Homeに戻る  一覧に戻る 

Integer Points on A^4+20010*B^4=C^4+20010*D^4

[2026.07.02]A^4+20010*B^4=C^4+20010*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=20010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、13個の整数解
    379^4+20010*40^4=517^4+20010*12^4
    1843^4+20010*3477^4=9059^4+20010*3475^4
    2078^4+20010*4253^4=45946^4+20010*3197^4
    2762^4+20010*4441^4=51062^4+20010*2649^4
    3375^4+20010*3487^4=40785^4+20010*1759^4
    3623^4+20010*1854^4=21977^4+20010*638^4
    4450^4+20010*781^4=8890^4+20010*531^4
    5003^4+20010*2753^4=11971^4+20010*2741^4
    5903^4+20010*1311^4=15671^4+20010*163^4
    14951^4+20010*1243^4=16193^4+20010*1097^4
    20031^4+20010*1636^4=23481^4+20010*362^4
    53540^4+20010*6369^4=77560^4+20010*3967^4
    78881^4+20010*6180^4=87713^4+20010*4568^4
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をいくつか求めると、
    u = 1104/13, 1794, 2300/39, 5106/41, 5658, 8625/68, 12053/40, 13800/89, 51185/36, 74037/410, 290145/1418
となる。
[gp2cによる計算]
gp > sss(20010,100000)
1:379^4+20010*40^4=517^4+20010*12^4
2:1843^4+20010*3477^4=9059^4+20010*3475^4
3:2078^4+20010*4253^4=45946^4+20010*3197^4
4:2762^4+20010*4441^4=51062^4+20010*2649^4
5:3375^4+20010*3487^4=40785^4+20010*1759^4
6:3623^4+20010*1854^4=21977^4+20010*638^4
7:4450^4+20010*781^4=8890^4+20010*531^4
8:5003^4+20010*2753^4=11971^4+20010*2741^4
9:5903^4+20010*1311^4=15671^4+20010*163^4
10:14951^4+20010*1243^4=16193^4+20010*1097^4
11:20031^4+20010*1636^4=23481^4+20010*362^4
12:53540^4+20010*6369^4=77560^4+20010*3967^4
13:78881^4+20010*6180^4=87713^4+20010*4568^4
13:78881^4+20010*6180^4=87713^4+20010*4568^4
13 solutions,
time = 24h, 40min, 15,384 ms.
%1 = 13
{Pari/GPによる計算]
gp > uu(20010, 379, 40, 517, 12)
[u, -1; 13*u - 1104, 1; 56*u + 15805, 1]
[u, -1; 13*u - 1104, 1; 56*u - 15805, 1]
[u, -1; 13*u + 1104, 1; 56*u + 15805, 1]
time = 1 ms.
%1 = [1104/13, -15805/56]
gp > uu(20010, 1843, 3477, 9059, 3475)
[u - 5658, 1; u, -1; 6952*u + 42731065, 1]
[u - 5658, 1; u, -1; 6952*u - 42731065, 1]
[u, -1; u + 5658, 1; 6952*u + 42731065, 1]
time = 1 ms.
%2 = [5658, -42731065/6952]
gp > uu(20010, 2078, 4253, 45946, 3197)
[u, -1; 264*u + 70985, 1; 7450*u - 1994997, 1]
[u, -1; 264*u - 70985, 1; 7450*u - 1994997, 1]
[u, -1; 264*u + 70985, 1; 7450*u + 1994997, 1]
time = 1 ms.
%3 = [-70985/264, 1994997/7450]
gp > uu(20010,2762,4441,51062,2649)
[u, -1; 1418*u - 290145, 1; 2240*u + 461029, 1]
[u, -1; 1418*u - 290145, 1; 2240*u - 461029, 1]
[u, -1; 1418*u + 290145, 1; 2240*u + 461029, 1]
time = 1 ms.
%4 = [290145/1418, -461029/2240]
gp > uu(20010,3375,3487,40785,1759)
[u, -1; 183*u - 33350, 1; 8256*u + 1525325, 1]
[u, -1; 183*u - 33350, 1; 8256*u - 1525325, 1]
[u, -1; 183*u + 33350, 1; 8256*u + 1525325, 1]
time = 1 ms.
