Homeに戻る  一覧に戻る 

Integer Points on A^4+9*B^4=C^4+9*D^4

[2026.05.23]A^4+9*B^4=C^4+9*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=9とする。

■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100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307個のuが抽出される。
[MAGMAによる計算]
> PP(9,1,100);
**u= 1 , ht= 1 ; ( 1 , -3 ), ( 0 , 2 )
**u= 1/4 , ht= 17 ; ( 17 , 72 ), ( 3/17 , 73/17 )
**u= 1/8 , ht= 5 ; ( -3/5 , -7 ), ( 5/3 , -35/3 )
**u= 1/9 , ht= 1 ; ( -1 , -9 ), ( -6/7 , 176/21 )
**u= 1/16 , ht= 43 ; ( 43/21 , -58/3 ), ( 21/43 , -406/43 )
**u= 1/17 , ht= 100 ; ( 189/100 , 187/10 ), ( -100/53 , -990/53 )
**u= 1/25 , ht= 1 ; ( -1 , -15 ), ( -78/109 , -7108/545 )
**u= 1/33 , ht= 280 ; ( 13/1120 , -195/16 ), ( -280/247 , -350/19 )
**u= 1/36 , ht= 149 ; ( 23/149 , -1919/149 ), ( 21/151 , -5821/453 )
**u= 1/49 , ht= 1 ; ( -1 , -21 ), ( -36/37 , 5366/259 )
**u= 1/56 , ht= 85 ; ( 165/52 , 845/16 ), ( 13/85 , 273/17 )
**u= 1/60 , ht= 437 ; ( 2381/2591 , -57821/2591 ), ( 437/367 , -9377/367 )
**u= 1/64 , ht= 267 ; ( 37/267 , -9149/534 ), ( 35/269 , -9207/538 )
**u= 1/65 , ht= 328 ; ( -153/328 , -3095/164 ), ( 952/3 , -16282/3 )
**u= 1/80 , ht= 7 ; ( 987/37 , -18740/37 ), ( -7/3 , 289/6 )
**u= 1/81 , ht= 99 ; ( -25/99 , -1595/81 ), ( 132/185 , -7810/333 )
**u= 1/100 , ht= 303 ; ( -2099/297 , -224849/1485 ), ( 303/299 , 45151/1495 )
**u= 3/2 , ht= 1 ; ( 7/3 , -4 ), ( 0 , 3/2 )
**u= 3/26 , ht= 90 ; ( 1/90 , -25/4 ), ( 135/187 , 1440/187 )
**u= 3/38 , ht= 34 ; ( -1598/21 , -8041/14 ), ( -15/34 , 33/4 )
**u= 3/49 , ht= 1 ; ( 1801/60 , -1802/7 ), ( 0 , 60/7 )
**u= 3/62 , ht= 681 ; ( -1372/411 , -9207/274 ), ( 681/404 , 15273/808 )
**u= 3/74 , ht= 511 ; ( 1289/1005 , -1148/67 ), ( 30/511 , 10785/1022 )
**u= 4 , ht= 1 ; ( -5/3 , -5/6 ), ( -1 , -3/2 )
**u= 4/5 , ht= 7 ; ( -3/7 , -37/14 ), ( -263/27 , -1297/54 )
**u= 4/9 , ht= 1 ; ( 31/15 , -653/90 ), ( 1 , -9/2 )
**u= 4/21 , ht= 827 ; ( 671/2099 , 21457/4198 ), ( -827/223 , -8341/446 )
**u= 4/25 , ht= 1 ; ( 79/63 , -5357/630 ), ( 1 , -15/2 )
**u= 4/49 , ht= 1 ; ( -1 , -21/2 ), ( 43/205 , 4353/574 )
**u= 4/81 , ht= 1 ; ( -1 , 27/2 ), ( -2765/533 , 161303/3198 )
**u= 4/85 , ht= 5119 ; ( 5119/2059 , 107901/4118 ), ( -237/5713 , 111815/11426 )
**u= 5/3 , ht= 2 ; ( -23/104 , -99/52 ), ( -2 , -4 )
**u= 5/19 , ht= 82 ; ( 51/82 , 200/41 ), ( 74/97 , -504/97 )
**u= 5/51 , ht= 754 ; ( -7571/1388 , -26061/694 ), ( 754/113 , 5168/113 )
**u= 5/83 , ht= 4076 ; ( -20539/4852 , -91185/2426 ), ( 4076/323 , 35346/323 )
**u= 6/11 , ht= 6 ; ( -5/6 , -15/4 ), ( 399/410 , -657/164 )
**u= 6/35 , ht= 29 ; ( 83/78 , 389/52 ), ( 6/29 , 303/58 )
**u= 6/71 , ht= 68 ; ( -343/12 , 1669/8 ), ( -3/68 , -993/136 )
**u= 9 , ht= 1 ; ( 1/2 , 2 ), ( 1 , 1 )
