309117/29774 ), ( 42682/29971 , 14884697/59942 )
**u= 28/81 , ht= 1325 ; ( 2890/9919 , 811145/4578 ), ( 182/1325 , -273013/1590 )
**u= 28/117 , ht= 1640 ; ( -38539/14320 , -3362193/5728 ), ( 1640/587 , 712245/1174 )
**u= 28/125 , ht= 144 ; ( 144/71 , -339279/710 ), ( -67181/154944 , 23792021/103296 )
**u= 28/137 , ht= 105122 ; ( 217814/14369 , -96583891/28738 ), ( 105122/73475 , 56747807/146950 )
**u= 28/181 , ht= 562335 ; ( 562335/61508 , -287695305/123016 ), ( -1469837/281700 , -50741891/37560 )
**u= 29/4 , ht= 2347 ; ( -18655/589 , -692369/589 ), ( 527/2347 , -89243/2347 )
**u= 29/6 , ht= 9553 ; ( 9553/3220 , -183361/1288 ), ( -37525/3149 , -1714100/3149 )
**u= 29/20 , ht= 71519 ; ( 200859/11221 , 16708074/11221 ), ( -71519/31119 , 2290/11 )
**u= 29/34 , ht= 25061 ; ( -25061/10061 , 2924464/10061 ), ( -114431/19350 , 8378909/12900 )
**u= 29/42 , ht= 74360 ; ( -74360/3151 , 17916145/6302 ), ( -105583/133705 , 315472/2057 )
**u= 29/100 , ht= 71449 ; ( 71449/3593 , 66432382/17965 ), ( 569971/64813 , 532696244/324065 )
**u= 29/108 , ht= 15377 ; ( -8755/15377 , -10245725/46131 ), ( 1029151/156095 , 40181552/31219 )
**u= 29/114 , ht= 1212434 ; ( 1212434/60673 , 11196457/2822 ), ( -21727513/935858 , 8625052789/1871716 )
**u= 29/120 , ht= 127567 ; ( 127567/9502 , -104101243/38008 ), ( -364862/346213 , 409323335/1384852 )
**u= 29/136 , ht= 77947 ; ( 640301/120781 , 141127462/120781 ), ( 77947/76910 , -94868311/307640 )
**u= 29/142 , ht= 39785805 ; ( 39785805/3571474 , 17681162415/7142948 ), ( -176833274/11943585 , 5230001605/1592478 )
**u= 29/146 , ht= 935629 ; ( 1684062/1105487 , -904143387/2210974 ), ( 935629/239574 , -144492507/159716 )
**u= 29/156 , ht= 43531 ; ( -43531/40489 , -13790517/40489 ), ( -906275/920537 , -299654305/920537 )
**u= 29/160 , ht= 3499 ; ( 141/3499 , -1645329/6998 ), ( -3499/141 , 11669/2 )
**u= 29/192 , ht= 264265 ; ( -264265/164023 , -80042130/164023 ), ( -3752851/129895 , 773086867/103916 )
**u= 29/196 , ht= 36377 ; ( -15299/36377 , -71826527/254639 ), ( -36377/15299 , 71826527/107093 )
**u= 32/27 , ht= 9817 ; ( -224213/132565 , 19143151/106052 ), ( -9817/1657 , -10976059/19884 )
**u= 32/163 , ht= 17981 ; ( 1355/17981 , 16281253/71924 ), ( 27159/10207 , -26196737/40828 )
**u= 34/71 , ht= 363917 ; ( 363917/148503 , 56806949/148503 ), ( 683983/65005 , -99301351/65005 )
**u= 34/95 , ht= 535979 ; ( 535979/27409 , -89722423/27409 ), ( 1267557/13367 , 211924181/13367 )
**u= 34/143 , ht= 116819 ; ( -31175/116819 , 24799705/116819 ), ( 946513/1317125 , 66536443/263425 )
**u= 34/151 , ht= 338763 ; ( -338763/255713 , 89460339/255713 ), ( 3340195/316119 , -235722785/105373 )
**u= 34/159 , ht= 6065261 ; ( -6065261/490097 , -1316092603/490097 ), ( 82911463/22140179 , 18560781113/22140179 )
**u= 35/12 , ht= 4643 ; ( 46607/4019 , 2738991/4019 ), ( -1321/4643 , 282647/4643 )
**u= 35/24 , ht= 1916 ; ( 1916/1657 , 839169/6628 ), ( -133291/27083 , -11265322/27083 )
**u= 35/34 , ht= 882 ; ( -98849/4562 , 19508479/9124 ), ( 882/559 , -205873/1118 )
**u= 35/48 , ht= 1919 ; ( 1919/1343 , 274338/1343 ), ( 2749/3103 , 971099/6206 )
**u= 35/66 , ht= 23141 ; ( 10843/23141 , -3509838/23141 ), ( -239374/9413 , 65803443/18826 )
**u= 35/82 , ht= 42741 ; ( 42741/36427 , -8597046/36427 ), ( 106103/42399 , 5830628/14133 )
**u= 35/94 , ht= 69 ; ( -321454/988577 , -340770649/1977154 ), ( -69/22 , -23741/44 )
**u= 35/96 , ht= 172 ; ( -172/31 , 231593/248 ), ( -31/172 , -231593/1376 )
**u= 36/31 , ht= 2638 ; ( -6630/10327 , -6833923/61962 ), ( 2638/3 , 1469047/18 )
**u= 36/47 , ht= 91495 ; ( 622630/21077 , -427155935/126462 ), ( -65438/91495 , 531923/3786 )
**u= 36/95 , ht= 5078 ; ( -5078/2503 , 5518835/15018 ), ( 1091014/67541 , -1065588239/405246 )
**u= 36/199 , ht= 439501 ; ( -439501/269934 , -727701283/1619604 ), ( 465903/65714 , 6445073/3828 )
**u= 37/2 , ht= 1334 ; ( 1334/173 , 62039/346 ), ( 4561/2086 , 234547/4172 )
**u= 37/8 , ht= 2473 ; ( 1166/4881 , -933977/19524 ), ( -2473/297 , -3512/9 )
**u= 37/20 , ht= 14831 ; ( -12431/14831 , -1423000/14831 ), ( 229093/23413 , 16934942/23413 )
**u= 37/24 , ht= 7055 ; ( -8305/9463 , 1014185/9463 ), ( -3887/7055 , 129762/1411 )
**u= 37/28 , ht= 8441 ; ( 1413/8441 , 744657/8441 ), ( -652573/307827 , 1609706/7893 )
**u= 37/38 , ht= 99163 ; ( -924443/130228 , -189244547/260456 ), ( 97624/99163 , 28208539/198326 )
**u= 37/50 , ht= 9857 ; ( -9857/4049 , 6194666/20245 ), ( -75293/27701 , -46638512/138505 )
**u= 37/78 , ht= 743 ; ( 743/167 , -110586/167 ), ( 120782/64037 , -39704187/128074 )
**u= 37/92 , ht= 1085561 ; ( -1165543/319383 , 190592962/319383 ), ( -1085561/57125 , -171440981/57125 )
**u= 37/100 , ht= 115259 ; ( 101541/115259 , 25256688/115259 ), ( -251077/116025 , -75796447/193375 )
**u= 37/110 , ht= 11059 ; ( 11059/10754 , -5320267/21508 ), ( -21743157/1654283 , 3760432414/1654283 )
**u= 37/118 , ht= 6759 ; ( -573183/362911 , 121170942/362911 ), ( -3920/6759 , -310125/1502 )
**u= 37/128 , ht= 21397 ; ( -3190/21397 , 64389731/342352 ), ( -293437/513722 , -1760890939/8219552 )
**u= 37/138 , ht= 120407 ; ( -10195156/12811307 , -6324973873/25622614 ), ( -120407/31393 , 24034616/31393 )
**u= 37/182 , ht= 691189 ; ( -664435/691189 , 212671256/691189 ), ( -10854640/670991 , -4824741155/1341982 )
**u= 37/200 , ht= 175351 ; ( 604147/130803 , 143737196/130803 ), ( -175351/52033 , 212658497/260165 )
**u= 40/23 , ht= 14351 ; ( -41991/22823 , 7248897/45646 ), ( 14351/13983 , 1013067/9322 )
**u= 40/39 , ht= 32293 ; ( 150071/42977 , 30831923/85954 ), ( 11389/32293 , -6763223/64586 )
**u= 40/71 , ht= 190911 ; ( 2923493/385811 , 785854769/771622 ), ( -190911/75623 , -54723925/151246 )
**u= 40/103 , ht= 1302609 ; ( -1302609/31177 , 418235163/62354 ), ( -2379907/379029 , 257847637/252686 )
**u= 40/119 , ht= 94621 ; ( 14310589/180053 , -4937751961/360106 ), ( 94621/35597 , -34879555/71194 )
**u= 40/143 , ht= 4573 ; ( -38422301/49712603 , 23762975683/99425206 ), ( -2109/4573 , -1904623/9146 )
**u= 40/191 , ht= 786743 ; ( 786743/152241 , 350265103/304482 ), ( 1148977/583999 , 563369653/1167998 )
**u= 41/4 , ht= 267 ; ( -25/267 , 8399/267 ), ( -367/59 , -11641/59 )
**u= 41/16 , ht= 1207 ; ( -55273/147719 , -758113/11363 ), ( -39/1207 , 75439/1207 )
**u= 41/24 , ht= 1171 ; ( -1109/1171 , -123413/1171 ), ( -120310/17323 , -37207185/69292 )
**u= 41/34 , ht= 63879 ; ( 36580/70223 , 14423263/140446 ), ( -63879/3968 , -11658769/7936 )
**u= 41/46 , ht= 31288 ; ( -103283/907 , -10941808/907 ), ( -31288/1595 , -6637921/3190 )
**u= 41/50 , ht= 1805 ; ( 1221/1805 , -1203444/9025 ), ( -54299/33099 , -212838/1003 )
**u= 41/54 , ht= 3253 ; ( -55657/882878 , 609241519/5297268 ), ( 202/3253 , -748193/6506 )
**u= 41/84 , ht= 6401 ; ( 2341/6401 , 975713/6401 ), ( -2784671/1337615 , -442252938/1337615 )
**u= 41/86 , ht= 55979 ; ( -54861/55979 , -11353404/55979 ), ( 151247/980169 , 47886116/326723 )
**u= 41/90 , ht= 23461 ; ( -28567/3313 , 4261456/3313 ), ( 23371/23461 , -4907078/23461 )
**u= 41/126 , ht= 44207 ; ( -44207/27225 , -5461672/16335 ), ( -205259/11597 , -36045736/11597 )
**u= 41/130 , ht= 5523 ; ( 507492/2190043 , -800730627/4380086 ), ( -5523/67 , -983678/67 )
**u= 41/132 , ht= 822067 ; ( -15077635/2016739 , -2729876560/2016739 ), ( 822067/54355 , 29569616/10871 )
**u= 41/144 , ht= 346241 ; ( -172533/346241 , -435056107/2077446 ), ( 2898449/118527 , 1631191951/355581 )
**u= 41/156 , ht= 989 ; ( -701/989 , 236496/989 ), ( -17429449/11377147 , -4060622769/11377147 )
**u= 41/184 , ht= 97272 ; ( 88831/97272 , -111641791/389088 ), ( -236927/99287 , -54428851/99287 )
**u= 41/186 , ht= 159049 ; ( -14222074/3990953 , 6293347529/7981906 ), ( -159049/11513 , 33970034/11513 )
**u= 45/2 , ht= 19 ; ( 19/3 , 1202/9 ), ( 3/19 , 1202/57 )
**u= 45/14 , ht= 14601 ; ( 6943/14601 , 2706220/43803 ), ( 112843/5421 , -18911716/16263 )
**u= 45/32 , ht= 258 ; ( -14697/4361 , 3878731/13083 ), ( -191/258 , -649759/6192 )
**u= 45/38 , ht= 43 ; ( 16983/1004 , 9381209/6024 ), ( -24/43 , -27155/258 )
**u= 45/46 , ht= 8117 ; ( 54889/1598 , 33315625/9588 ), ( 8117/3701 , -2706302/11103 )
**u= 45/58 , ht= 689 ; ( 457/689 , -281636/2067 ), ( 5897/12364 , -9332249/74184 )
**u= 45/88 , ht= 797 ; ( -731/797 , -453769/2391 ), ( -15452/289 , 25938505/3468 )
**u= 45/98 , ht= 4029 ; ( 4029/283 , -1788368/849 ), ( -233859/86957 , 8498194/20067 )
**u= 45/118 , ht= 39271 ; ( 35023/39271 , 25566356/117813 ), ( -2424169/14537 , -1177859302/43611 )
**u= 45/136 , ht= 159731 ; ( 335489/6913 , -175032568/20739 ), ( 159731/121702 , -418985573/1460424 )
**u= 45/148 , ht= 4057481 ; ( -4057481/1135913 , -2292722632/3407739 ), ( -9873373/1237949 , 5414578792/3713847 )
**u= 45/188 , ht= 574181 ; ( 1920331/44937 , 1178022007/134811 ), ( -574181/484353 , -460688239/1453059 )
**u= 50/19 , ht= 1085 ; ( 3197/5971 , 417619/5971 ), ( 1085/971 , 448859/4855 )
**u= 50/27 , ht= 719 ; ( -719/233 , -55545/233 ), ( -4879/18689 , -21292357/280335 )
**u= 50/51 , ht= 85585 ; ( 121579/21653 , -12473683/21653 ), ( -85585/30457 , -45881119/152285 )
**u= 50/99 , ht= 265085 ; ( 265085/105013 , -200634253/525065 ), ( -1110675/142357 , -2363833951/2135355 )
**u= 50/123 , ht= 965671 ; ( -965671/662111 , -918346867/3310555 ), ( -2708401/1454657 , -482262171/1454657 )
**u= 52/27 , ht= 11927 ; ( 30715/9262 , 13871137/55572 ), ( 11927/2786 , 5296637/16716 )
**u= 52/43 , ht= 11811 ; ( -2540541/591394 , -474458949/1182788 ), ( 3454/11811 , 746103/7874 )
**u= 52/75 , ht= 37261 ; ( 37261/24286 , 10684549/48572 ), ( -224506/150131 , -64880725/300262 )
**u= 52/115 , ht= 542573 ; ( 471342/542573 , 213795393/1085146 ), ( -901729/5046 , 89414135/3364 )
**u= 52/123 , ht= 78229 ; ( -58670/78229 , 30082899/156458 ), ( 189914/40961 , -59769311/81922 )
**u= 53/6 , ht= 36992 ; ( -63655/21988 , 4525585/43976 ), ( -36992/16105 , -543797/6442 )
**u= 53/10 , ht= 396 ; ( -396/379 , -47625/758 ), ( -719 , -31258 )
**u= 53/22 , ht= 30348 ; ( -774779/1618612 , 231285607/3237224 ), ( 30348/23803 , -4970777/47606 )
**u= 53/30 , ht= 4094 ; ( -1079/4094 , -637203/8188 ), ( -8926/11159 , -2150477/22318 )
**u= 53/40 , ht= 5939 ; ( 154604/37349 , 55276139/149396 ), ( -5939/16 , -2064197/64 )
**u= 53/42 , ht= 44323 ; ( -89216/1835 , 3177819/734 ), ( -44323/14125 , -4141894/14125 )
**u= 53/52 , ht= 58783 ; ( -58783/23299 , -6264083/23299 ), ( -119081/47735 , 12709768/47735 )
**u= 53/58 , ht= 416557 ; ( 416557/283445 , -10543036/56689 ), ( -1616015/1715108 , -493103855/3430216 )
**u= 53/88 , ht= 7651 ; ( -7651/808 , -3965987/3232 ), ( 368988/207343 , -218187977/829372 )
**u= 53/90 , ht= 296569 ; ( 1949644/95669 , -508806035/191338 ), ( -296569/132911 , 42356588/132911 )
**u= 53/100 , ht= 3851 ; ( 15135/45581 , 32990949/227905 ), ( 3349/3851 , -701128/3851 )
**u= 53/108 , ht= 89455 ; ( 54479/89455 , 8972180/53673 ), ( -841015/295993 , -127292350/295993 )
**u= 53/120 , ht= 95443 ; ( 1837/95443 , 14366284/95443 ), ( 672641/655991 , -141397485/655991 )
**u= 53/142 , ht= 1091585 ; ( -420857/1091585 , 38304800/218317 ), ( 6508735/2711513 , -1154304100/2711513 )
**u= 53/150 , ht= 8680 ; ( -3403/8680 , 15687029/86800 ), ( 13016/10841 , 5700347/21682 )
**u= 53/152 , ht= 363061 ; ( 35787/363061 , 61791612/363061 ), ( -1106629/1096635 , 87959704/365545 )
**u= 53/156 , ht= 348739 ; ( 238265/348739 , 72472645/348739 ), ( -776929/3755905 , -131622820/751181 )
**u= 53/170 , ht= 204481 ; ( -250029/37781 , -45294288/37781 ), ( -204481/80721 , 13125994/26907 )
**u= 53/172 , ht= 727877 ; ( -1787339/1474901 , 417517688/1474901 ), ( -9691/727877 , 131155612/727877 )
**u= 53/178 , ht= 4681141 ; ( -68653862/784739 , 25168629997/1569478 ), ( -2414651/4681141 , -965424266/4681141 )
**u= 53/198 , ht= 112949 ; ( -112949/28016 , -44992123/56032 ), ( -357479/394129 , 102861256/394129 )
**u= 56/129 , ht= 25343 ; ( -6907/25343 , 7974697/50686 ), ( -54169/10933 , 16777179/21866 )
**u= 58/15 , ht= 27787 ; ( -22519/30311 , 1920801/30311 ), ( 27787/23227 , -1842163/23227 )
**u= 58/23 , ht= 31559 ; ( -1189539/166403 , 75637233/166403 ), ( -31559/11313 , -78213/419 )
**u= 58/47 , ht= 72005 ; ( 55453/79893 , -8755961/79893 ), ( -26037/72005 , 6893411/72005 )
**u= 58/103 , ht= 677827 ; ( -301053/677827 , 98852103/677827 ), ( -1565791/1137105 , 85972313/379035 )
**u= 58/143 , ht= 36681 ; ( 23239/36681 , -6819281/36681 ), ( -189191/236377 , 47547319/236377 )
**u= 63/8 , ht= 1127 ; ( -767/1127 , -145841/3381 ), ( -62996/26197 , -29210887/314364 )
**u= 63/32 , ht= 5804 ; ( -3935/5804 , -11996363/139296 ), ( 15505/16049 , -4771925/48147 )
**u= 63/58 , ht= 3467 ; ( -651/3467 , 1015594/10401 ), ( 6614/8673 , -6280121/52038 )
**u= 63/74 , ht= 6917 ; ( 2555/6917 , -2397890/20751 ), ( -73381/19045 , 4930690/11427 )
**u= 63/92 , ht= 15407 ; ( 15407/3953 , -5767232/11859 ), ( 160507/24853 , 58891642/74559 )
**u= 63/128 , ht= 3451 ; ( 3451/2053 , 1717351/6159 ), ( 27919/55535 , 1833361/11490 )
**u= 63/170 , ht= 6146 ; ( 961/6146 , -6132107/36876 ), ( 33409/191 , 16466930/573 )
**u= 63/200 , ht= 33107 ; ( -33107/3724 , -356213297/223440 ), ( -94253/111797 , -78173057/335391 )
**u= 64/95 , ht= 14242 ; ( 8333/14242 , 16085405/113936 ), ( 128918/90253 , 153409331/722024 )
**u= 64/111 , ht= 8482 ; ( 8482/731 , -8970739/5848 ), ( 24598/21823 , 34649833/174584 )
**u= 65/4 , ht= 877 ; ( 6901/8427 , -270592/8427 ), ( 877/529 , -25489/529 )
**u= 65/6 , ht= 2767 ; ( 226757/27251 , 6920180/27251 ), ( -2767/881 , 88450/881 )
**u= 65/12 , ht= 673 ; ( -513973/80939 , 22343720/80939 ), ( 673/431 , -34354/431 )
**u= 65/16 , ht= 24229 ; ( 8283/24229 , 1271004/24229 ), ( -305671/27927 , -1693017/3103 )
**u= 65/22 , ht= 122 ; ( -4243/2549 , -287980/2549 ), ( -122 , -14201/2 )
**u= 65/56 , ht= 157592 ; ( -40629/157592 , -60434241/630368 ), ( 247649/171777 , 9326498/57259 )
**u= 65/58 , ht= 37121 ; ( -37121/10903 , -3655100/10903 ), ( -102201/390103 , 38098226/390103 )
**u= 65/64 , ht= 294059 ; ( -1718689/188003 , 686321617/752012 ), ( 289957/294059 , 81968831/588118 )
**u= 65/74 , ht= 460423 ; ( 437271/460423 , 67761414/460423 ), ( 2474911/527343 , 90014358/175781 )
**u= 65/82 , ht= 154493 ; ( 154493/38381 , -17882278/38381 ), ( -209419/113948 , 53564603/227896 )
**u= 65/102 , ht= 294458 ; ( 158339/294458 , 83775439/588916 ), ( 829846/225037 , -215451039/450074 )
**u= 65/106 , ht= 152046 ; ( -123866/293933 , -81477709/587866 ), ( 83327/152046 , -14762923/101364 )
**u= 65/108 , ht= 47 ; ( -1/47 , -18182/141 ), ( -337306843/7756799 , 130492475855/23270397 )
**u= 65/138 , ht= 125951 ; ( 621058/200089 , -190174673/400178 ), ( -125951/89743 , 22537530/89743 )
**u= 65/148 , ht= 401621 ; ( 401621/296837 , -75369746/296837 ), ( -3198467/78031 , 482850275/78031 )
**u= 65/154 , ht= 1195582 ; ( -1195582/462459 , 394687871/924918 ), ( 15209783/1234591 , -2349198796/1234591 )
**u= 65/164 , ht= 140021 ; ( -7453569/91087 , 1184200878/91087 ), ( 140021/131973 , -10189282/43991 )
**u= 65/166 , ht= 89242 ; ( -2434849/748782 , 814301995/1497564 ), ( -89242/10441 , 28722061/20882 )
**u= 65/168 , ht= 1615646 ; ( -1615646/675853 , -1126380585/2703412 ), ( 34503533/7416781 , -5674621090/7416781 )
**u= 65/174 , ht= 24253 ; ( -24253/3964 , -8042699/7928 ), ( 70876628/5392511 , 23263281831/10785022 )
**u= 68/27 , ht= 463 ; ( 463/220 , -12921/88 ), ( 865/1148 , 543505/6888 )
**u= 68/43 , ht= 232324 ; ( 232324/62603 , 38270719/125206 ), ( -548628/284239 , 98287819/568478 )
**u= 68/123 , ht= 10579 ; ( 4468/10579 , -3089457/21158 ), ( -30479/29996 , 11504507/59992 )
**u= 68/147 , ht= 785 ; ( -785/148 , -234937/296 ), ( 1776479/14024 , -3657407183/196336 )
**u= 70/29 , ht= 13509 ; ( 13509/8753 , -1036167/8753 ), ( -24371/1287 , -523763/429 )
**u= 70/33 , ht= 24311 ; ( 24311/17447 , -2054809/17447 ), ( 82403/61811 , -7073949/61811 )
**u= 70/141 , ht= 27331 ; ( 5993/27331 , 3971743/27331 ), ( -380951/115933 , 56523671/115933 )
**u= 72/19 , ht= 88295 ; ( -513595/101131 , 161310905/606786 ), ( -58271/88295 , -6521389/105954 )
**u= 72/115 , ht= 1061 ; ( -57059/1027621 , -780557147/6165726 ), ( 181/1061 , 816281/6366 )
**u= 72/163 , ht= 127877 ; ( -1820675/268333 , 1661653769/1609998 ), ( -127877/85979 , -139132969/515874 )
**u= 73/10 , ht= 6762 ; ( -6762/4087 , -584595/8174 ), ( 32569/23574 , 992607/15716 )
**u= 73/18 , ht= 26164 ; ( 26164/17819 , 3143793/35638 ), ( 12002299/3881664 , 3760119659/23289984 )
**u= 73/20 , ht= 31279 ; ( -31279/369 , 1637032/369 ), ( 1166327/475027 , 65943125/475027 )
**u= 73/42 , ht= 9089 ; ( -9089/4841 , -781188/4841 ), ( -4229/10450 , 1710351/20900 )
**u= 73/62 , ht= 495708 ; ( -1033049/202311 , 97023962/202311 ), ( -495708/119213 , -93989203/238426 )
**u= 73/78 , ht= 217000 ; ( -70981/905620 , 37566115/362248 ), ( 217000/52069 , 46142905/104138 )
**u= 73/104 , ht= 6849 ; ( 484870/2159959 , 1057076365/8639836 ), ( 6849/2030 , -682223/1624 )
**u= 73/128 , ht= 831855 ; ( -1060204/87071 , -2254105931/1393136 ), ( 831855/661336 , -2251870285/10581376 )
**u= 73/132 , ht= 213347 ; ( -68969/274693 , 38090431/274693 ), ( -213347/64447 , 29973819/64447 )
**u= 73/150 , ht= 207155 ; ( -207155/109429 , -167940792/547145 ), ( 33946/213479 , 61980403/426958 )
**u= 73/200 , ht= 13063 ; ( 13063/4199 , 11357486/20995 ), ( -96003/37379 , 85275548/186895 )
**u= 74/135 , ht= 377347 ; ( -37133/377347 , -153664721/1132041 ), ( -549841/1269559 , -560683895/3808677 )
**u= 74/159 , ht= 44185 ; ( 222155/135341 , 38136861/135341 ), ( -30863/44185 , -7901473/44185 )
**u= 74/199 , ht= 15162143 ; ( -21085325/2493389 , -3482331581/2493389 ), ( 15162143/1862177 , 2505468113/1862177 )
**u= 80/43 , ht= 13867 ; ( -1054/13867 , -4079293/55468 ), ( -197477/151726 , -73043693/606904 )
**u= 80/171 , ht= 82087 ; ( 1292258/473681 , -805009975/1894724 ), ( 82087/22066 , 4519725/8024 )
**u= 80/179 , ht= 976722 ; ( 976722/109109 , 45240267/33572 ), ( 9254869/6468018 , -2252270501/8624024 )
**u= 81/4 , ht= 767 ; ( 3833/471 , -764762/4239 ), ( 417/767 , 173656/6903 )
**u= 81/10 , ht= 3946 ; ( -3946/1239 , -2612671/22302 ), ( 21243/7918 , -14358115/142524 )
**u= 81/16 , ht= 495 ; ( 443/495 , 106309/1782 ), ( 645/187 , 268810/1683 )
**u= 81/20 , ht= 677 ; ( 167/677 , -311999/6093 ), ( -677/167 , 311999/1503 )
**u= 81/34 , ht= 4752 ; ( 4752/1895 , -1193281/6822 ), ( 24435/51227 , -33096320/461043 )
**u= 81/40 , ht= 426 ; ( 1878/367 , -4841141/13212 ), ( 161/426 , 1152187/15336 )
**u= 81/44 , ht= 19535 ; ( -19535/11573 , 15062929/104157 ), ( -144811/30991 , -98254163/278919 )
**u= 81/64 , ht= 2287 ; ( -41781/31487 , 167438111/1133532 ), ( 1119/2287 , -4074241/41166 )
**u= 81/74 , ht= 10807 ; ( 11567/6239 , 11306962/56151 ), ( -10807/9247 , 12237116/83223 )
**u= 81/86 , ht= 294011 ; ( -294011/99181 , -287788376/892629 ), ( 451849/189377 , -454417376/1704393 )
**u= 81/104 , ht= 5002 ; ( 5002/4729 , 28083727/170244 ), ( -437633/3769 , 446390101/33921 )
**u= 81/116 , ht= 119575 ; ( -579457/174095 , -130348910/313371 ), ( -119575/21497 , 130871990/193473 )
**u= 81/136 , ht= 60929 ; ( -106268/131511 , 788837431/4734396 ), ( -60929/1767 , 71095979/15903 )
**u= 81/140 , ht= 10897 ; ( 10897/2767 , -13304576/24903 ), ( -187301/217439 , 339617789/1956951 )
**u= 81/146 , ht= 58823 ; ( -10351/58823 , -72179492/529407 ), ( -220307/97900 , 52971511/160200 )
**u= 81/194 , ht= 1119193 ; ( -431255/1119193 , -1670834686/10072737 ), ( 5928839/285961 , 8268792862/2573649 )
**u= 82/3 , ht= 3457 ; ( 4991/5231 , 138433/5231 ), ( 3457/2249 , -79491/2249 )
**u= 82/11 , ht= 91855 ; ( 1123135/1391711 , -65530115/1391711 ), ( -12271/91855 , -678025/18371 )
**u= 82/83 , ht= 134693 ; ( -134693/70981 , -15319787/70981 ), ( -482347/46861 , 48764881/46861 )
**u= 82/147 , ht= 1619369 ; ( -1645021/1169477 , -1891960599/8186339 ), ( 1619369/220385 , -218852251/220385 )
**u= 90/23 , ht= 3341 ; ( 3341/397 , 510139/1191 ), ( 2581169/260423 , -393647549/781269 )
**u= 90/31 , ht= 15219 ; ( -551001/39377 , -7481189/9087 ), ( 10387/15219 , 3244423/45657 )
**u= 90/119 , ht= 547 ; ( 547/139 , -194719/417 ), ( 118081/15503 , -41089967/46509 )
**u= 90/143 , ht= 2980819 ; ( -26105311/443783 , 9874621823/1331349 ), ( -2980819/891907 , 1176770903/2675721 )
**u= 91/6 , ht= 1124 ; ( -1124/451 , -61947/902 ), ( -1147/596 , 66611/1192 )
**u= 91/20 , ht= 4097 ; ( 4097/1297 , 201410/1297 ), ( -12609/11471 , 799294/11471 )
**u= 91/24 , ht= 343 ; ( 83032/26525 , 3580629/21220 ), ( -209/343 , 20627/343 )
**u= 91/68 , ht= 323075 ; ( -323075/202769 , -32976995/202769 ), ( -542871/130825 , -9656084/26165 )
**u= 91/132 , ht= 1456735 ; ( -1456735/276337 , -178600970/276337 ), ( 2275763/235735 , 55120231/47147 )
**u= 91/146 , ht= 21298 ; ( -9289/21298 , 5887169/42596 ), ( 25943/16745 , 3911722/16745 )
**u= 91/194 , ht= 13891 ; ( 13891/893 , 2032694/893 ), ( 73106/111809 , -39015899/223618 )
**u= 97/8 , ht= 41585 ; ( 50524/2905 , -1158611/2324 ), ( 41585/713 , -1198985/713 )
**u= 97/18 , ht= 48233 ; ( 38649/48233 , -57482/1041 ), ( -1601419/66875 , -69105518/66875 )
**u= 97/32 , ht= 244 ; ( 2693/1573 , 179137/1573 ), ( 33/244 , -113129/1952 )
**u= 97/44 , ht= 15519 ; ( 2125415/13436451 , -916446035/13436451 ), ( -15519/12685 , -270037/2537 )
**u= 97/80 , ht= 153709 ; ( 405809/204849 , 41288228/204849 ), ( 44119/153709 , 29048993/307418 )
**u= 97/90 , ht= 149679 ; ( 211013/813 , -60984800/2439 ), ( 149679/1279 , 43262450/3837 )
**u= 97/120 , ht= 88621 ; ( 5187809/944831 , -586587822/944831 ), ( 47104/88621 , -44657445/354484 )
**u= 97/128 , ht= 381 ; ( -30664/1331 , 56420323/21296 ), ( -251/381 , -69891/508 )
**u= 97/132 , ht= 52093 ; ( 174395/630257 , 76297297/630257 ), ( -52093/3677 , 6092992/3677 )
**u= 97/150 , ht= 7691 ; ( -4205/7691 , 5450956/38455 ), ( 32002/25723 , 352177/1774 )
**u= 97/156 , ht= 855169 ; ( 1837655/1158691 , 275543155/1158691 ), ( 855169/259295 , -22668644/51859 )
**u= 97/162 , ht= 271111 ; ( -266935/271111 , -147528382/813333 ), ( -2794753/11599212 , -27757999021/208785816 )
**u= 97/170 , ht= 141574 ; ( 3112898/609053 , -839948659/1218106 ), ( -45341/141574 , 39365629/283148 )
**u= 97/178 , ht= 3135119 ; ( 6931764/1312571 , 147050433/201934 ), ( 3135119/2698920 , 124549529/599760 )
**u= 97/188 , ht= 420241 ; ( 8761703/21207 , 1219958626/21207 ), ( 286479/420241 , 70816172/420241 )
**u= 97/200 , ht= 726707 ; ( -1064631/442097 , 827764521/2210485 ), ( -726707/701324 , -2900785291/14026480 )
**u= 100/39 , ht= 21514 ; ( 181/21514 , 13440187/215140 ), ( 5606/23533 , 3021529/47066 )
**u= 100/47 , ht= 8927 ; ( -21110/54513 , 40085923/545130 ), ( -4750/8927 , 6933083/89270 )
**u= 100/87 , ht= 47657 ; ( -111851/43270 , -111876269/432700 ), ( 11030/47657 , 45632017/476570 )
**u= 100/143 , ht= 51614 ; ( -408510/10909 , 488748393/109090 ), ( -51614/27135 , -7749133/30150 )
**u= 100/159 , ht= 3158 ; ( -4262/63737 , -80561651/637370 ), ( 3158/31 , -796583/62 )
**u= 100/183 , ht= 137119 ; ( 137119/122558 , -189219/932 ), ( -505486/41825 , -686253527/418250 )
**u= 100/191 , ht= 300038 ; ( -182237/300038 , 485229193/3000380 ), ( -300107/89342 , 432812407/893420 )
**u= 104/55 , ht= 617567 ; ( -617567/425233 , -109067675/850466 ), ( -183453703/16951513 , 724384049/916298 )
**u= 104/175 , ht= 4577 ; ( 483849/185351 , -134443047/370702 ), ( -4577/3177 , 481897/2118 )
**u= 106/15 , ht= 9799 ; ( -3689/9799 , -394301/9799 ), ( -754087/20633 , -28417269/20633 )
**u= 106/39 , ht= 3549973 ; ( 3549973/205411 , -215682897/205411 ), ( -14837923/1400507 , -904320441/1400507 )
**u= 106/119 , ht= 4634817 ; ( -234793245/4171301 , 24884630703/4171301 ), ( -4634817/502423 , 494041579/502423 )
**u= 112/5 , ht= 8 ; ( -59/44 , -6199/176 ), ( 8/3 , 243/4 )
**u= 112/13 , ht= 629 ; ( -4857/1552 , 693921/6208 ), ( -629/60 , -28759/80 )
**u= 112/57 , ht= 3074 ; ( 3074/175 , 175731/140 ), ( -68125/5158 , -19500365/20632 )
**u= 112/101 , ht= 803 ; ( -158000/31173 , -61180625/124692 ), ( 803/500 , -2875/16 )
**u= 112/137 , ht= 7477 ; ( 9646/6221 , 5078581/24884 ), ( 7477/490 , 3315443/1960 )
**u= 116/19 , ht= 1673 ; ( -3027/538 , -248673/1076 ), ( 1673/398 , 139333/796 )
**u= 116/75 , ht= 45790 ; ( -393247/263810 , 380813573/2638100 ), ( -45790/13589 , 38413849/135890 )
**u= 116/91 , ht= 66999 ; ( 66999/61994 , -16171983/123988 ), ( -1024909/139150 , 183254803/278300 )
**u= 116/99 , ht= 222534 ; ( -722554/613665 , 8085169/56646 ), ( -222534/189545 , 162054113/1137270 )
**u= 116/107 , ht= 591403 ; ( -398910/591403 , 137048205/1182806 ), ( -591403/398910 , -9136547/53188 )
**u= 116/139 , ht= 656427 ; ( 2991913/580958 , 667347913/1161916 ), ( 656427/285158 , 156712397/570316 )
**u= 116/179 , ht= 1015641 ; ( -14326230/1950419 , -3592594089/3900838 ), ( -438734/1015641 , -91635023/677094 )
**u= 117/2 , ht= 17 ; ( 1954/943 , -160913/5658 ), ( -7/17 , -676/51 )
**u= 117/16 , ht= 8263 ; ( -13775/1743 , -1538620/5229 ), ( 8263/285 , 19339/18 )
**u= 117/20 , ht= 101 ; ( -164153/789077 , 100061743/2367231 ), ( -101/11 , 12614/33 )
**u= 117/28 , ht= 46489 ; ( 10551/46489 , 7000034/139467 ), ( 95731/236361 , 37419361/709083 )
**u= 117/32 , ht= 3577 ; ( 1394/3577 , 4820443/85848 ), ( 3577/1394 , 4820443/33456 )
**u= 117/40 , ht= 18741 ; ( 22827/15517 , 4841989/46551 ), ( -12221/18741 , -3924839/56223 )
**u= 117/58 , ht= 20911 ; ( 20911/11457 , 5036878/34371 ), ( -47511/79904 , 3570503/43584 )
**u= 117/68 , ht= 71969 ; ( -71969/71443 , 23196427/214329 ), ( 436199/138019 , 104659421/414057 )
**u= 117/80 , ht= 292793 ; ( 292793/72263 , 74820136/216789 ), ( -1873607/543373 , 968061403/3260238 )
**u= 117/82 , ht= 19609 ; ( 367041/120713 , -97050776/362139 ), ( 19609/16329 , -6409802/48987 )
**u= 117/112 , ht= 99425 ; ( -61878261/13714075 , -3721108873/8228445 ), ( 99425/96351 , -40644410/289053 )
**u= 117/128 , ht= 66503 ; ( -29654/66503 , -365632427/3192144 ), ( -1438579/153938 , 7264961561/7389024 )
**u= 117/134 , ht= 281989 ; ( 1088755/209236 , 711989585/1255416 ), ( 281989/170395 , 21159194/102237 )
**u= 117/142 , ht= 17519 ; ( -17519/9177 , -6537256/27531 ), ( 3214877/606621 , -1081448494/1819863 )
**u= 117/170 , ht= 3038708 ; ( 3038708/626663 , 2244276367/3759978 ), ( 973691/6847276 , 5002740547/41083656 )
**u= 117/184 , ht= 342705 ; ( -342705/44546 , -520135825/534552 ), ( 977126/506385 , 971513/3564 )
**u= 117/190 , ht= 137369 ; ( 8903743/530082 , -6820827655/3180492 ), ( -137369/95001 , -63861374/285003 )
**u= 119/30 , ht= 3133 ; ( 46313/33113 , 2858650/33113 ), ( -677/3133 , -160904/3133 )
**u= 119/46 , ht= 12227 ; ( -643361/128657 , 40791806/128657 ), ( -12227/3620 , 1586087/7240 )
**u= 119/106 , ht= 5063 ; ( 3473/5063 , 579556/5063 ), ( 129938/2403 , 8178571/1602 )
**u= 119/184 , ht= 61189 ; ( 2186405/112859 , 272272936/112859 ), ( 61189/20852 , -2473727/6416 )
**u= 121/4 , ht= 113 ; ( 27/113 , -23709/1243 ), ( -107/807 , -53067/2959 )
**u= 121/14 , ht= 715 ; ( -2091/6347 , 2504514/69817 ), ( -477/715 , -321416/7865 )
**u= 121/74 , ht= 34703 ; ( -7958/34703 , 61267903/763466 ), ( -34703/7958 , -61267903/175076 )
**u= 121/76 , ht= 56967 ; ( 56967/1667 , -49688247/18337 ), ( -3845447/1308795 , -1180646281/4798915 )
**u= 121/86 , ht= 27806 ; ( 60291/28762 , -123910509/632764 ), ( 27806/6809 , -53106473/149798 )
**u= 121/90 , ht= 6009 ; ( 1894/6009 , -35869337/396594 ), ( 17757/1037 , 50633278/34221 )
**u= 121/96 , ht= 22574 ; ( 93319/5593 , 91607773/61523 ), ( -22574/6947 , 185166437/611336 )
**u= 121/114 , ht= 87199 ; ( -87199/21649 , 95941522/238139 ), ( -217753/24668 , -468050393/542696 )
**u= 121/120 , ht= 40003 ; ( 40003/1693 , -43865786/18623 ), ( -487/73703 , -80749412/810733 )
**u= 121/126 , ht= 341 ; ( -75/341 , 1175962/11253 ), ( -923/1821 , 6875876/60093 )
**u= 121/144 , ht= 6329 ; ( -6329/5195 , 3930843/22858 ), ( 5195/6329 , 19654215/139238 )
**u= 121/146 , ht= 4496 ; ( 597344/14761 , -49799293/11198 ), ( 2233/4496 , -12133049/98912 )
**u= 121/166 , ht= 239371 ; ( 336621/188051 , -496864386/2068561 ), ( 239371/101259 , 111640198/371283 )
**u= 121/174 , ht= 193877 ; ( -193877/26059 , -258076798/286649 ), ( -797419/98507 , -1060034782/1083577 )
**u= 121/184 , ht= 14871 ; ( 18291/87728 , -486313959/3860032 ), ( 5372/14871 , -28601153/218108 )
**u= 121/192 , ht= 16541 ; ( -422185/17449 , -585578150/191939 ), ( 16541/6695 , -19784007/58916 )
**u= 122/19 , ht= 62135 ; ( -1017155/632729 , -47258785/632729 ), ( -2237/62135 , 490301/12427 )
**u= 122/75 , ht= 132869 ; ( 29287489/16301123 , 13141776989/81505615 ), ( -10247/132869 , -4749809/60395 )
**u= 122/147 , ht= 26309 ; ( -26309/5017 , 20582381/35119 ), ( -71689/74147 , -79259823/519029 )
**u= 125/4 , ht= 23 ; ( 19/23 , -2681/115 ), ( -8981/5833 , 193489/5833 )
**u= 125/8 , ht= 199 ; ( 46938/4591 , -4744515/18364 ), ( 199/177 , -11239/295 )
**u= 125/22 , ht= 856 ; ( -2473/19824 , -8388907/198240 ), ( -856/13 , -359497/130 )
**u= 125/34 , ht= 25711 ; ( 65978/229841 , 124765907/2298410 ), ( 13063/25711 , -7519964/128555 )
**u= 125/42 , ht= 6536 ; ( 6536/397 , -3795343/3970 ), ( 8846521/1087252 , 5168398491/10872520 )
**u= 125/44 , ht= 181 ; ( 27/181 , -54306/905 ), ( 12847/94401 , 3140082/52445 )
**u= 125/46 , ht= 136882 ; ( -256699/145157 , 17890390/145157 ), ( 136882/44721 , 29127403/149070 )
**u= 125/54 , ht= 167 ; ( -167/76 , 120603/760 ), ( 289112/30779 , -573453883/923370 )
**u= 125/62 , ht= 20494 ; ( 521853/167246 , -77193195/334492 ), ( -20494/6513 , -5049409/21710 )
**u= 125/76 , ht= 11563 ; ( -10591/11563 , 1222853/11563 ), ( 174303/40771 , -13960961/40771 )
**u= 125/88 , ht= 41267 ; ( 211723/50771 , 91350499/253855 ), ( -27219/41267 , 4148386/41267 )
**u= 125/94 , ht= 22073 ; ( 22073/4166 , -19481309/41660 ), ( -1917206/565757 , 1733763521/5657570 )
**u= 125/108 , ht= 699713 ; ( 699713/174431 , 1005570004/2616465 ), ( -807497/997861 , -1790026991/14967915 )
**u= 125/132 , ht= 74939 ; ( 539819/208763 , 297422334/1043815 ), ( 74939/293 , 38511288/1465 )
**u= 125/136 , ht= 28687 ; ( -23214/28687 , -76996953/573740 ), ( -1643099/305403 , -290586798/509005 )
**u= 125/154 , ht= 34509 ; ( 13147/34509 , -20497816/172545 ), ( -350183/73361 , -198595288/366805 )
**u= 125/162 , ht= 5583 ; ( 3679/5583 , -34257874/251235 ), ( 18581/2578 , 64077347/77340 )
**u= 125/172 , ht= 343763 ; ( 60042063/31243841 , 7940758062/31243841 ), ( -343763/194381 , 231659284/971905 )
**u= 125/178 , ht= 5654 ; ( -5654/2623 , 7438753/26230 ), ( 1359451/511637 , -866808134/2558185 )
**u= 125/198 , ht= 32939 ; ( 32939/4868 , -41912227/48680 ), ( 201868351/18893977 , -127608371668/94469885 )
**u= 126/29 , ht= 12459 ; ( 76849/231 , -11056949/693 ), ( -12459/4187 , 1892701/12561 )
**u= 126/73 , ht= 42083 ; ( 1718285/52279 , 392578385/156837 ), ( -42083/34345 , 2481139/20607 )
**u= 126/185 , ht= 383493 ; ( -383493/71977 , 141859189/215931 ), ( -71977/383493 , -141859189/1150479 )
**u= 128/27 , ht= 569 ; ( 295/1247 , 1413391/29928 ), ( 311/569 , 714653/13656 )
**u= 128/67 , ht= 8877 ; ( -301/8877 , 5142127/71016 ), ( 1215105/194039 , -712372175/1552312 )
**u= 128/115 , ht= 3989 ; ( -381/3989 , 3039123/31912 ), ( -3989/381 , -1013041/1016 )
**u= 130/3 , ht= 1073 ; ( -943/5191 , -2883/179 ), ( 1073/1049 , -22823/1049 )
**u= 130/11 , ht= 29441 ; ( -49859/2347 , -1447103/2347 ), ( -29441/3647 , -865993/3647 )
**u= 130/27 , ht= 287843 ; ( 149509/379957 , 1053899/21507 ), ( 287843/13141 , -39423467/39423 )
**u= 130/67 , ht= 1482331 ; ( -28961577/12229801 , 2257125789/12229801 ), ( 1482331/1025613 , -43143103/341871 )
**u= 130/83 , ht= 1982119 ; ( 11986171/1251117 , 963030373/1251117 ), ( -1500063/1982119 , 198647833/1982119 )
**u= 130/99 , ht= 48437 ; ( 44829/48437 , 17280919/145311 ), ( 190061/35027 , 16868415/35027 )
**u= 130/179 , ht= 358039 ; ( 11137/358039 , 42040321/358039 ), ( -32770677/17439181 , 4356667811/17439181 )
**u= 136/71 , ht= 17575 ; ( 17575/4653 , -2627401/9306 ), ( -383015/547227 , 32178385/364818 )
**u= 136/159 , ht= 12649 ; ( -12649/1237 , 2748777/2474 ), ( -3947059/2826977 , 1050065711/5653954 )
**u= 137/10 , ht= 7723 ; ( 30679/42959 , -1428592/42959 ), ( -7723/517 , 19102/47 )
**u= 137/18 , ht= 631 ; ( -5788/2715 , 277865/3258 ), ( -631/217 , -24218/217 )
**u= 137/20 , ht= 60033 ; ( -60033/4013 , 2296533/4013 ), ( 7771/100731 , 1285424/33577 )
**u= 137/24 , ht= 2195 ; ( 2195/2029 , -125120/2029 ), ( 376066/359525 , -17423987/287620 )
**u= 137/58 , ht= 9269 ; ( 9269/7493 , -775606/7493 ), ( 7493/9269 , -2594/31 )
**u= 137/72 , ht= 107663 ; ( 58351/309753 , 68567251/929259 ), ( 55193/107663 , 8771683/107663 )
**u= 137/76 , ht= 110965 ; ( 1485695/1262331 , 145225105/1262331 ), ( -24201/110965 , -1691945/22193 )
**u= 137/90 , ht= 92882 ; ( -92882/22183 , 15481423/44366 ), ( -2571821/156234 , 1253265073/937404 )
**u= 137/124 , ht= 78283 ; ( 1209805/435613 , -122346830/435613 ), ( -78283/24095 , -1558759/4819 )
**u= 137/132 , ht= 56615 ; ( 56615/43591 , -7014636/43591 ), ( 3231535/428581 , -320035555/428581 )
**u= 137/138 , ht= 337919 ; ( 68198/337919 , 69209541/675838 ), ( 10330873/15612823 , 1879214174/15612823 )
**u= 137/142 , ht= 1418392 ; ( -847431/1418392 , 336483021/2836784 ), ( 32717999/971343 , 1111004702/323781 )
**u= 137/148 , ht= 2409283 ; ( -14341759/7093007 , -1663210934/7093007 ), ( -1918237/2409283 , -320136916/2409283 )
**u= 137/150 , ht= 171826 ; ( 31781/172594 , 36732905/345188 ), ( -171826/119815 , -219222927/1198150 )
**u= 137/182 , ht= 70313 ; ( -55131/198430 , 9496419/79372 ), ( 70313/10911 , -2734184/3637 )
**u= 137/190 , ht= 1353397 ; ( -50078959/1581984 , -11802605575/3163968 ), ( -1353397/275093 , -162668378/275093 )
**u= 140/13 , ht= 427 ; ( -714/1873 , 122445/3746 ), ( -427/314 , 32335/628 )
**u= 140/57 , ht= 48212 ; ( -230287/4496 , 29393769/8992 ), ( 5861/48212 , -6197917/96424 )
**u= 144/71 , ht= 32254 ; ( 29059/32254 , 36586879/387048 ), ( 2377/48314 , -40761283/579768 )
**u= 144/175 , ht= 40386 ; ( -1586185/66942 , 10502364731/4016520 ), ( 40386/24811 , -62712163/297732 )
**u= 144/191 , ht= 126346 ; ( -126346/20533 , 176928613/246396 ), ( -263285/980698 , 1403538245/11768376 )
**u= 145/8 , ht= 1407 ; ( -3033/38279 , -909366/38279 ), ( 1407/1241 , 44119/1241 )
**u= 145/14 , ht= 1912 ; ( 13957/8604 , 1017859/17208 ), ( -1912/1861 , 165851/3722 )
**u= 145/26 , ht= 79 ; ( -38/79 , -7429/158 ), ( 1383933/571501 , -3339334/30079 )
**u= 145/56 , ht= 12487 ; ( -1382711/211273 , 86925760/211273 ), ( 4409/12487 , -822980/12487 )
**u= 145/66 , ht= 906628 ; ( -19560424/3500867 , -2681383703/7001734 ), ( 182509/906628 , 124792913/1813256 )
**u= 145/72 , ht= 49897 ; ( -23489/49897 , 3887007/49897 ), ( -74507/910701 , 193174415/2732103 )
**u= 145/74 , ht= 1673 ; ( 1081/1673 , -142322/1673 ), ( 1673/1081 , -142322/1081 )
**u= 145/78 , ht= 28813 ; ( 965971/341152 , 150285571/682304 ), ( -8276/28813 , -4397739/57626 )
**u= 145/86 , ht= 859347 ; ( -662509/859347 , -83579426/859347 ), ( -1754114/609037 , 286060733/1218074 )
**u= 145/92 , ht= 91123 ; ( -3648749/928527 , 299930783/928527 ), ( -19921/91123 , 7430471/91123 )
**u= 145/96 , ht= 13153 ; ( -93379/307562 , 209269989/2460496 ), ( -359/13153 , -2141453/26306 )
**u= 145/102 , ht= 39199253 ; ( -485314111/34160873 , -40808971420/34160873 ), ( 39199253/16021 , 3288372290/16021 )
**u= 145/104 , ht= 361701 ; ( 274757/361701 , -38474510/361701 ), ( -2122733/247726 , 724120957/990904 )
**u= 145/132 , ht= 47191 ; ( 47191/36937 , -5718640/36937 ), ( 11743489/16949173 , -1967668595/16949173 )
**u= 145/134 , ht= 1070637 ; ( -42006451/6392333 , 4085155382/6392333 ), ( 181459/1070637 , -34801050/356879 )
**u= 145/138 , ht= 58421 ; ( -58421/31877 , -6493438/31877 ), ( 679583/971246 , 231317343/1942492 )
**u= 145/156 , ht= 263969 ; ( -289097/370601 , 48760188/370601 ), ( -93973/263969 , -29066999/263969 )
**u= 145/168 , ht= 83474 ; ( -116129/241697 , -28867920/241697 ), ( -53593/83474 , 42716195/333896 )
**u= 146/19 , ht= 2595 ; ( -21203/1445 , -153131/289 ), ( 2595/277 , -94295/277 )
**u= 146/75 , ht= 68501 ; ( 343391/55745 , 124677377/278725 ), ( 63949/68501 , -6717533/68501 )
**u= 146/83 , ht= 1158403 ; ( -416715/1158403 , -92840745/1158403 ), ( -2790139/4943325 , -28535857/329555 )
**u= 146/91 , ht= 41123 ; ( 41123/26563 , -3865487/26563 ), ( 217807/348119 , 32423537/348119 )
**u= 146/155 , ht= 73729 ; ( 1841499/839381 , 208550481/839381 ), ( 73729/39009 , -2865303/13003 )
**u= 148/35 , ht= 138826 ; ( 1019314/178309 , -2454059/8698 ), ( -138826/90379 , 16116731/180758 )
**u= 148/147 , ht= 170918 ; ( 20664631/5159030 , 5944265003/14445284 ), ( 5455/170918 , -238626615/2392852 )
**u= 148/195 , ht= 138622 ; ( -39803/138622 , 33114795/277244 ), ( -5973569/2226766 , -1463773607/4453532 )
**u= 157/12 , ht= 28127 ; ( -25907/46573 , -1476928/46573 ), ( 13301/28127 , -857969/28127 )
**u= 157/14 , ht= 335 ; ( 335/274 , 25835/548 ), ( -807/145 , -4912/29 )
**u= 157/32 , ht= 51563 ; ( 58650/108821 , 44669553/870568 ), ( 51563/24186 , 6860357/64496 )
**u= 157/40 , ht= 46189 ; ( -4941/46189 , 2345946/46189 ), ( -80981/152364 , -11611803/203152 )
**u= 157/48 , ht= 196157 ; ( 196157/98975 , -4859391/39590 ), ( 218695/73801 , 12766925/73801 )
**u= 157/68 , ht= 5323 ; ( 253290099/62950483 , -17177169744/62950483 ), ( 1307/5323 , 360734/5323 )
**u= 157/78 , ht= 378301 ; ( 53899/378301 , -26940532/378301 ), ( 2860358/130709 , -403747417/261418 )
**u= 157/92 , ht= 260201 ; ( 673313/712827 , 75072263/712827 ), ( 174227/260201 , -23974097/260201 )
**u= 157/102 , ht= 7505 ; ( -1487/7505 , 123362/1501 ), ( -87733/21611 , 7284382/21611 )
**u= 157/132 , ht= 649897 ; ( 185543/649897 , -61983302/649897 ), ( 1813307/1924715 , -242505178/1924715 )
**u= 157/152 , ht= 407817 ; ( 6626557/6822960 , 748799281/5458368 ), ( -407817/163703 , -43246568/163703 )
**u= 157/158 , ht= 4854 ; ( -2405/4854 , -1087051/9708 ), ( -2692121/1358127 , -100846136/452709 )
**u= 157/186 , ht= 176651 ; ( 120205/176651 , 23260440/176651 ), ( 176651/120205 , 4652088/24041 )
**u= 157/188 , ht= 5875429 ; ( -5875429/1377799 , 660467477/1377799 ), ( 21910117/6347477 , 2496581666/6347477 )
**u= 157/198 , ht= 141101 ; ( -141101/6200 , -6345269/2480 ), ( -702200/9566971 , -2154826495/19133942 )
**u= 160/23 , ht= 11611 ; ( -11611/7387 , -2086301/29548 ), ( -367193/163279 , -61003451/653116 )
**u= 160/39 , ht= 964807 ; ( -1151263/534569 , -250639939/2138276 ), ( 127711/964807 , 192146843/3859228 )
**u= 160/119 , ht= 340979 ; ( -340979/122747 , -125029145/490988 ), ( -5851159/2731863 , -742665915/3642484 )
**u= 160/151 , ht= 646637 ; ( 821253/245411 , -333114417/981644 ), ( 646637/65781 , -84205499/87708 )
**u= 162/107 , ht= 2311 ; ( -1455/2311 , 1997821/20799 ), ( 4716865/3360351 , -4236787625/30243159 )
**u= 162/155 , ht= 899429 ; ( -899429/352971 , 850704355/3176739 ), ( 12125901/251 , 10676818505/2259 )
**u= 162/187 , ht= 239027 ; ( 166963/419413 , -436577953/3774717 ), ( 239027/160861 , 278637557/1447749 )
**u= 164/39 , ht= 11114 ; ( -25885/73838 , 7634929/147676 ), ( 5303/11114 , -1200869/22228 )
**u= 164/47 , ht= 570 ; ( -570/241 , -66255/482 ), ( 15536818/37373385 , -288879121/4983118 )
**u= 164/119 , ht= 48326 ; ( -2837514/347929 , 487085517/695858 ), ( 48326/6771 , -2771709/4514 )
**u= 164/175 , ht= 6285 ; ( -4894/6285 , 8229791/62850 ), ( 5419/44794 , -9323045/89588 )
**u= 169/6 , ht= 745 ; ( -115/3893 , -972848/50609 ), ( 742/745 , 515149/19370 )
**u= 169/16 , ht= 627 ; ( 1821/1565 , 14772/313 ), ( 205/627 , 87770/2717 )
**u= 169/38 , ht= 14737 ; ( -3375/14737 , 9325230/191581 ), ( -25946/25965 , -3017449/45006 )
**u= 169/44 , ht= 941 ; ( 941/479 , 700357/6227 ), ( -8619/4207 , -6364303/54691 )
**u= 169/86 , ht= 24451 ; ( -99205/48586 , -204898267/1263236 ), ( -2811/24451 , -1755812/24451 )
**u= 169/106 , ht= 108273 ; ( 108273/37889 , 9085698/37889 ), ( -1679639/177831 , 579776078/770601 )
**u= 169/110 , ht= 14933 ; ( -147997/161917 , -230105822/2104921 ), ( 14933/5363 , -1280348/5363 )
**u= 169/114 , ht= 5279 ; ( -43112/26827 , 108444237/697502 ), ( -5279/92 , -11276881/2392 )
**u= 169/120 , ht= 3941 ; ( -108079/35396 , 498383815/1840592 ), ( 2339/3941 , 5020881/51233 )
**u= 169/126 , ht= 18632 ; ( 25721/703 , -28885568/9139 ), ( 18632/16539 , 167819077/1290042 )
**u= 169/134 , ht= 11994 ; ( 149131/463115 , -112661350/1204099 ), ( -11994/3619 , 617233/2002 )
**u= 169/138 , ht= 267415 ; ( 267415/146447 , -358210360/1903811 ), ( -966857/69305 , -227785410/180193 )
**u= 169/140 , ht= 5611 ; ( -451481/39709 , 41255611/39709 ), ( -5611/1711 , -6942085/22243 )
**u= 169/170 , ht= 1179051 ; ( -530299/1179051 , -1685908850/15327663 ), ( -25416089/5423441 , -33890432864/70504733 )
**u= 169/174 , ht= 74171 ; ( -74171/45019 , 114465922/585247 ), ( -45019/74171 , -114465922/964223 )
**u= 170/107 , ht= 204439 ; ( -573/204439 , -16222785/204439 ), ( 735353/4003989 , -107666839/1334663 )
**u= 