%5 = [33350/183, -1525325/8256]
gp > uu(20010,3623,1854,21977,638)
[u, -1; 89*u - 13800, 1; 17024*u + 2787161, 1]
[u, -1; 89*u - 13800, 1; 17024*u - 2787161, 1]
[u, -1; 89*u + 13800, 1; 17024*u + 2787161, 1]
time = 1 ms.
%6 = [13800/89, -2787161/17024]
gp > uu(20010,4450,781,8890,531)
[u, -1; 40*u + 12053, 1; 410*u - 74037, 1]
[u, -1; 40*u - 12053, 1; 410*u - 74037, 1]
[u, -1; 40*u + 12053, 1; 410*u + 74037, 1]
time = 1 ms.
%7 = [-12053/40, 74037/410]
gp > uu(20010,5003,2753,11971,2741)
[u - 1794, 1; u, -1; 32964*u + 84167425, 1]
[u - 1794, 1; u, -1; 32964*u - 84167425, 1]
[u, -1; u + 1794, 1; 32964*u + 84167425, 1]
time = 1 ms.
%8 = [1794, -84167425/32964]
gp > uu(20010,5903,1311,15671,163)
[u, -1; 41*u - 5106, 1; 20636*u + 3419825, 1]
[u, -1; 41*u - 5106, 1; 20636*u - 3419825, 1]
[u, -1; 41*u + 5106, 1; 20636*u + 3419825, 1]
time = 1 ms.
%9 = [5106/41, -3419825/20636]
gp > uu(20010,14951,1243,16193,1097)
[u, -1; 36*u + 51185, 1; 4745*u - 537234, 1]
[u, -1; 36*u - 51185, 1; 4745*u - 537234, 1]
[u, -1; 36*u + 51185, 1; 4745*u + 537234, 1]
time = 1 ms.
%10 = [-51185/36, 537234/4745]
gp > uu(20010,20031,1636,23481,362)
[u, -1; 39*u - 2300, 1; 10878*u + 4070933, 1]
[u, -1; 39*u - 2300, 1; 10878*u - 4070933, 1]
[u, -1; 39*u + 2300, 1; 10878*u + 4070933, 1]
time = 1 ms.
%11 = [2300/39, -4070933/10878]
gp > uu(20010,53540,6369,77560,3967)
[u, -1; 68*u - 8625, 1; 91276*u + 32654725, 1]
[u, -1; 68*u - 8625, 1; 91276*u - 32654725, 1]
[u, -1; 68*u + 8625, 1; 91276*u + 32654725, 1]
time = 1 ms.
%12 = [8625/68, -32654725/91276]
gp > uu(20010,78881,6180,87713,4568)
[u, -1; 13*u - 1104, 1; 666376*u + 535222405, 1]
[u, -1; 13*u - 1104, 1; 666376*u - 535222405, 1]
[u, -1; 13*u + 1104, 1; 666376*u + 535222405, 1]
time = 1 ms.
