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Integer Points on A^4+11*B^4=C^4+11*D^4

[2026.05.26]A^4+11*B^4=C^4+11*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=11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352個のuが抽出される。
[MAGMAによる計算]
> PP(11,1,100);
**u= 1/3 , ht= 7 ; ( -25/7 , 15 ), ( -7/5 , 7 )
**u= 1/11 , ht= 1 ; ( 1 , -11 ), ( 1 , -11 )
**u= 1/16 , ht= 75 ; ( 183/73 , 1848/73 ), ( 1/75 , -469/50 )
**u= 1/35 , ht= 157 ; ( 157/137 , -2891/137 ), ( -267/13 , 3709/13 )
**u= 1/36 , ht= 167 ; ( 25/169 , 2404/169 ), ( 27/167 , 7141/501 )
**u= 1/44 , ht= 1 ; ( 1 , 22 ), ( 1 , 22 )
**u= 1/51 , ht= 173 ; ( -479/31 , 8039/31 ), ( 173/37 , 2963/37 )
**u= 1/56 , ht= 4 ; ( 331/572 , -46393/2288 ), ( 4/3 , 117/4 )
**u= 1/60 , ht= 199 ; ( 199/61 , 3781/61 ), ( 61/199 , -19 )
**u= 1/72 , ht= 211 ; ( 120/211 , 57965/2532 ), ( -713/225 , 8927/135 )
**u= 1/75 , ht= 35 ; ( 35/23 , -4253/115 ), ( 1535/211 , -157347/1055 )
**u= 1/84 , ht= 589 ; ( 433/589 , -827/31 ), ( 589/433 , -15713/433 )
**u= 1/96 , ht= 19 ; ( -271/151 , 14257/302 ), ( -19/13 , 529/13 )
**u= 1/99 , ht= 297567 ; ( -297567/296477 , 29405227/889431 ), ( -590789/592965 , 58595921/1778895 )
**u= 2/3 , ht= 13 ; ( 2/13 , -77/26 ), ( -1/14 , 79/28 )
**u= 2/7 , ht= 95 ; ( 5/124 , -1093/248 ), ( -24/95 , 857/190 )
**u= 2/15 , ht= 23 ; ( 109/64 , -1623/128 ), ( 23/8 , 313/16 )
**u= 2/19 , ht= 14 ; ( 1675/554 , 25495/1108 ), ( 14/5 , -43/2 )
**u= 2/23 , ht= 171 ; ( 171/148 , 3597/296 ), ( 592/891 , -5671/594 )
**u= 2/31 , ht= 16 ; ( 2940/1121 , -58095/2242 ), ( -16/15 , 27/2 )
**u= 2/43 , ht= 182 ; ( -53/182 , -589/52 ), ( -763/118 , -16793/236 )
**u= 2/75 , ht= 67 ; ( 67/26 , 10321/260 ), ( 26/67 , 10321/670 )
**u= 2/87 , ht= 52 ; ( 23/52 , 1759/104 ), ( -5512/6755 , -269709/13510 )
**u= 4/11 , ht= 1 ; ( -5147/4819 , -54821/9638 ), ( -1 , -11/2 )
**u= 4/15 , ht= 473 ; ( -1009/251 , 9417/502 ), ( 473/293 , -5061/586 )
**u= 4/35 , ht= 123 ; ( -759/199 , -10881/398 ), ( -123/53 , -1859/106 )
**u= 4/39 , ht= 193 ; ( -193/67 , 2991/134 ), ( -1691/961 , 28493/1922 )
**u= 4/59 , ht= 585 ; ( -113/585 , -2147/234 ), ( 3655/2599 , 715/46 )
**u= 4/63 , ht= 9 ; ( -5/9 , 575/54 ), ( 9/5 , -115/6 )
**u= 4/71 , ht= 1177 ; ( -81/2333 , 46137/4666 ), ( -1177/811 , -28247/1622 )
**u= 4/87 , ht= 23675 ; ( -24565/31177 , -868265/62354 ), ( -8711/23675 , -110357/9470 )
**u= 4/91 , ht= 29875 ; ( -73803/13925 , -336021/5570 ), ( -29875/4341 , -225145/2894 )
**u= 4/99 , ht= 21495 ; ( 47529/43063 , -4489717/258378 ), ( 18151/21495 , 1969421/128970 )
**u= 5/4 , ht= 119 ; ( 119/27 , -239/27 ), ( -2299/667 , -5314/667 )
**u= 5/7 , ht= 11 ; ( -1/13 , -37/13 ), ( -3/11 , 31/11 )
**u= 5/12 , ht= 733 ; ( -733/109 , 2672/109 ), ( -1067/571 , -4416/571 )
**u= 5/16 , ht= 133 ; ( -173/119 , -1759/238 ), ( -1/133 , -1111/266 )
**u= 5/24 , ht= 31 ; ( 1453/376 , 30793/1504 ), ( -7/31 , 163/31 )
**u= 5/28 , ht= 1237 ; ( 341/1237 , 7130/1237 ), ( 8209/6417 , -622/69 )
**u= 5/31 , ht= 193 ; ( -193/179 , -1537/179 ), ( 2297/499 , -13741/499 )
**u= 5/39 , ht= 401 ; ( 1637/4679 , 32487/4679 ), ( -67/401 , 2661/401 )
**u= 5/48 , ht= 197 ; ( 197/11 , -1433/11 ), ( 157/439 , -6773/878 )
**u= 5/64 , ht= 217 ; ( -61/217 , 1081/124 ), ( 1001/103 , -16891/206 )
**u= 5/72 , ht= 154 ; ( -154/117 , -20653/1404 ), ( 117/154 , -20653/1848 )
**u= 5/76 , ht= 4639 ; ( 4949/1833 , 48247/1833 ), ( 4639/877 , 43175/877 )
**u= 5/84 , ht= 5707 ; ( 5851/13957 , -145490/13957 ), ( 5707/4279 , -68569/4279 )
**u= 5/88 , ht= 2364 ; ( -118193/13224 , 4679815/52896 ), ( -2364/1057 , -101921/4228 )
**u= 9/4 , ht= 7 ; ( 3/7 , 41/21 ), ( 11/27 , -112/81 )
**u= 9/11 , ht= 271 ; ( -271/125 , 91/15 ), ( -625/277 , -5425/831 )
**u= 9/16 , ht= 15 ; ( -15 , 139/3 ), ( -31/75 , -1507/450 )
**u= 9/19 , ht= 351 ; ( 351/199 , -4103/597 ), ( 1401/793 , -16543/2379 )
**u= 9/40 , ht= 243 ; ( -114/301 , -19129/3612 ), ( -243/77 , 3788/231 )
**u= 9/44 , ht= 4667 ; ( -91415/93341 , -2032507/280023 ), ( 4667/4271 , 98431/12813 )
**u= 9/52 , ht= 1503 ; ( -2385/1697 , -49480/5091 ), ( -1503/385 , -5254/231 )
**u= 9/56 , ht= 67 ; ( 823/284 , -61061/3408 ), ( -67/59 , -1567/177 )
**u= 9/59 , ht= 349 ; ( -241/349 , -7643/1047 ), ( -9347/3515 , 180031/10545 )
**u= 9/64 , ht= 59 ; ( 1957/795 , -7921/477 ), ( -15/59 , -4565/708 )
**u= 9/80 , ht= 43 ; ( -43/17 , 1939/102 ), ( -17/43 , 1939/258 )
**u= 10/3 , ht= 53 ; ( -53/4 , -5/8 ), ( 32/179 , 107/358 )
**u= 10/7 , ht= 9 ; ( -9/2 , 33/4 ), ( 382/2481 , 2979/1654 )
**u= 10/11 , ht= 12 ; ( -87/56 , -501/112 ), ( 1/12 , 19/8 )
**u= 10/27 , ht= 143 ; ( -143/4 , -3287/24 ), ( -17411/10148 , -155795/20296 )
**u= 10/39 , ht= 478 ; ( -1789/1682 , 22741/3364 ), ( 478/361 , 5553/722 )
**u= 10/47 , ht= 142 ; ( 142 , 1441/2 ), ( 2122/571 , 