**u= 9/4 , ht= 1 ; ( 1 , -2 ), ( -11/5 , -4 )
**u= 9/16 , ht= 559 ; ( -559/493 , -4207/986 ), ( -595/529 , -4513/1058 )
**u= 9/25 , ht= 1 ; ( -103/76 , -1129/190 ), ( -1 , -5 )
**u= 9/40 , ht= 22 ; ( 17/47 , 224/47 ), ( 17/22 , 497/88 )
**u= 9/47 , ht= 337 ; ( 337/180 , -185/18 ), ( 580/219 , 3010/219 )
**u= 9/49 , ht= 1 ; ( 1 , 7 ), ( 194/73 , -7192/511 )
**u= 9/64 , ht= 1 ; ( 67/55 , 1961/220 ), ( -1 , -8 )
**u= 9/95 , ht= 294 ; ( 181/294 , -170/21 ), ( 10192/543 , -70378/543 )
**u= 9/100 , ht= 1 ; ( 103/91 , 4859/455 ), ( -1 , -10 )
**u= 13/4 , ht= 35 ; ( -19/35 , 12/7 ), ( 45/113 , -55/113 )
**u= 13/5 , ht= 407 ; ( -407/227 , -477/227 ), ( -458/847 , -972/847 )
**u= 13/8 , ht= 14 ; ( 41/23 , -72/23 ), ( -5/14 , 87/56 )
**u= 13/12 , ht= 343 ; ( 1193/119 , -326/17 ), ( 343/181 , 819/181 )
**u= 13/20 , ht= 289 ; ( -83/547 , 1488/547 ), ( -289/271 , 1044/271 )
**u= 13/21 , ht= 11 ; ( -11/10 , 4 ), ( -10/11 , -40/11 )
**u= 13/28 , ht= 123 ; ( -17/123 , 391/123 ), ( 1943/3077 , -663/181 )
**u= 13/29 , ht= 1407 ; ( 3213/835 , 2083/167 ), ( 275/1407 , 4495/1407 )
**u= 13/32 , ht= 187 ; ( 187/155 , -807/155 ), ( 217/162 , -7229/1296 )
**u= 13/37 , ht= 1918 ; ( -2839/4837 , 2877/691 ), ( 1918/919 , -7644/919 )
**u= 13/53 , ht= 1695 ; ( -1695/794 , -4000/397 ), ( -14497/9727 , -74871/9727 )
**u= 13/60 , ht= 1291 ; ( -1291/539 , 6364/539 ), ( -17791/3311 , -82672/3311 )
**u= 13/77 , ht= 5171 ; ( -5171/2641 , -29949/2641 ), ( 76624/15941 , -404646/15941 )
**u= 13/92 , ht= 413 ; ( 13553/17 , 4494 ), ( 323/413 , 2958/413 )
**u= 15/2 , ht= 3 ; ( -53/66 , 67/44 ), ( 3 , -6 )
**u= 15/11 , ht= 317 ; ( -317/156 , -50/13 ), ( -156/317 , 600/317 )
**u= 15/13 , ht= 102 ; ( -971/438 , -333/73 ), ( 102/89 , 270/89 )
**u= 15/59 , ht= 3528 ; ( -2761/3528 , 262/49 ), ( -6552/5549 , -36144/5549 )
**u= 15/74 , ht= 564 ; ( -527/564 , -2425/376 ), ( 2028/1931 , -26391/3862 )
**u= 16 , ht= 31 ; ( -23/31 , 249/124 ), ( -33/23 , -281/92 )
**u= 16/9 , ht= 17 ; ( -17/9 , -329/108 ), ( -49/41 , -419/164 )
**u= 16/21 , ht= 65 ; ( 65/37 , -715/148 ), ( -667/2665 , 399/164 )
**u= 16/25 , ht= 1 ; ( 91/27 , -4937/540 ), ( 1 , -15/4 )
**u= 16/29 , ht= 47 ; ( 47/11 , 543/44 ), ( 247/275 , 843/220 )
**u= 16/49 , ht= 209 ; ( 7/241 , 25209/6748 ), ( 39/209 , 21977/5852 )
**u= 16/53 , ht= 305 ; ( -1883/25 , 5787/20 ), ( -305/93 , -4945/372 )
**u= 16/65 , ht= 97 ; ( 2153/983 , -40389/3932 ), ( 97/33 , -1757/132 )
**u= 16/81 , ht= 337 ; ( 71/369 , 64697/13284 ), ( 39/337 , -58169/12132 )
**u= 16/85 , ht= 79 ; ( -3181/3279 , 89353/13116 ), ( 11/79 , -1557/316 )
**u= 17/7 , ht= 67 ; ( 67/4 , 27/2 ), ( 794/717 , 1504/717 )
**u= 17/23 , ht= 42 ; ( 39719/3360 , -47683/1680 ), ( -42/19 , 116/19 )
**u= 17/39 , ht= 740 ; ( 1571/2420 , -931/242 ), ( 740/287 , -2570/287 )
**u= 17/55 , ht= 262 ; ( 1361/2104 , -4791/1052 ), ( 262/57 , -1028/57 )
**u= 17/87 , ht= 73 ; ( -31711/5872 , 77229/2936 ), ( -14/73 , 356/73 )
**u= 25 , ht= 1 ; ( -1/18 , -32/9 ), ( 1 , 3/5 )
**u= 25/4 , ht= 1 ; ( -13/87 , -847/435 ), ( 1 , -6/5 )
**u= 25/8 , ht= 24 ; ( -24/7 , 5/4 ), ( -7/25 , -42/125 )
**u= 25/9 , ht= 1 ; ( -41/12 , -43/18 ), ( 1 , 9/5 )
**u= 25/16 , ht= 73 ; ( -3551/3357 , -16459/6714 ), ( 27/73 , 1177/730 )
**u= 25/17 , ht= 15 ; ( 407/101 , 651/101 ), ( 2/15 , 116/75 )
**u= 25/28 , ht= 15 ; ( -1/15 , 176/75 ), ( 15 , 176/5 )
**u= 25/36 , ht= 1 ; ( 1 , 18/5 ), ( 1 , -18/5 )
**u= 25/41 , ht= 91 ; ( -291/119 , -601/85 ), ( -91/9 , -1267/45 )
**u= 25/44 , ht= 1873 ; ( -1873/613 , 27327/3065 ), ( 15569/433 , 44610/433 )
**u= 25/49 , ht= 1 ; ( 1 , 21/5 ), ( -1 , 21/5 )
**u= 25/64 , ht= 1 ; ( 1 , 24/5 ), ( -267/167 , -21457/3340 )
**u= 25/68 , ht= 373 ; ( -373/239 , 1545/239 ), ( -239/373 , -1545/373 )
**u= 25/81 , ht= 1 ; ( 1 , 27/5 ), ( 42/67 , -13556/3015 )
**u= 25/88 , ht= 1385 ; ( -1385/714 , 7277/840 ), ( 97733/7405 , 1959063/37025 )
**u= 25/92 , ht= 491 ; ( -645/229 , -13882/1145 ), ( -491/337 , -2427/337 )
**u= 25/96 , ht= 7 ; ( -1/7 , -59/14 ), ( 7 , -59/2 )
**u= 29/11 , ht= 74 ; ( 317/24 , -101/12 ), ( 74/47 , -132/47 )
**u= 29/43 , ht= 2701 ; ( -2701/2330 , -918/233 ), ( 43610/8221 , 117300/8221 )
**u= 29/59 , ht= 5633 ; ( -8473/4874 , -14688/2437 ), ( 90/5633 , 16816/5633 )
**u= 29/75 , ht= 5152 ; ( 5233/30758 , -7662/2197 ), ( 5152/373 , -17766/373 )
**u= 29/91 , ht= 2637 ; ( -2637/730 , 5116/365 ), ( 22894/417 , -86528/417 )
**u= 30/47 , ht= 74 ; ( 31/78 , -151/52 ), ( 27/74 , 411/148 )
**u= 30/71 , ht= 263 ; ( 2029921/292338 , -1473081/64964 ), ( -234/263 , -2295/526 )
**u= 30/83 , ht= 1828 ; ( -1828/81 , 4273/54 ), ( -7071/2308 , 52797/4616 )
**u= 36 , ht= 289 ; ( -289/287 , 1/574 ), ( -287/289 , -1/578 )
**u= 36/13 , ht= 31 ; ( -77/25 , 23/10 ), ( 5/31 , 35/62 )
**u= 36/25 , ht= 1 ; ( -1 , -5/2 ), ( -1439/1537 , 36937/15370 )
**u= 36/49 , ht= 1 ; ( 1 , -7/2 ), ( -13/61 , 2101/854 )
**u= 36/61 , ht= 959 ; ( -1613/2345 , -2261/670 ), ( 959/349 , 5719/698 )
**u= 36/85 , ht= 127 ; ( -43/127 , 881/254 ), ( -127/43 , 881/86 )
**u= 37/5 , ht= 27 ; ( 27/23 , 5/23 ), ( -337/383 , -189/383 )
**u= 37/8 , ht= 269 ; ( -51/269 , -484/269 ), ( 1655/726 , -11519/2904 )
**u= 37/13 , ht= 17 ; ( -17/9 , 17/9 ), ( -1002/833 , 104/49 )
**u= 37/20 , ht= 33 ; ( 1101/461 , -1594/461 ), ( 13/33 , -47/33 )
**u= 37/28 , ht= 1111 ; ( -635/1111 , -225/101 ), ( 1903/2605 , -1155/521 )
**u= 37/32 , ht= 143 ; ( -441/167 , -877/167 ), ( -143/118 , -2967/944 )
**u= 37/45 , ht= 91 ; ( 1107/1498 , 944/321 ), ( -91/9 , -665/27 )
**u= 37/48 , ht= 1349 ; ( -1349/139 , 3169/139 ), ( 1337/1931 , -11217/3862 )
**u= 37/60 , ht= 31 ; ( 31 , 82 ), ( 1/31 , -82/31 )
**u= 37/68 , ht= 12153 ; ( -35031/22633 , -119126/22633 ), ( -971/12153 , 34487/12153 )
**u= 37/93 , ht= 17177 ; ( -17177/10010 , 3026/455 ), ( 136114/18779 , 5236/211 )
**u= 37/96 , ht= 80 ; ( 4331/9040 , -55085/14464 ), ( 80 , 2205/8 )
**u= 37/100 , ht= 2999 ; ( 5755/1989 , 105538/9945 ), ( 2999/25 , -52692/125 )
**u= 39/14 , ht= 19 ; ( -19/9 , -2 ), ( 477/71 , -828/71 )
**u= 39/61 , ht= 924 ; ( -1083671/176040 , -237437/14670 ), ( 924/619 , -2976/619 )
**u= 39/62 , ht= 735 ; ( 113/1995 , 52/19 ), ( -735/394 , 4515/788 )
**u= 39/85 , ht= 294 ; ( -209/294 , -27/7 ), ( 1686/71 , -5346/71 )
**u= 40 , ht= 42 ; ( 41/42 , -181/168 ), ( -297/106 , 4981/424 )
**u= 40/13 , ht= 292 ; ( 292/149 , 981/596 ), ( -2417/12676 , 9231/50704 )
**u= 40/17 , ht= 23 ; ( 273/86 , 1115/344 ), ( 23/6 , 163/24 )
**u= 40/21 , ht= 128 ; ( -541/68 , -2617/272 ), ( -61/128 , -753/512 )
**u= 40/29 , ht= 21 ; ( -21/8 , 149/32 ), ( 41/12 , 335/48 )
**u= 40/37 , ht= 688 ; ( 59/688 , -5985/2752 ), ( 1497/424 , -13381/1696 )
**u= 40/41 , ht= 1203 ; ( -3414/4007 , -45601/16028 ), ( 506/1203 , 10789/4812 )
**u= 40/57 , ht= 1409 ; ( 5207/4694 , -70809/18776 ), ( 1409/98 , -14689/392 )
**u= 40/61 , ht= 3813 ; ( -11917/2296 , 124317/9184 ), ( -3813/884 , -41885/3536 )
**u= 40/69 , ht= 3641 ; ( -3641/568 , -40331/2272 ), ( 15524/3173 , -179591/12692 )
**u= 40/73 , ht= 2282 ; ( 2282/1479 , 30959/5916 ), ( -11037/2614 , -131941/10456 )
**u= 40/89 , ht= 1122 ; ( -1122/569 , 15817/2276 ), ( 3842/3219 , -63565/12876 )
**u= 41/31 , ht= 233 ; ( 26091/1136 , -21629/568 ), ( -170/233 , -516/233 )
**u= 41/47 , ht= 35220 ; ( -54223/83410 , -22980/8341 ), ( 35220/12391 , -87550/12391 )
**u= 41/95 , ht= 671 ; ( 671/234 , -1138/117 ), ( 106104/21899 , -353146/21899 )
**u= 45/4 , ht= 3 ; ( 53/351 , -853/351 ), ( 3 , 7 )
**u= 45/11 , ht= 7 ; ( 7/4 , 1/2 ), ( 4/7 , -2/7 )
**u= 45/13 , ht= 17 ; ( 17/9 , 11/9 ), ( -9/17 , -11/17 )
**u= 45/28 , ht= 601 ; ( 601/237 , 961/237 ), ( 811/27 , 1540/27 )
**u= 45/37 , ht= 489 ; ( 677/174 , 622/87 ), ( -489/103 , 1031/103 )
**u= 45/52 , ht= 77 ; ( 51/77 , 214/77 ), ( 1593/1411 , -4877/1411 )
**u= 45/61 , ht= 279 ; ( -639/212 , -811/106 ), ( 279/157 , -803/157 )
**u= 45/64 , ht= 161 ; ( 123/161 , -295/92 ), ( -91/327 , -3353/1308 )
**u= 49 , ht= 301 ; ( 301/300 , 151/1050 ), ( 303/301 , 1507/2107 )
**u= 49/4 , ht= 1 ; ( -1 , 6/7 ), ( 1 , 6/7 )
**u= 49/9 , ht= 7 ; ( -19/30 , 526/315 ), ( -6/7 , 124/147 )
**u= 49/16 , ht= 77 ; ( -77/39 , 907/546 ), ( -99/97 , -2377/1358 )
**u= 49/25 , ht= 1 ; ( 1 , -15/7 ), ( 1 , -15/7 )
**u= 49/33 , ht= 197 ; ( -337/95 , 759/133 ), ( 100/197 , -2490/1379 )
**u= 49/36 , ht= 1 ; ( 1379/569 , -17547/3983 ), ( 1 , 18/7 )
**u= 49/40 , ht= 121 ; ( 283/13 , 267/7 ), ( 39/121 , -1586/847 )
**u= 49/41 , ht= 59 ; ( -23/59 , 909/413 ), ( -277/687 , 9469/4809 )
**u= 49/64 , ht= 1 ; ( 1 , 24/7 ), ( 1 , -24/7 )
**u= 49/76 , ht= 27 ; ( -5051/3081 , -108253/21567 ), ( -27/11 , -548/77 )
**u= 49/80 , ht= 493 ; ( -2337/527 , 2623/217 ), ( -493/97 , -19431/1358 )
**u= 49/81 , ht= 1 ; ( 1325/168 , 112465/5292 ), ( 1 , -27/7 )
**u= 49/89 , ht= 6843 ; ( 6843/4618 , -82084/16163 ), ( -34718/9337 , -729888/65359 )
**u= 49/92 , ht= 217 ; ( -217/125 , -1011/175 ), ( 7145/15993 , -352640/111951 )
**u= 49/96 , ht= 19 ; ( -559/47 , 11492/329 ), ( 19/13 , 37/7 )
**u= 49/100 , ht= 1 ; ( -1 , 30/7 ), ( -186003/184801 , -27812851/6468035 )
**u= 51 , ht= 24 ; ( 53/60 , -12/5 ), ( -24/11 , 108/11 )
**u= 51/14 , ht= 19 ; ( 19/9 , -2/3 ), ( 111/253 , -12/253 )
**u= 51/25 , ht= 54 ; ( 29/54 , 17/9 ), ( 6/197 , 1074/985 )
**u= 51/26 , ht= 249 ; ( 197/249 , -168/83 ), ( 30/1291 , -2961/2582 )
**u= 51/50 , ht= 979 ; ( 281/1800 , 13439/6000 ), ( -96/979 , -3891/1958 )
**u= 51/86 , ht= 491 ; ( 4159/3237 , -4816/1079 ), ( 111/491 , 1362/491 )
**u= 51/95 , ht= 161 ; ( -161/66 , -83/11 ), ( -762/2453 , 7338/2453 )
**u= 52/5 , ht= 29 ; ( -1307/1213 , -759/2426 ), ( -29/11 , -129/22 )
**u= 52/9 , ht= 179 ; ( -179/145 , -39/58 ), ( -1425/547 , 15665/3282 )
**u= 52/17 , ht= 513 ; ( -1527/581 , 1873/1162 ), ( 139/513 , 413/1026 )
**u= 52/25 , ht= 83 ; ( -437/337 , 1521/674 ), ( 83/25 , 1509/250 )
**u= 52/37 , ht= 591 ; ( -295/591 , -2515/1182 ), ( -1921/743 , 7947/1486 )
**u= 52/53 , ht= 3873 ; ( -3873/473 , -15821/946 ), ( 151535/166799 , 960297/333598 )
**u= 52/73 , ht= 4141 ; ( 199/4141 , -21417/8282 ), ( -13523/4873 , 73809/9746 )
**u= 52/77 , ht= 4083 ; ( -1830923/446275 , -9507933/892550 ), ( 541/4083 , -20737/8166 )
**u= 52/97 , ht= 9877 ; ( -5449/9877 , 9417/2822 ), ( -306999/56933 , 1836023/113866 )
**u= 53/3 , ht= 20 ; ( 17/20 , 17/10 ), ( 20/17 , 2 )
**u= 53/35 , ht= 6833 ; ( 6833/92 , -5085/46 ), ( 10068/1703 , 19606/1703 )
**u= 53/51 , ht= 2851 ; ( -4031/4150 , 1206/415 ), ( -1790/2851 , -6820/2851 )
**u= 53/67 , ht= 10660 ; ( -110737/28126 , -132024/14063 ), ( 10660/797 , -26358/797 )
**u= 53/83 , ht= 7798 ; ( -7051/7798 , 1998/557 ), ( -4760/11483 , 32466/11483 )
**u= 61/4 , ht= 35 ; ( 19/35 , 12/5 ), ( 35/19 , -84/19 )
**u= 61/5 , ht= 243 ; ( -243/233 , -113/233 ), ( 1126/1101 , -1120/1101 )
**u= 61/12 , ht= 473 ; ( 473/445 , -111/89 ), ( 13445/2837 , -24620/2837 )
**u= 61/13 , ht= 180 ; ( -89/180 , -31/18 ), ( 4260/2123 , 7330/2123 )
**u= 61/16 , ht= 35 ; ( 35/19 , 15/19 ), ( -81/85 , -49/34 )
**u= 61/20 , ht= 597 ; ( -597/143 , -19/13 ), ( -341/1189 , -528/1189 )
**u= 61/21 , ht= 179 ; ( -1237/65 , 401/65 ), ( -17/179 , 63/179 )
**u= 61/29 , ht= 6088 ; ( -7671/5264 , -6173/2632 ), ( 6088/3101 , 11346/3101 )
**u= 61/40 , ht= 527 ; ( -697/103 , 1050/103 ), ( -303/527 , -974/527 )
**u= 61/44 , ht= 493 ; ( -189/493 , -1024/493 ), ( -13/2583 , 4127/2583 )
**u= 61/45 , ht= 363 ; ( 363/28 , 127/6 ), ( 679/149 , -1379/149 )
**u= 61/56 , ht= 1517 ; ( 1517/698 , 12981/2792 ), ( -186470/143517 , 1945061/574068 )
**u= 61/69 , ht= 2437 ; ( 31939/22601 , 86991/22601 ), ( -2437/277 , -5763/277 )
**u= 61/84 , ht= 145 ; ( 145/131 , -485/131 ), ( 28541/9805 , 15368/1961 )
**u= 61/96 , ht= 1999 ; ( -1999/1676 , -55319/13408 ), ( 10304/5423 , 251023/43384 )
**u= 64 , ht= 1 ; ( -1 , -3/8 ), ( 1 , -3/8 )
**u= 64/5 , ht= 1351 ; ( -2747/5537 , -14517/6328 ), ( 1351/171 , 27881/1368 )
**u= 64/9 , ht= 33 ; ( 605/677 , -7375/5416 ), ( -33/25 , -251/120 )
**u= 64/21 , ht= 101 ; ( 101/17 , 157/136 ), ( -1649/157 , -22849/1256 )
**u= 64/25 , ht= 139 ; ( -589/267 , 24733/10680 ), ( -117/139 , -9017/5560 )
**u= 64/29 , ht= 207 ; ( 73/595 , -8457/4760 ), ( 91/207 , 2057/1656 )
**u= 64/37 , ht= 769 ; ( -769/45 , -1627/72 ), ( 45/769 , -8135/6152 )
**u= 64/49 , ht= 1 ; ( -1 , -21/8 ), ( 45/211 , 20393/11816 )
**u= 64/65 , ht= 2567 ; ( -2567/2113 , 56271/16904 ), ( -1597/3083 , -57507/24664 )
**u= 64/81 , ht= 1 ; ( 1 , -27/8 ), ( 51/307 , -51689/22104 )
**u= 65/7 , ht= 7702 ; ( 191/7702 , -8718/3851 ), ( 614798/75819 , 1383532/75819 )
**u= 65/23 , ht= 812 ; ( 191/812 , 705/406 ), ( -194/2543 , -1128/2543 )
**u= 72 , ht= 15 ; ( -48/65 , 211/52 ), ( 15/8 , -305/32 )
**u= 72/13 , ht= 15 ; ( 2/15 , 113/60 ), ( 34/21 , 229/84 )
**u= 72/25 , ht= 1 ; ( 600/7 , -601/20 ), ( 0 , 7/20 )
**u= 72/37 , ht= 934 ; ( 934/231 , -4637/924 ), ( 1806/1991 , 16013/7964 )
**u= 72/73 , ht= 336 ; ( 29/336 , 433/192 ), ( 2639/1836 , -27979/7344 )
**u= 72/85 , ht= 481 ; ( -766/339 , 7513/1356 ), ( -481/114 , -4717/456 )
**u= 72/97 , ht= 229 ; ( -408/77 , -361/28 ), ( -132/229 , -2563/916 )
**u= 73/8 , ht= 213 ; ( 213/203 , -22/29 ), ( 427/141 , -917/141 )
**u= 73/12 , ht= 11 ; ( -5/11 , 20/11 ), ( 11/5 , 4 )
**u= 73/20 , ht= 43 ; ( 903/583 , 746/583 ), ( -43/7 , -75/7 )
**u= 73/25 , ht= 125 ; ( -157/68 , 63/34 ), ( -17/125 , -231/625 )
**u= 73/28 , ht= 1165 ; ( 2203/789 , 1984/789 ), ( 1165/941 , 2118/941 )
**u= 73/32 , ht= 59 ; ( 59/13 , -9/2 ), ( -221/419 , -546/419 )
**u= 73/41 , ht= 560 ; ( 137/560 , -15/8 ), ( -280/1647 , 2170/1647 )
**u= 73/57 , ht= 953 ; ( 953/35 , 231/5 ), ( 2107/971 , 4599/971 )
**u= 73/65 , ht= 2361 ; ( 1841/4009 , -9225/4009 ), ( 931/2361 , 