%13 = [1104/13, -535222405/666376]

■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1000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、
このように16564個のuが抽出される。
[MAGMAによる計算]
> PP(20010,1,1000);
**u= 1/4 , ht= 661 ; ( -661/453 , 160306/453 ), ( -663/451 , -160411/451 )
**u= 1/6 , ht= 1273 ; ( -3839/103 , 940932/103 ), ( -1273/119 , -313258/119 )
**u= 1/14 , ht= 1669 ; ( -4103/9736 , 7908279/19472 ), ( 1669/459 , 647830/459 )
**u= 1/16 , ht= 3159 ; ( 6463/7761 , 310837/597 ), ( -3159/2749 , -3350945/5498 )
**u= 1/20 , ht= 2741 ; ( 18223/21533 , 12618603/21533 ), ( 2741/1161 , 1331573/1161 )
**u= 1/22 , ht= 9673 ; ( 2215/9673 , -4655634/9673 ), ( 9673/2215 , -4655634/2215 )
**u= 1/36 , ht= 549 ; ( -73/549 , -997147/1647 ), ( -947/7139 , 4322002/7139 )
**u= 1/46 , ht= 4756 ; ( -62113/4656 , 84511673/9312 ), ( 4756/317 , -6467283/634 )
**u= 1/58 , ht= 1367 ; ( 5980621/2944205 , -5077978458/2944205 ), ( 663/1367 , 1157350/1367 )
**u= 1/72 , ht= 1795 ; ( -46664/7435 , -160420729/29740 ), ( -1795/381 , 4672288/1143 )
**u= 1/74 , ht= 3653 ; ( 55826/65927 , 148664937/131854 ), ( 3653/2506 , -7623477/5012 )
**u= 1/78 , ht= 231311 ; ( 231311/71996 , 428017225/143992 ), ( 255713/19217 , -226533050/19217 )
**u= 1/84 , ht= 482773 ; ( 583313/155459 , -553414097/155459 ), ( 482773/91381 , 450438024/91381 )
**u= 1/86 , ht= 3647 ; ( 25213/4186 , 47415117/8372 ), ( 3647/1583 , 3687870/1583 )
**u= 1/92 , ht= 75459 ; ( 75459/65 , -72395849/65 ), ( 189433/38553 , -185468840/38553 )
**u= 1/116 , ht= 724617 ; ( -724617/115339 , -112922657/16477 ), ( 21768187/675609 , -23462207503/675609 )
**u= 1/142 , ht= 10197 ; ( -2245/10197 , 1131386/927 ), ( -776644/77365 , -1860582141/154730 )
**u= 1/144 , ht= 2273 ; ( -1301/2273 , 6287157/4546 ), ( -14309/25001 , 69152175/50002 )
**u= 1/146 , ht= 202404 ; ( -3912732/1163 , 9457908125/2326 ), ( -115511/202404 , -563320495/404808 )
**u= 1/156 , ht= 25513 ; ( -172909/108593 , -255085983/108593 ), ( -8183/25513 , -33473040/25513 )
**u= 1/160 , ht= 8001 ; ( 26302/12997 , 296953719/103976 ), ( -8001/4999 , 23873065/9998 )
**u= 1/162 , ht= 229661 ; ( -1321616/116701 , -3378219855/233402 ), ( -223829/229661 , 408276780/229661 )
**u= 1/166 , ht= 75814 ; ( 12817/75814 , 198180627/151628 ), ( 336093/39451 , 436107508/39451 )
**u= 1/178 , ht= 127573 ; ( -96872/524517 , 203373035/149862 ), ( 7245/127573 , 170520476/127573 )
**u= 1/182 , ht= 58185 ; ( -58185/43553 , -98074964/43553 ), ( -43553/58185 , 98074964/58185 )
**u= 1/200 , ht= 2525 ; ( 2525/1189 , 19739859/5945 ), ( 144155/6371 , -1020579843/31855 )
**u= 1/202 , ht= 546531 ; ( -1246558/341263 , -3674685825/682526 ), ( 546531/8605 , 777056962/8605 )
**u= 1/218 , ht= 3846 ; ( -864914/162629 , 2599467225/325258 ), ( -1931/3846 , 12711385/7692 )
**u= 1/222 , ht= 234595 ; ( 394805/172028 , 1283646873/344056 ), ( 213653/234595 , 472891876/234595 )
**u= 1/224 , ht= 118444 ; ( 680263/29551 , 