22385/1142 )
**u= 10/71 , ht= 4561 ; ( -14003/694 , 175067/1388 ), ( -622/4561 , 57481/9122 )
**u= 10/79 , ht= 6753 ; ( 6753/3514 , -14331/1004 ), ( 27146/2103 , -119735/1402 )
**u= 10/83 , ht= 864 ; ( 864/283 , -12279/566 ), ( -1172/6049 , 83209/12098 )
**u= 11 , ht= 1 ; ( -1/3 , -7/3 ), ( -1 , 1 )
**u= 11/4 , ht= 3 ; ( -3 , 3 ), ( -1/3 , -1 )
**u= 11/6 , ht= 2 ; ( 2 , -7/2 ), ( -5/14 , 45/28 )
**u= 11/9 , ht= 1 ; ( 1 , 3 ), ( 1 , 3 )
**u= 11/10 , ht= 7 ; ( 31/21 , -82/21 ), ( 2/7 , -31/14 )
**u= 11/16 , ht= 7 ; ( 37/21 , 17/3 ), ( -7/5 , -49/10 )
**u= 11/25 , ht= 1 ; ( -1 , 5 ), ( 59/103 , 2089/515 )
**u= 11/30 , ht= 88 ; ( 88/13 , 685/26 ), ( -157/128 , -1571/256 )
**u= 11/36 , ht= 1 ; ( -1 , -6 ), ( -2759/2613 , -48377/7839 )
**u= 11/46 , ht= 83 ; ( -55/83 , -478/83 ), ( -83/55 , -478/55 )
**u= 11/49 , ht= 61 ; ( -91/131 , -5531/917 ), ( -61/39 , -837/91 )
**u= 11/64 , ht= 37 ; ( -37/71 , 1813/284 ), ( 21/37 , 13/2 )
**u= 11/81 , ht= 395 ; ( -411/325 , -6001/585 ), ( -395/73 , 7675/219 )
**u= 11/90 , ht= 23 ; ( -23/22 , 427/44 ), ( 202/157 , 3433/314 )
**u= 11/94 , ht= 575 ; ( -252/1615 , -4485/646 ), ( 575/224 , 8465/448 )
**u= 11/100 , ht= 6793 ; ( -6793/6659 , -336313/33295 ), ( -20423/20021 , -1011161/100105 )
**u= 13/2 , ht= 5 ; ( 1/5 , 2 ), ( 5 , 10 )
**u= 13/14 , ht= 12 ; ( -5/12 , 65/24 ), ( 15761/7045 , 8620/1409 )
**u= 13/38 , ht= 71 ; ( 10807/7482 , -105209/14964 ), ( -43/71 , 332/71 )
**u= 13/42 , ht= 37 ; ( 23/37 , 184/37 ), ( 37/23 , 8 )
**u= 13/62 , ht= 467 ; ( -281/467 , 2794/467 ), ( -1164/4853 , -51029/9706 )
**u= 13/82 , ht= 5004 ; ( 2427/5297 , 34344/5297 ), ( -2257/5004 , -21539/3336 )
**u= 13/86 , ht= 281 ; ( -11705/1858 , 142835/3716 ), ( 281/235 , 442/47 )
**u= 13/98 , ht= 91385 ; ( -21477/91385 , 846858/127939 ), ( -228067/45852 , 6995461/213976 )
**u= 16/11 , ht= 1 ; ( 1 , 11/4 ), ( 1 , 11/4 )
**u= 16/15 , ht= 97 ; ( 97/67 , 1051/268 ), ( 677/97 , 6513/388 )
**u= 16/23 , ht= 45 ; ( 45 , 495/4 ), ( 1/45 , 11/4 )
**u= 16/39 , ht= 755 ; ( 727/755 , 3071/604 ), ( 755/937 , -17595/3748 )
**u= 16/59 , ht= 103 ; ( -15/103 , -1881/412 ), ( 43567/5217 , 264327/6956 )
**u= 16/95 , ht= 449 ; ( -1619/289 , 37547/1156 ), ( 349/449 , -12995/1796 )
**u= 16/99 , ht= 1267 ; ( 1899/763 , 20449/1308 ), ( 1267/515 , -31941/2060 )
**u= 17/2 , ht= 77 ; ( 15/77 , 24/11 ), ( -217/237 , -56/79 )
**u= 17/10 , ht= 4 ; ( 4 , -13/2 ), ( -121/229 , -430/229 )
**u= 17/18 , ht= 122 ; ( -122/5 , 113/2 ), ( 505/37 , -1270/37 )
**u= 17/70 , ht= 4581 ; ( 4581/529 , 21888/529 ), ( 12028/5217 , -41671/3478 )
**u= 17/74 , ht= 2700 ; ( -5820/877 , 57465/1754 ), ( -1331/2700 , 1961/360 )
**u= 17/78 , ht= 275 ; ( 275/158 , -3183/316 ), ( -35842/15443 , -392687/30886 )
**u= 17/82 , ht= 2801 ; ( -2801/1325 , 3188/265 ), ( -4075/3182 , -53285/6364 )
**u= 17/90 , ht= 708 ; ( -708/367 , -25795/2202 ), ( -253/1828 , -19885/3656 )
**u= 18/7 , ht= 13 ; ( -8/13 , -151/78 ), ( 44/65 , -607/390 )
**u= 18/23 , ht= 20 ; ( 7056/593 , 109493/3558 ), ( -3/20 , -313/120 )
**u= 18/35 , ht= 38 ; ( 38/3 , 739/18 ), ( -18/907 , -17585/5442 )
**u= 18/43 , ht= 1699 ; ( 710/1699 , -40271/10194 ), ( -2102/607 , 47903/3642 )
**u= 18/55 , ht= 673 ; ( -152/673 , 17045/4038 ), ( -673/152 , -17045/912 )
**u= 18/67 , ht= 5522 ; ( -5522/307 , -149651/1842 ), ( 10673/7390 , 352831/44340 )
**u= 18/83 , ht= 7658 ; ( -7658/1887 , -237863/11322 ), ( 16677/1654 , -507443/9924 )
**u= 20/3 , ht= 43 ; ( -43/41 , 97/82 ), ( -157/41 , 627/82 )
**u= 20/7 , ht= 29 ; ( 127 , -179/2 ), ( -27/29 , 107/58 )
**u= 20/23 , ht= 911 ; ( 1623/151 , 7917/302 ), ( 911/543 , -1807/362 )
**u= 20/31 , ht= 909 ; ( -909/227 , 5379/454 ), ( -5567/1289 , 33931/2578 )
**u= 20/43 , ht= 169 ; ( -169/3 , -1151/6 ), ( 3743/389 , 26129/778 )
**u= 20/87 , ht= 761 ; ( 761/7 , 1061/2 ), ( -2891/67 , 28357/134 )
**u= 20/91 , ht= 1733 ; ( 1733/1551 , -23263/3102 ), ( -4233/2501 , -49243/5002 )
**u= 22/17 , ht= 10 ; ( -10/7 , -7/2 ), ( 14/3 , -21/2 )
**u= 22/41 , ht= 355 ; ( -355/14 , -2245/28 ), ( 14/355 , 449/142 )
**u= 22/57 , ht= 83 ; ( 83/26 , 653/52 ), ( 274/2155 , 16293/4310 )
**u= 25/3 , ht= 19 ; ( -7/65 , 711/325 ), ( -19/17 , -27/17 )
**u= 25/4 , ht= 11 ; ( -9/11 , -87/55 ), ( 41/39 , -96/65 )
**u= 25/8 , ht= 4 ; ( -4/3 , -23/12 ), ( 13/5 , 119/25 )
**u= 25/11 , ht= 1 ; ( -1 , 11/5 ), ( -10517/10757 , 116393/53785 )
**u= 25/24 , ht= 7 ; ( -1/7 , 17/7 ), ( -72074/10441 , -3501553/208820 )
**u= 25/27 , ht= 5 ; ( -241/737 , -5845/2211 ), ( 5 , -193/15 )
**u= 25/32 , ht= 8 ; ( -243/235 , 8961/2350 ), ( -5/8 , -989/320 )
**u= 25/44 , ht= 1 ; ( -1 , 22/5 ), ( -1 , -22/5 )
**u= 25/51 , ht= 1561 ; ( 1561/541 , -27431/2705 ), ( 1451/1757 , -7617/1757 )
**u= 25/56 , ht= 271 ; ( 893/501 , 3577/501 ), ( -79/271 , -4916/1355 )
**u= 25/72 , ht= 