170/163 , ht= 265843 ; ( -265843/4391 , -26037971/4391 ), ( -3578707/1069159 , 365792891/1069159 )
**u= 170/171 , ht= 1845563 ; ( -1845563/490639 , 191552199/490639 ), ( 10804397/534479 , -1085127275/534479 )
**u= 170/179 , ht= 801487 ; ( -1466997/101401 , -150911289/101401 ), ( -801487/138621 , 27826077/46207 )
**u= 175/36 , ht= 35 ; ( 881/1925 , -480353/9625 ), ( 1/35 , 7937/175 )
**u= 175/128 , ht= 2111 ; ( -2111/473 , -740147/1892 ), ( 6924/1651 , -48710687/132080 )
**u= 175/144 , ht= 1983 ; ( -1981/1983 , 1525799/11898 ), ( 1983/1981 , -1525799/11886 )
**u= 175/158 , ht= 123313 ; ( -2596188/246017 , -2478214593/2460170 ), ( 123313/12488 , 117791033/124880 )
**u= 175/176 , ht= 149 ; ( 105/149 , 182829/1490 ), ( -149/105 , -60943/350 )
**u= 175/198 , ht= 3849 ; ( 551239/19191 , 880165766/287865 ), ( -3849/2311 , 7164254/34665 )
**u= 178/15 , ht= 24499 ; ( -24499/2329 , -712027/2329 ), ( 2389801/221371 , 69922199/221371 )
**u= 178/63 , ht= 3013 ; ( 2495/3013 , 232773/3013 ), ( 2637321/2809829 , -687880813/8429487 )
**u= 180/11 , ht= 102 ; ( 338106/122957 , 53096009/737742 ), ( -41/102 , 16229/612 )
**u= 180/19 , ht= 137 ; ( -34/137 , 27575/822 ), ( 202082/10619 , -39541213/63714 )
**u= 180/83 , ht= 76482 ; ( 76482/45191 , -36197657/271146 ), ( 45191/76482 , -36197657/458892 )
**u= 180/107 , ht= 264966 ; ( 992182/505011 , -39621149/233082 ), ( -264966/204443 , -154844729/1226658 )
**u= 180/131 , ht= 2291586 ; ( -2757627/923866 , -1488792469/5543196 ), ( -2291586/784687 , -1240062913/4708122 )
**u= 182/37 , ht= 401 ; ( 179/401 , 19811/401 ), ( -93867/113 , 4235513/113 )
**u= 182/73 , ht= 1207 ; ( 460573/54515 , 5874553/10903 ), ( 185/1207 , 77335/1207 )
**u= 185/22 , ht= 10363 ; ( -1979/10363 , 364474/10363 ), ( 83406/14293 , -5846897/28586 )
**u= 185/32 , ht= 325753 ; ( 397884/50273 , -133348563/402184 ), ( 325753/283524 , -47908599/756064 )
**u= 185/34 , ht= 119981 ; ( -49492/119981 , 11135525/239962 ), ( -200961/47 , -8622866/47 )
**u= 185/36 , ht= 153461 ; ( -520259/95657 , -23323923/95657 ), ( 153461/126297 , -26309471/378891 )
**u= 185/42 , ht= 9911 ; ( 1658/9911 , 958173/19822 ), ( 9911/1658 , 958173/3316 )
**u= 185/66 , ht= 6887 ; ( 4951/6887 , -506726/6887 ), ( -6887/4951 , 506726/4951 )
**u= 185/68 , ht= 102669 ; ( -102669/68227 , -7474206/68227 ), ( 242929/547707 , -12108887/182569 )
**u= 185/72 , ht= 86553 ; ( -12409/86553 , 16369514/259659 ), ( -177297/102596 , -153384355/1231152 )
**u= 185/78 , ht= 846133 ; ( 846133/92861 , -55272028/92861 ), ( -1014071/853847 , 2968750/29443 )
**u= 185/82 , ht= 447491 ; ( -447491/169723 , -31864996/169723 ), ( -1938187/1476211 , 162233048/1476211 )
**u= 185/84 , ht= 3407519 ; ( -5986877/1220701 , 411733104/1220701 ), ( -3407519/967537 , 238747634/967537 )
**u= 185/86 , ht= 680309 ; ( -1369089/5857343 , 410227746/5857343 ), ( 680309/56883 , 15519432/18961 )
**u= 185/94 , ht= 765467 ; ( -765467/405491 , 61752592/405491 ), ( -2929372/271939 , -419519683/543878 )
**u= 185/108 , ht= 1163437 ; ( 1354019/387773 , 322869668/1163319 ), ( 1163437/576739 , 297706262/1730217 )
**u= 185/118 , ht= 35866 ; ( 207109/54122 , 34195607/108244 ), ( 35866/5953 , -5808443/11906 )
**u= 185/156 , ht= 66283 ; ( -25471/66283 , 6521737/66283 ), ( -2690357/1637291 , 289252348/1637291 )
**u= 185/162 , ht= 88791 ; ( -330769/232257 , 340436386/2090313 ), ( 88791/79012 , -200231255/1422216 )
**u= 185/176 , ht= 20697 ; ( -183867/18379 , 18025467/18379 ), ( -17881/20697 , -889388/6899 )
**u= 194/115 , ht= 220747 ; ( 220747/453 , 16997225/453 ), ( -48469237/2535563 , -3737692175/2535563 )
**u= 194/131 , ht= 554303 ; ( -5054781/2066117 , 448780731/2066117 ), ( 554303/2793 , 15186413/931 )
**u= 194/139 , ht= 1226637 ; ( -4445037/616699 , 379897761/616699 ), ( -500779/1226637 , 37387427/408879 )
**u= 194/171 , ht= 3075025 ; ( -3075025/1232807 , 311075779/1232807 ), ( -15367451/1096125 , -4340114741/3288375 )
**u= 194/187 , ht= 4049465 ; ( 4049465/2131831 , 34566325/163987 ), ( -5044533/845635 , 2714959/4571 )
**u= 197/6 , ht= 295 ; ( -295/7 , -5010/7 ), ( 10603/3445 , 39758/689 )
**u= 197/28 , ht= 3139 ; ( -22151/4889 , 854368/4889 ), ( 3139/2349 , 49294/783 )
**u= 197/30 , ht= 377 ; ( -377/353 , 20156/353 ), ( 235693/4973 , -9210922/4973 )
**u= 197/58 , ht= 104780 ; ( -227468/2116615 , -46223597/846646 ), ( 60027/104780 , -13104563/209560 )
**u= 197/60 , ht= 63833 ; ( 50351987/3707777 , -2786010609/3707777 ), ( 43607/63833 , -4266566/63833 )
**u= 197/70 , ht= 749009 ; ( -749009/117031 , -45188072/117031 ), ( -1157721/238289 , -70481264/238289 )
**u= 197/78 , ht= 287249 ; ( -287249/82471 , 18805248/82471 ), ( -2557717/103955 , 161123624/103955 )
**u= 197/80 , ht= 793799 ; ( 113367/843463 , -54249321/843463 ), ( 793799/51789 , 33805229/34526 )
**u= 197/98 , ht= 17163 ; ( -17163/3013 , 8603718/21091 ), ( -617507/132531 , 103963914/309239 )
**u= 197/100 , ht= 277545 ; ( -176711/277545 , 117232994/1387725 ), ( 381417/61705 , 137674076/308525 )
**u= 197/130 , ht= 1797589 ; ( -218386/1797589 , 294259055/3595178 ), ( 2161915123/165019378 , 352337169161/330038756 )
**u= 197/132 , ht= 312751 ; ( 1807789/602167 , -155989437/602167 ), ( 80959/312751 , 26447254/312751 )
**u= 197/168 , ht= 114707 ; ( -39643/114707 , 11209569/114707 ), ( 941777/3287 , -86986591/3287 )
**u= 197/180 , ht= 103289 ; ( -53321/103289 , -11113006/103289 ), ( 3570363/160283 , 1025066492/480849 )
**u= 200/43 , ht= 25881 ; ( -25881/18409 , 14726247/184090 ), ( -18409/25881 , -4908749/86270 )
**u= 200/91 , ht= 2643139 ; ( -179522639/7630327 , -24241513855/15260654 ), ( 2643139/74573 , 356817343/149146 )
587
>
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。

[2026.06.21追記]u=854/9のときの整数解を追加した。

[参考文献]

Last Update: 2026.06.21
H.Nakao

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