2038678719/59102 ), ( -118444/46357 , 1523296071/370856 )
**u= 1/240 , ht= 457219 ; ( -68531/457219 , -16660745/10633 ), ( 934003/319117 , -1529458228/319117 )
**u= 1/242 , ht= 74135 ; ( -116760/25519 , -4091340349/561418 ), ( 74135/1848 , 2538616459/40656 )
**u= 1/244 , ht= 75339 ; ( 28613/75339 , 125916302/75339 ), ( -716021/140053 , 1139940192/140053 )
**u= 1/250 , ht= 4489 ; ( 305903/60233 , -2465426712/301165 ), ( -2271/4489 , -39781604/22445 )
**u= 1/266 , ht= 242101 ; ( 242101/151243 , -465687342/151243 ), ( 34558543/14142777 , 60915694120/14142777 )
**u= 1/270 , ht= 801959 ; ( 1951999/30169 , -9625939724/90507 ), ( 124364/801959 , -8003023139/4811754 )
**u= 1/278 , ht= 282155 ; ( -423433/84890 , 1440464217/169780 ), ( -282155/53669 , 479000922/53669 )
**u= 1/286 , ht= 944496 ; ( 1327519/121457 , -2254978602/121457 ), ( 944496/37447 , -3197884357/74894 )
**u= 1/298 , ht= 103883 ; ( 117107/2441805 , 4221108746/2441805 ), ( -103883/49885 , 198984312/49885 )
**u= 1/312 , ht= 469033 ; ( 469033/75640 , 3357565333/302560 ), ( -6772619/630379 , 12017541410/630379 )
**u= 1/330 , ht= 36689 ; ( 771209/433289 , 1607342518/433289 ), ( 36689/17519 , -73875714/17519 )
**u= 1/342 , ht= 2591 ; ( 2591/1774 , -11617105/3548 ), ( 3508505/305751 , -19543746442/917253 )
**u= 1/350 , ht= 5169 ; ( -5169/1193 , 49635064/5965 ), ( 214864/213177 , 5663909609/2131770 )
**u= 1/352 , ht= 46927 ; ( -46265/46927 , -247334367/93854 ), ( 991950/182999 , 15143530507/1463992 )
**u= 1/368 , ht= 146167 ; ( 198441/484055 , 1003830512/484055 ), ( 87287/146167 , -326670720/146167 )
**u= 1/382 , ht= 267335 ; ( -3570403/619755 , -7084408228/619755 ), ( 192081/267335 , -643548028/267335 )
**u= 1/390 , ht= 2535059 ; ( 874703473/53552818 , 3462136000469/107105636 ), ( -1337381/2535059 , -5661709684/2535059 )
**u= 1/400 , ht= 57129 ; ( -57131/56031 , 320165603/112062 ), ( -57129/56029 , 320154287/112058 )
**u= 1/424 , ht= 6568567 ; ( -6568567/309879 , 13543934668/309879 ), ( -50457117/300506 , -415701508601/1202024 )
**u= 1/426 , ht= 695198 ; ( 54982/1881031 , 7770066593/3762062 ), ( -92689/695198 , 2895862953/1390396 )
**u= 1/434 , ht= 641709 ; ( -641709/73957 , -1346036476/73957 ), ( -2695187/1993749 , -6985838824/1993749 )
**u= 1/464 , ht= 50619 ; ( 93164677/20872607 , 411418255125/41745214 ), ( -11963/50619 , 112068356/50619 )
**u= 1/482 , ht= 46247 ; ( -651160/3067711 , 13773558411/6135422 ), ( 46247/38535 , 132193486/38535 )
**u= 1/484 , ht= 95801 ; ( -14039/95803 , 2343779181/1053833 ), ( -14041/95801 , -2343738300/1053811 )
**u= 1/498 , ht= 25721 ; ( 21340/25721 , 149201463/51442 ), ( 22138303/19294640 , 131100534791/38589280 )
**u= 1/502 , ht= 2425 ; ( -1591/2425 , -6499914/2425 ), ( -2425/1591 , 6499914/1591 )
**u= 1/510 , ht= 336797 ; ( -4373681/917569 , 10094707510/917569 ), ( -336797/67123 , 775746798/67123 )
**u= 1/512 , ht= 52536 ; ( -112024/137761 , -12859913985/4408352 ), ( -15671/52536 , -3970634665/1681152 )
**u= 1/520 , ht= 125471 ; ( 37279/125471 , -298554120/125471 ), ( 125471/37279 , -298554120/37279 )
**u= 1/524 , ht= 262163 ; ( -706921/24583 , 1619599026/24583 ), ( 262163/90291 , 634872116/90291 )
**u= 1/540 , ht= 7909607 ; ( -9478403/19210867 , 149377278662/57632601 ), ( 7909607/121607 , -55161098900/364821 )
**u= 1/546 , ht= 42067 ; ( 3317284/666181 , 15816227505/1332362 ), ( -42067/25988 , -231139975/51976 )
**u= 1/552 , ht= 276350 ; ( -218066/619801 , 6176353565/2479204 ), ( -8827/276350 , 2599075991/1105400 )
**u= 1/554 , ht= 3661321 ; ( 7892429/1560763 , 18941057004/1560763 ), ( -3661321/1854857 , 9662929482/1854857 )
**u= 1/564 , ht= 267031 ; ( -267031/4583 , -634415426/4583 ), ( 20753629/1599343 , 49445630217/1599343 )
**u= 1/572 , ht= 110843 ; ( -751515/876683 , -2762347414/876683 ), ( -96023/110843 , -350826435/110843 )
**u= 1/580 , ht= 1190809 ; ( 777559/1190809 , -3425942970/1190809 ), ( -19071843/1609217 , 46105813471/1609217 )
**u= 1/588 , ht= 1723 ; ( -1723/103 , 29305905/721 ), ( -103/1723 , -29305905/12061 )
**u= 1/602 , ht= 3379485 ; ( -3379485/1600861 , 9177353278/1600861 ), ( 10765259/1843619 , -26804537880/1843619 )
**u= 1/606 , ht= 222967 ; ( -6059867/693131 , 15018634434/693131 ), ( -107303/222967 , -609285250/222967 )
**u= 1/608 , ht= 189497 ; ( 161985/189497 , -1229715919/378994 ), ( 545422/543123 , -15187373765/4344984 )
**u= 1/610 , ht= 3707213 ; ( 274133/3707213 , 9183434388/3707213 ), ( -39246229/189356 , -193912786959/378712 )
**u= 1/618 , ht= 270713 ; ( 952121/991055 , -3417331272/991055 ), ( 270713/156383 , 777394660/156383 )
**u= 1/634 , ht= 112461 ; ( -1911409/303153 , 4874178986/303153 ), ( 112461/59522 , -640930919/119044 )
**u= 1/636 , ht= 2752589 ; ( 3074323/55549 , -7756351474/55549 ), ( -2752589/1520377 , -7932274869/1520377 )
**u= 1/664 , ht= 681674 ; ( -302623/681674 , 7689383847/2726696 ), ( -688053/805283 , 2730041215/805283 )
**u= 1/672 , ht= 377150 ; ( -377150/340777 , -10543993403/2726216 ), ( 898885/1391986 , -34371883167/11135888 )
**u= 1/676 , ht= 16507 ; ( -81677231/5659603 , -212923217907/5659603 ), ( 10037/16507 , 38420250/12623 )
**u= 1/684 , ht= 25901 ; ( -7273/25901 , 70377427/25901 ), ( -1529149/165613 , 4023635818/165613 )
**u= 1/686 , ht= 30399 ; ( 30399/14191 , 87890150/14191 ), ( 1603652/180097 , 59187549495/2521358 )
**u= 1/718 , ht= 438985 ; ( 20971649/9910500 , -124337815289/19821000 ), ( -186393/438985 , 1278244438/438985 )
**u= 1/722 , ht= 66961 ; ( 2576043/46475 , 131569287022/883025 ), ( -66961/4551 , 3427310260/86469 )
**u= 1/740 , ht= 354989 ; ( 20717397/771067 , -56410529966/771067 ), ( -354989/254771 , -1188930096/254771 )
**u= 1/766 , ht= 9899568 ; ( 14326831/550704 , 11340328355/157344 ), ( -9899568/2922889 , -57150351629/5845778 )
**u= 1/774 , ht= 141727 ; ( 141727/36121 , 407002712/36121 ), ( -552281/294367 , -91660768/15493 )
**u= 1/776 , ht= 148793 ; ( 