5 ; ( 3/5 , 349/75 ), ( -5/3 , -349/45 )
**u= 25/84 , ht= 1741 ; ( 271/1741 , -38017/8705 ), ( 10693/2293 , 235924/11465 )
**u= 25/99 , ht= 1 ; ( -1 , 33/5 ), ( -1 , -33/5 )
**u= 29/6 , ht= 25 ; ( 25/19 , -22/19 ), ( -19/25 , -22/25 )
**u= 29/14 , ht= 37 ; ( -2799/197 , 3582/197 ), ( 37/9 , 8 )
**u= 29/30 , ht= 47 ; ( 47/13 , -112/13 ), ( 1376/2831 , 14617/5662 )
**u= 29/34 , ht= 724 ; ( -724/75 , 143/6 ), ( -1975/28427 , -69950/28427 )
**u= 29/46 , ht= 2447 ; ( 2447/216 , 14251/432 ), ( 5581/337 , 16808/337 )
**u= 29/54 , ht= 427 ; ( 3473/1157 , -34774/3471 ), ( 281/427 , -4882/1281 )
**u= 29/86 , ht= 831 ; ( 831/373 , 3666/373 ), ( -1409/291 , 1946/97 )
**u= 29/90 , ht= 797 ; ( -383/797 , -3664/797 ), ( -1589/1146 , -48637/6876 )
**u= 36/11 , ht= 1 ; ( -1 , 11/6 ), ( -1 , -11/6 )
**u= 36/19 , ht= 217 ; ( -217/87 , 2089/522 ), ( -975/3167 , 28945/19002 )
**u= 36/35 , ht= 41 ; ( 2941/1051 , -41689/6306 ), ( -1/41 , -541/246 )
**u= 36/67 , ht= 2655 ; ( -2655/913 , -53335/5478 ), ( 8101/5835 , -38489/7002 )
**u= 36/71 , ht= 2299 ; ( 1447/2299 , -53951/13794 ), ( -14377/2237 , -290707/13422 )
**u= 36/83 , ht= 215 ; ( 215/49 , -4675/294 ), ( 22939/1795 , -99149/2154 )
**u= 36/91 , ht= 111 ; ( 97/111 , 3301/666 ), ( -14309/6075 , -349489/36450 )
**u= 37/7 , ht= 501 ; ( -501/361 , 279/361 ), ( -773/801 , -361/267 )
**u= 37/8 , ht= 29 ; ( 1/31 , -58/31 ), ( -29/26 , 187/104 )
**u= 37/12 , ht= 7 ; ( 371/179 , -374/179 ), ( -7/5 , -13/5 )
**u= 37/15 , ht= 31 ; ( -34379 , 34317 ), ( -1/31 , 1 )
**u= 37/23 , ht= 169 ; ( 169/15 , -55/3 ), ( 375/821 , 1535/821 )
**u= 37/32 , ht= 8271 ; ( 9442/2659 , 162037/21272 ), ( -8271/305 , -19114/305 )
**u= 37/36 , ht= 1197 ; ( 625/1197 , -1375/513 ), ( -1239/1625 , 112/39 )
**u= 37/39 , ht= 4825 ; ( 139595/12949 , -323695/12949 ), ( -1237/4825 , 2311/965 )
**u= 37/47 , ht= 1059 ; ( -1059/1009 , 3861/1009 ), ( -26131/5829 , 24183/1943 )
**u= 37/52 , ht= 1077 ; ( -6573/5065 , 4587/1013 ), ( -431/1077 , 1057/359 )
**u= 37/60 , ht= 167 ; ( -167/77 , 543/77 ), ( -1129/151 , 3458/151 )
**u= 37/63 , ht= 319 ; ( 61/929 , 2893/929 ), ( 249/319 , -3701/957 )
**u= 37/72 , ht= 557 ; ( 4257/2689 , -49162/8067 ), ( -557/123 , 5659/369 )
**u= 37/80 , ht= 27 ; ( -23/27 , -245/54 ), ( -1273/1607 , 7054/1607 )
**u= 37/88 , ht= 9589 ; ( 3627/9589 , -37302/9589 ), ( 51736/24813 , -278089/33084 )
**u= 37/92 , ht= 87 ; ( 