4813/2361 )
**u= 73/80 , ht= 427 ; ( 957/5837 , 13699/5837 ), ( -427/297 , -2347/594 )
**u= 73/88 , ht= 2177 ; ( -31505/11714 , -303945/46856 ), ( 351/2177 , -4946/2177 )
**u= 73/97 , ht= 3438 ; ( -66361/13981 , -160989/13981 ), ( 3438/43 , 8668/43 )
**u= 75 , ht= 6 ; ( -5/6 , -17/5 ), ( 6/5 , -102/25 )
**u= 75/14 , ht= 4 ; ( 4/3 , 1/2 ), ( 3/4 , 3/8 )
**u= 75/23 , ht= 17 ; ( -17/6 , -1 ), ( -6/17 , -6/17 )
**u= 75/38 , ht= 93 ; ( -118/213 , -1359/710 ), ( -93/17 , -846/85 )
**u= 75/97 , ht= 731 ; ( -731/246 , -1509/205 ), ( 4434/1621 , -58374/8105 )
**u= 78/23 , ht= 502 ; ( 502/441 , 67/42 ), ( 399/682 , -1113/1364 )
**u= 78/35 , ht= 47 ; ( 179/24 , 117/16 ), ( -12/47 , -99/94 )
**u= 81 , ht= 1 ; ( 1 , 1/3 ), ( 1 , 1/3 )
**u= 81/4 , ht= 1 ; ( -1 , 2/3 ), ( -1 , -2/3 )
**u= 81/16 , ht= 185 ; ( -881/947 , -8033/5682 ), ( -185/179 , -1523/1074 )
**u= 81/25 , ht= 1 ; ( -1 , -5/3 ), ( 1 , -5/3 )
**u= 81/49 , ht= 1 ; ( 67/16 , 1019/168 ), ( -1 , -7/3 )
**u= 81/52 , ht= 383 ; ( -383/305 , -484/183 ), ( -865/929 , 6410/2787 )
**u= 81/64 , ht= 41 ; ( 25/41 , -565/246 ), ( -101/55 , -545/132 )
**u= 81/71 , ht= 99 ; ( 3041/2142 , -1544/459 ), ( -28/99 , -574/297 )
**u= 81/76 , ht= 41 ; ( -2497/49 , -2056/21 ), ( -35/41 , -329/123 )
**u= 81/100 , ht= 1 ; ( 727/181 , -25621/2715 ), ( -1 , -10/3 )
**u= 85/4 , ht= 253 ; ( 179/253 , -600/253 ), ( -377/199 , -1062/199 )
**u= 85/8 , ht= 7 ; ( -639/737 , 1058/737 ), ( -7/6 , -41/24 )
**u= 85/12 , ht= 8239 ; ( -5923/8239 , 175/107 ), ( 329329/76157 , -660429/76157 )
**u= 85/13 , ht= 7 ; ( 7/6 , 2/3 ), ( 4874/5953 , -1596/5953 )
**u= 85/21 , ht= 59 ; ( -59/32 , -3/16 ), ( -359/467 , -453/467 )
**u= 85/24 , ht= 19 ; ( -19/8 , 15/32 ), ( 217/409 , 244/409 )
**u= 85/28 , ht= 1159 ; ( -1159/403 , 51/31 ), ( 533/3611 , -624/3611 )
**u= 85/37 , ht= 13063 ; ( 13063/2884 , -915/206 ), ( -239932/130771 , 439026/130771 )
**u= 85/44 , ht= 13973 ; ( 13973/5189 , 18840/5189 ), ( 20711/48827 , -435/311 )
**u= 85/48 , ht= 889 ; ( -889/377 , 1340/377 ), ( 1027/5939 , -7875/5939 )
**u= 85/53 , ht= 209 ; ( 3/209 , -397/209 ), ( -712/4429 , 6414/4429 )
**u= 85/61 , ht= 1259 ; ( -1447/1386 , -256/99 ), ( 693/1259 , 2429/1259 )
**u= 85/64 , ht= 29641 ; ( -29641/3083 , 197259/12332 ), ( -15533/30529 , 237207/122116 )
**u= 85/69 , ht= 38983 ; ( 38983/19916 , -39713/9958 ), ( 254404/50647 , 532990/50647 )
**u= 85/72 , ht= 97 ; ( 10698/31 , 230417/372 ), ( 97/54 , 2705/648 )
**u= 85/76 , ht= 65397 ; ( 65687/11309 , -124635/11309 ), ( 65397/13489 , 142231/13489 )
**u= 85/77 , ht= 1154 ; ( -64893/5779 , -122461/5779 ), ( -1154/43 , -2484/43 )
**u= 85/84 , ht= 2273 ; ( -10567/73531 , 164985/73531 ), ( -751/2273 , -4812/2273 )
**u= 85/92 , ht= 10161 ; ( -10161/133 , 1123/7 ), ( -17993/89431 , -192318/89431 )
**u= 85/93 , ht= 7478 ; ( 53467/21599 , 123607/21599 ), ( -7478/3341 , 18728/3341 )
**u= 87/2 , ht= 18 ; ( -91/135 , -52/15 ), ( 18/13 , 9/2 )
**u= 87/13 , ht= 12 ; ( 1/12 , 2 ), ( -12 , -24 )
**u= 87/35 , ht= 396 ; ( 199/396 , -59/33 ), ( -9816/3271 , -17328/3271 )
**u= 87/38 , ht= 596 ; ( -1256/21 , 759/14 ), ( 69/596 , 1107/1192 )
**u= 87/62 , ht= 334 ; ( 334/255 , -487/170 ), ( -9573/9418 , -48183/18836 )
**u= 87/83 , ht= 979 ; ( -347/1728 , -107/48 ), ( -432/979 , -2124/979 )
**u= 87/98 , ht= 285 ; ( 1243/570 , 1993/380 ), ( -285/89 , 696/89 )
**u= 89/15 , ht= 43 ; ( -43/38 , -18/19 ), ( 38/43 , 36/43 )
**u= 89/31 , ht= 3607 ; ( 3607/538 , 804/269 ), ( 7366/12729 , -13576/12729 )
**u= 89/63 , ht= 867 ; ( 867/820 , -635/246 ), ( -7820/22821 , 117310/68463 )
**u= 97/8 , ht= 697 ; ( -697/671 , -33/61 ), ( 4675/1291 , -11451/1291 )
**u= 97/12 , ht= 151 ; ( 5467/6505 , -9509/6505 ), ( -151/121 , -232/121 )
**u= 97/17 , ht= 4737 ; ( -4737/3916 , 1543/1958 ), ( 5665/1373 , -10629/1373 )
**u= 97/20 , ht= 47 ; ( 5219/3619 , 1770/3619 ), ( 47/33 , -76/33 )
**u= 97/24 , ht= 241 ; ( 241/130 , -1/8 ), ( 9490/17327 , 12545/69308 )
**u= 97/25 , ht= 125 ; ( 1493/1007 , 1185/1007 ), ( -63/125 , 53/625 )
**u= 97/32 , ht= 13 ; ( -5/13 , -45/26 ), ( 293/635 , -99/127 )
**u= 97/40 , ht= 5426 ; ( 5426/1781 , -21417/7124 ), ( -12647/38 , -88611/152 )
**u= 97/48 , ht= 4957 ; ( 4957/847 , 11351/1694 ), ( 5683/4663 , -22891/9326 )
**u= 97/57 , ht= 310 ; ( 127/310 , 302/155 ), ( 1678/24485 , -6588/4897 )
**u= 97/60 , ht= 527 ; ( -32453/32347 , -76410/32347 ), ( -337/527 , -978/527 )
**u= 97/73 , ht= 127 ; ( 1117/5205 , -10637/5205 ), ( -19/127 , -213/127 )
**u= 97/76 , ht= 493 ; ( -46313/28035 , -19459/5607 ), ( 115/493 , -30/17 )
**u= 97/92 , ht= 141 ; ( 1031/1859 , -4533/1859 ), ( 95/141 , 343/141 )
**u= 100 , ht= 1 ; ( -1 , -3/10 ), ( -303/97 , -20309/970 )
**u= 100/9 , ht= 1 ; ( -1 , -9/10 ), ( 273/127 , 18029/3810 )
**u= 100/17 , ht= 49 ; ( 15/49 , 131/70 ), ( -49/15 , -917/150 )
**u= 100/33 , ht= 295 ; ( 295/31 , 161/310 ), ( -305/289 , 5259/2890 )
**u= 100/49 , ht= 217 ; ( -217/73 , 18909/5110 ), ( -153/247 , -26909/17290 )
**u= 100/53 , ht= 2205 ; ( 2205/1039 , -32561/10390 ), ( 395/16719 , -200869/167190 )
**u= 100/57 , ht= 205 ; ( 205/53 , 2839/530 ), ( -48511/12023 , 182697/24046 )
**u= 100/77 , ht= 44635 ; ( 57375/2243 , -963821/22430 ), ( -44635/2489 , 906333/24890 )
**u= 100/81 , ht= 21 ; ( -19/21 , -701/270 ), ( -4403/3629 , 66409/21774 )
307
>

■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。

[2026.05.24追記]u=4/9,4/25のときの整数解を追加した。

[参考文献]

Last Update: 2026.06.16
H.Nakao

Homeに戻る[Homeに戻る]  一覧に戻る[一覧に戻る]