1137367/1022519 , 4261557294/1022519 ), ( -148793/5049 , 414831568/5049 )
**u= 1/788 , ht= 265225 ; ( -135079/265225 , 835729182/265225 ), ( -4922509/8044031 , 26479768635/8044031 )
**u= 1/816 , ht= 291217 ; ( 291217/76459 , 1720582407/152918 ), ( 766811/835423 , 6480259593/1670846 )
**u= 1/818 , ht= 546435 ; ( -546435/65429 , -1574399264/65429 ), ( -873515/841013 , -3468908166/841013 )
**u= 1/820 , ht= 47649351 ; ( -52813591/619659 , 151283325085/619659 ), ( -47649351/5366519 , 137343949334/5366519 )
**u= 1/826 , ht= 4881862 ; ( -4567298/7866991 , -4754657607/1430362 ), ( -813989/4881862 , -28455659727/9763724 )
**u= 1/828 , ht= 1073777 ; ( 397485/1073777 , -9886552711/3221331 ), ( -11050303/1051745 , 31948917432/1051745 )
**u= 1/842 , ht= 385361 ; ( -1330746/164225 , -7783445537/328450 ), ( -385361/116050 , 2336215071/232100 )
**u= 1/844 , ht= 549581 ; ( 428399/662187 , -2291822669/662187 ), ( 334783/549581 , 1870003041/549581 )
**u= 1/852 , ht= 521957 ; ( 1747241/2196245 , 8193900011/2196245 ), ( -521957/274663 , -1722036575/274663 )
**u= 1/868 , ht= 1516141 ; ( 111731/1516141 , -407278440/137831 ), ( 7427915/261077 , 21902991264/261077 )
**u= 1/872 , ht= 298903 ; ( -238066/537251 , 6942788085/2149004 ), ( 187505/298903 , 1042205673/298903 )
**u= 1/894 , ht= 3342709 ; ( -3342709/1037444 , -20935109215/2074888 ), ( 3694592/1316999 , -23461130973/2633998 )
**u= 1/896 , ht= 597171 ; ( -597171/513211 , -9430135735/2052844 ), ( 10321111/390729 , -61848615515/781458 )
**u= 1/900 , ht= 16303 ; ( -11003/16303 , -495971/137 ), ( -16303/11003 , 59020549/11003 )
**u= 1/904 , ht= 18098681 ; ( -93067999/56391903 , 327265159924/56391903 ), ( 18098681/317792 , -217754238687/1271168 )
**u= 1/910 , ht= 98863 ; ( -6929423/4959272 , 51423459489/9918544 ), ( 98863/66887 , -360165900/66887 )
**u= 1/916 , ht= 3278349 ; ( 2421889/3278349 , -12339053995/3278349 ), ( -55559853/17834189 , -176649335272/17834189 )
**u= 1/926 , ht= 8713456 ; ( -23458121/18214657 , -90398746248/18214657 ), ( -8713456/132159 , -53049882445/264318 )
**u= 1/950 , ht= 173882 ; ( -56645/173882 , 5638023711/1738820 ), ( -173882/56645 , -5638023711/566450 )
**u= 1/952 , ht= 45 ; ( -836107633/3425526912 , -43529171729275/13702107648 ), ( -45/11 , -142969/11 )
**u= 1/966 , ht= 236849 ; ( 3304051/1274002 , 22017753279/2548004 ), ( -236849/151897 , -874737684/151897 )
**u= 1/968 , ht= 352896 ; ( 984616/112451 , -135699994275/4947844 ), ( -137531/352896 , 51862055675/15527424 )
**u= 1/980 , ht= 363191 ; ( 363191/97111 , 8240422464/679777 ), ( -69277/747633 , -2351075407/747633 )
**u= 1/988 , ht= 204521 ; ( 849089/221215 , -2758676988/221215 ), ( 7415/204521 , -643443132/204521 )
**u= 1/994 , ht= 1915527 ; ( -103784/1915527 , -12099196147/3831054 ), ( 943650167/47265937 , -2979592463760/47265937 )
**u= 1/998 , ht= 87349 ; ( -87349/30914 , 45044715/4756 ), ( 402815/614391 , -2321478152/614391 )
**u= 2/13 , ht= 201 ; ( 201/7 , 7327 ), ( 3941/4899 , 1603385/4899 )
**u= 2/45 , ht= 5097 ; ( -4903/5097 , 10066715/15291 ), ( 69379/29869 , 35838563/29869 )
**u= 2/77 , ht= 74069 ; ( -23963/74069 , -48316011/74069 ), ( -83547/135067 , 3179635/4357 )
**u= 2/117 , ht= 100021 ; ( -100021/4837 , -76609903/4837 ), ( 125431/42959 , -101432305/42959 )
**u= 2/133 , ht= 290501 ; ( 1587819/105667 , 1298015621/105667 ), ( 172397/290501 , -275539701/290501 )
**u= 2/149 , ht= 215897 ; ( -49025/215897 , 191139759/215897 ), ( 7146523/1338205 , 6277185819/1338205 )
**u= 2/157 , ht= 320447 ; ( 320447/25601 , -284892591/25601 ), ( 360851/107139 , -333592565/107139 )
**u= 2/205 , ht= 785339 ; ( 785339/359021 , -874458063/359021 ), ( 3629723/553493 , -3718223211/553493 )
**u= 2/213 , ht= 1967581 ; ( -2746049/720313 , 2930495519/720313 ), ( 1734133/1967581 , 2707279741/1967581 )
**u= 2/245 , ht= 22377 ; ( -652013/31867 , 5058817077/223069 ), ( -22377/3103 , -25010039/3103 )
**u= 2/253 , ht= 10561 ; ( 44763/28571 , -59742107/28571 ), ( -10561/6183 , -13767587/6183 )
**u= 2/341 , ht= 1560793 ; ( -303113/1560793 , 2076611115/1560793 ), ( 1995225/220351 , 2621773073/220351 )
**u= 2/413 , ht= 1051559 ; ( 4194221/1193363 , 6267922941/1193363 ), ( -293169/1051559 , -1569121195/1051559 )
**u= 2/429 , ht= 531983 ; ( -3857141/1648645 , 6145027461/1648645 ), ( 85537/531983 , -789337135/531983 )
**u= 2/437 , ht= 263019 ; ( -52324283/1897803 , 77414626975/1897803 ), ( -263019/2677 , -388905319/2677 )
**u= 2/445 , ht= 5803 ; ( 65233703/303849823 , 463679071629/303849823 ), ( -5803/1317 , 8878349/1317 )
**u= 2/453 , ht= 487687 ; ( -437641/487687 , 986410339/487687 ), ( 566497/1196167 , -1992401095/1196167 )
**u= 2/501 , ht= 5984057 ; ( 5984057/1357823 , 9714266655/1357823 ), ( -47894531/2085229 , 75894370105/2085229 )
**u= 2/549 , ht= 35141 ; ( -14269/35141 , 62854125/35141 ), ( 60776455/25772337 , 328204198259/77317011 )
**u= 2/597 , ht= 14154403 ; ( 14154403/6762349 , -27109142119/6762349 ), ( -32389843/540677 , -55982212285/540677 )
**u= 2/613 , ht= 6543287 ; ( 6543287/2840767 , -12491563665/2840767 ), ( -290641259/2783451 , -508980293095/2783451 )
**u= 2/621 , ht= 601343 ; ( 4224785/695527 , -22639781609/2086581 ), ( 601343/334313 , 1212672845/334313 )
**u= 2/661 , ht= 1878505 ; ( 43209667/1922285 , -78651051831/1922285 ), ( -1878505/204967 , -3436185381/204967 )
**u= 2/701 , ht= 64535 ; ( 7596163/2544757 , -15001817895/2544757 ), ( -30801/64535 , 133909171/64535 )
**u= 2/733 , ht= 9214327 ; ( -5557089/9214327 , 20604961099/9214327 ), ( -13849717/2119421 , 26829525609/2119421 )
**u= 2/805 , ht= 204579 ; ( -10200331/1698059 , -1886465667/154369 ), ( 25421/204579 , -413694545/204579 )
**u= 2/949 , ht= 411461 ; ( 411461/252029 , -1051321755/252029 ), ( -60703/2550945 , 5559691291/2550945 )
**u= 2/957 , ht= 1183019 ; ( 8055739/3484933 , -19204666843/3484933 ), ( 1183019/216797 , -2631564231/216797 )

...省略...