87/41 , -354/41 ), ( -41/87 , -118/29 )
**u= 37/95 , ht= 4409 ; ( -7393/587 , -27679/587 ), ( 4409/4219 , -22939/4219 )
**u= 41/6 , ht= 5 ; ( 5/4 , 3/8 ), ( 88/95 , 191/190 )
**u= 41/10 , ht= 64 ; ( -7908/3653 , 1947/7306 ), ( 39/64 , -95/128 )
**u= 41/26 , ht= 259 ; ( 129/259 , 576/259 ), ( -1277/192 , 1773/128 )
**u= 41/30 , ht= 803 ; ( 1409/914 , -6501/1828 ), ( 803/503 , -1970/503 )
**u= 41/34 , ht= 50 ; ( 85625/32789 , 185932/32789 ), ( 1/50 , -199/100 )
**u= 41/54 , ht= 773 ; ( 20851/18124 , 148169/36248 ), ( 773/151 , -6512/453 )
**u= 41/66 , ht= 589 ; ( -589/193 , -1818/193 ), ( 3121/1603 , -10546/1603 )
**u= 41/90 , ht= 24 ; ( 24/11 , 547/66 ), ( -1319/716 , 10483/1432 )
**u= 41/98 , ht= 3195 ; ( -3195/2351 , -100440/16457 ), ( -4397/1935 , -2724/301 )
**u= 44 , ht= 1 ; ( -1/25 , 47/10 ), ( 1 , 1/2 )
**u= 44/9 , ht= 1 ; ( 1 , 3/2 ), ( -29/27 , 271/162 )
**u= 44/25 , ht= 73 ; ( 73/49 , -1469/490 ), ( -637/531 , 5009/1770 )
**u= 44/45 , ht= 43 ; ( -7/43 , 5/2 ), ( -387/377 , -7697/2262 )
**u= 44/49 , ht= 19 ; ( -19/13 , 787/182 ), ( -135/31 , 4955/434 )
**u= 44/81 , ht= 1 ; ( -13095/12769 , -1047221/229842 ), ( -1 , -9/2 )
**u= 45/7 , ht= 43 ; ( -41/43 , -179/129 ), ( -89/107 , -217/321 )
**u= 45/8 , ht= 17 ; ( 177/199 , 913/597 ), ( 17/6 , 385/72 )
**u= 45/68 , ht= 577 ; ( 1671/73 , -14176/219 ), ( -577/171 , -5291/513 )
**u= 45/71 , ht= 249 ; ( 369/227 , -3797/681 ), ( 249/107 , -2425/321 )
**u= 45/76 , ht= 1033 ; ( 8107/3319 , 79181/9957 ), ( 1033/311 , -9994/933 )
**u= 49/3 , ht= 47 ; ( -25/47 , 825/329 ), ( -47/25 , -33/7 )
**u= 49/11 , ht= 1 ; ( -1 , 11/7 ), ( -1 , -11/7 )
**u= 49/20 , ht= 143 ; ( 703/253 , -5965/1771 ), ( -143/47 , 1894/329 )
**u= 49/24 , ht= 71 ; ( 71/38 , 3289/1064 ), ( -46/341 , -12597/9548 )
**u= 49/36 , ht= 25 ; ( -25/7 , -338/49 ), ( 921/199 , 42718/4179 )
**u= 49/44 , ht= 1 ; ( 1 , 22/7 ), ( -18157/17373 , -130632/40537 )
**u= 49/51 , ht= 373 ; ( 20939/4823 , -346027/33761 ), ( 271/373 , -7621/2611 )
**u= 49/64 , ht= 15 ; ( -2031/1967 , 53001/13769 ), ( 1/15 , -183/70 )
**u= 49/72 , ht= 1155 ; ( -1155/197 , -68545/4137 ), ( 3689/2960 , -75639/16576 )
**u= 49/75 , ht= 1231 ; ( 1231/19 , 24515/133 ), ( 1517/365 , -161163/12775 )
**u= 49/99 , ht= 1 ; ( -1 , 33/7 ), ( 6857/5471 , 205199/38297 )
**u= 50/19 , ht= 30 ; ( 1838/841 , 4483/1682 ), ( -11/30 , 111/100 )
**u= 50/43 , ht= 6674 ; ( -2657/6674 , -32611/13348 ), ( -8346/4597 , -42785/9194 )
**u= 50/51 , ht= 314 ; ( -314/77 , -1471/154 ), ( -1237/310 , 31347/3100 )
**u= 50/79 , ht= 568 ; ( 2395/4844 , 161047/48440 ), ( -301/568 , -3751/1136 )
**u= 50/91 , ht= 565 ; ( -306/565 , 20481/5650 ), ( -565/306 , -6827/1020 )
**u= 53/7 , ht= 9 ; ( 5/9 , -17/9 ), ( -177/149 , -271/149 )
**u= 53/12 , ht= 67 ; ( 269/385 , -19/11 ), ( 49/67 , 63/67 )
**u= 53/15 , ht= 227 ; ( 857/217 , 47/217 ), ( 133/227 , -219/227 )
**u= 53/16 , ht= 175 ; ( 225/19 , -75/38 ), ( 3/175 , 1/14 )
**u= 53/28 , ht= 4763 ; ( 7393/939 , 10511/939 ), ( 3867/4763 , 10102/4763 )
**u= 53/52 , ht= 40147 ; ( -48297/13193 , 111474/13193 ), ( -40147/7329 , -32967/2443 )
**u= 53/60 , ht= 29579 ; ( 79529/30179 , -206545/30179 ), ( -29579/2341 , 76588/2341 )
**u= 53/72 , ht= 869 ; ( -869/59 , 2322/59 ), ( -2243/4790 , -56911/19160 )
**u= 53/84 , ht= 8065 ; ( 8065/6047 , 29610/6047 ), ( 37157/1735 , 22357/347 )
**u= 53/87 , ht= 11221 ; ( 23471/7501 , 73013/7501 ), ( -6175/11221 , 38135/11221 )
**u= 53/88 , ht= 7487 ; ( -7487/6780 , 24379/5424 ), ( 79955/7179 , 82165/2393 )
**u= 53/92 , ht= 3477 ; ( -3477/3239 , 14667/3239 ), ( 1177/18933 , -19243/6311 )
**u= 53/100 , ht= 55 ; ( -7/55 , -904/275 ), ( -279541/40505 , -4604939/202525 )
**u= 55/2 , ht= 43 ; ( 33/46 , -243/92 ), ( 43/39 , 24/13 )
**u= 55/6 , ht= 47 ; ( 176/167 , 291/334 ), ( 47/29 , -90/29 )
**u= 55/12 , ht= 73 ; ( -79/47 , -30/47 ), ( -73/29 , 133/29 )
**u= 55/16 , ht= 93 ; ( -687/191 , 255/191 ), ( -29/93 , 14/31 )
**u= 55/18 , ht= 11 ; ( -93/14 , 47/12 ), ( 7/11 , -14/11 )
**u= 55/36 , ht= 119 ; ( 119/83 , -265/83 ), ( 393/331 , -2977/993 )
**u= 55/48 , ht= 31 ; ( 409/283 , 2133/566 ), ( 13/31 , -141/62 )
**u= 55/54 , ht= 23 ; ( -16/23 , 397/138 ), ( 553/176 , 2799/352 )
**u= 55/56 , ht= 139 ; ( -83/139 , 391/139 ), ( 227/291 , -288/97 )
**u= 55/62 , ht= 1478 ; ( -1478/289 , 7247/578 ), ( -2719/1342 , -15425/2684 )
**u= 55/69 , ht= 541 ; ( 127/541 , -1497/541 ), ( 167/2099 , -5371/2099 )
**u= 55/92 , ht= 1039 ; ( -203/1039 , 3259/1039 ), ( -1021/1647 , -1945/549 )
**u= 61/6 , ht= 250 ; ( -269/335 , -110/67 ), ( 250/127 , 1055/254 )
**u= 61/14 , ht= 164 ; ( -157/164 , 529/328 ), ( -844/1635 , 27/1090 )
**u= 61/26 , ht= 3074 ; ( -3074/813 , -7319/1626 ), ( 48033/934 , -180179/1868 )
**u= 61/42 , ht= 50 ; ( 42275/3089 , 74250/3089 ), ( 1/50 , 7/4 )
**u= 