**u= 1000/37 , ht= 73255 ; ( 73255/31011 , 15109601/310110 ), ( 696867/120185 , -10644469/92450 )
**u= 1000/49 , ht= 2207 ; ( 44521/3519 , -68503349/246330 ), ( 801/2207 , -721957/30898 )
**u= 1000/77 , ht= 7393 ; ( -5235/7393 , -2517857/73930 ), ( 894793/1285 , -249401793/12850 )
**u= 1000/121 , ht= 6193 ; ( -6193/2873 , -26100093/316030 ), ( -32583/1919 , -25026593/42218 )
**u= 1000/169 , ht= 77713 ; ( 77713/46953 , -485159627/6103890 ), ( -130177/16439 , -140398599/427414 )
**u= 1000/201 , ht= 130393 ; ( 2482069/222829 , 1116860851/2228290 ), ( -130393/55063 , -63500997/550630 )
**u= 1000/213 , ht= 3755 ; ( 3755/1057 , 257137/1510 ), ( -1080989/617473 , 114971199/1234946 )
**u= 1000/253 , ht= 9767 ; ( 6731/9767 , 1193733/19534 ), ( 83453/284335 , -149039049/2843350 )
**u= 1000/329 , ht= 229763 ; ( -707037/288517 , 438048829/2885170 ), ( -229763/99237 , -143614301/992370 )
**u= 1000/369 , ht= 38721 ; ( 38721/15761 , 76194299/472830 ), ( -42887/27633 , -93004247/828990 )
**u= 1000/409 , ht= 208979 ; ( -39181/1609741 , 147170769/2299630 ), ( 78099/208979 , -142696183/2089790 )
**u= 1000/481 , ht= 64109 ; ( -36901/64109 , 51317289/641090 ), ( 9342603/6021779 , -1542214877/12043558 )
**u= 1000/493 , ht= 26319 ; ( 25333/26319 , -5131153/52638 ), ( 341613/62575 , 243935239/625750 )
**u= 1000/517 , ht= 46591 ; ( 211629155/3793087 , 152215361757/37930870 ), ( 3163/46591 , 6716517/93182 )
**u= 1000/569 , ht= 82153 ; ( 82153/27529 , -13073745/55058 ), ( 111583/635063 , -486465801/6350630 )
**u= 1000/681 , ht= 1066441 ; ( 1066441/347759 , 925856519/3477590 ), ( -400493/1578493 , -1344160537/15784930 )
**u= 1000/689 , ht= 4889 ; ( 17799/320959 , -266905307/3209590 ), ( -4889/2001 , 4386173/20010 )
**u= 1000/693 , ht= 391267 ; ( -215031/391267 , 223060615/2347602 ), ( -391267/215031 , 223060615/1290186 )
**u= 1000/721 , ht= 3101787 ; ( 1753517/3101787 , 3026347297/31017870 ), ( 44437389/4442077 , -7586373985/8884154 )
**u= 1000/733 , ht= 296215 ; ( 296215/101237 , 268065201/1012370 ), ( 43174171/23839097 , -8447163357/47678194 )
**u= 1000/761 , ht= 371687 ; ( -509251/2193821 , -1965238809/21938210 ), ( -371687/46863 , -326904697/468630 )
**u= 1000/797 , ht= 2156323 ; ( -843585/2156323 , -2067701887/21563230 ), ( 8453171/2998503 , -1601903015/5997006 )
**u= 1000/841 , ht= 12509 ; ( -5189/12509 , -360259959/3627610 ), ( 12509/5189 , -360259959/1504810 )
**u= 1000/849 , ht= 3412501 ; ( 3886793/1441673 , -3820613319/14416730 ), ( 3412501/775507 , -645079547/1551014 )
**u= 1000/853 , ht= 286061 ; ( 286061/2865 , 264270437/28650 ), ( 138615/401339 , 392241613/4013390 )
**u= 1000/917 , ht= 1167695 ; ( 434619/1167695 , -1193453591/11676950 ), ( -87026525/2226919 , -83387294841/22269190 )
16564
>
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。

[2026.07.03追記]u=230/87,345/2,12053/406のときの整数解を追加した。
[2026.07.04追記]u=598/3のときの整数解を追加した。

[参考文献]

Last Update: 2026.07.04
H.Nakao

Homeに戻る[Homeに戻る]  一覧に戻る[一覧に戻る]