61/66 , ht= 2993 ; ( 2993/749 , -7264/749 ), ( 43867/1132 , 222239/2264 )
**u= 61/74 , ht= 17883 ; ( 35231/33603 , -125576/33603 ), ( 17883/4526 , -97853/9052 )
**u= 61/90 , ht= 2686 ; ( -4613/4017 , -52124/12051 ), ( -39/2686 , 44945/16116 )
**u= 61/94 , ht= 7720 ; ( -2127/7720 , -9477/3088 ), ( 2447/12123 , 11788/4041 )
**u= 64/11 , ht= 43 ; ( 21/43 , -633/344 ), ( -367/513 , 43/456 )
**u= 64/15 , ht= 109 ; ( -31/109 , 1597/872 ), ( -109/31 , -1597/248 )
**u= 64/39 , ht= 19 ; ( -7/19 , -17/8 ), ( -59/175 , -2427/1400 )
**u= 64/51 , ht= 551 ; ( -781/749 , 18069/5992 ), ( 551/415 , -11779/3320 )
**u= 64/95 , ht= 16341 ; ( -8929/16341 , -430369/130728 ), ( -53721/1379 , -1253425/11032 )
**u= 64/99 , ht= 439 ; ( -351/439 , -39511/10536 ), ( -8021/5069 , -223207/40552 )
**u= 65/14 , ht= 87 ; ( -141/218 , 759/436 ), ( 59/87 , -20/29 )
**u= 65/18 , ht= 487 ; ( -1906/623 , 1031/1246 ), ( -487/386 , -1729/772 )
**u= 65/22 , ht= 7 ; ( -7/4 , 17/8 ), ( -5879/14493 , -4668/4831 )
**u= 65/42 , ht= 8 ; ( 8 , -27/2 ), ( 1499/1268 , -7535/2536 )
**u= 65/46 , ht= 422 ; ( 563/241 , -1130/241 ), ( -422/61 , 1823/122 )
**u= 65/58 , ht= 48 ; ( 387/271 , -1026/271 ), ( -19/48 , -73/32 )
**u= 65/94 , ht= 594 ; ( -487/594 , 4351/1188 ), ( -766/2007 , 3973/1338 )
**u= 73/6 , ht= 716 ; ( -425/716 , 3055/1432 ), ( -4928/3965 , -3351/1586 )
**u= 73/18 , ht= 486 ; ( -1407/701 , -1732/2103 ), ( 383/486 , -3487/2916 )
**u= 73/38 , ht= 169 ; ( 1111/67 , 1538/67 ), ( -36/169 , -487/338 )
**u= 73/42 , ht= 3316 ; ( 4601/5675 , 2672/1135 ), ( 2375/3316 , -13855/6632 )
**u= 73/50 , ht= 838 ; ( -7126/99 , 24841/198 ), ( -361/838 , 16489/8380 )
**u= 73/62 , ht= 1075 ; ( 1075/847 , -2914/847 ), ( 13537/4733 , 32486/4733 )
**u= 73/70 , ht= 10989 ; ( -371283/10888 , -1619667/21776 ), ( -1721/10989 , -8104/3663 )
**u= 74/15 , ht= 2438 ; ( -2438/1567 , -1743/3134 ), ( -22778/13717 , -80097/27434 )
**u= 74/31 , ht= 922 ; ( -4707/26 , -9921/52 ), ( 922/907 , -3943/1814 )
**u= 74/39 , ht= 2062 ; ( -4714/941 , 13673/1882 ), ( 2062/541 , -8229/1082 )
**u= 74/51 , ht= 968 ; ( 11936/3827 , 44847/7654 ), ( 968/815 , 5007/1630 )
**u= 74/55 , ht= 15302 ; ( -15302/3123 , 58175/6246 ), ( 794086/142831 , -3505169/285662 )
**u= 74/79 , ht= 513 ; ( 554/475 , -37/10 ), ( 513/118 , 2641/236 )
**u= 74/91 , ht= 669 ; ( -7685/49736 , 269161/99472 ), ( 669/500 , 4381/1000 )
**u= 80/7 , ht= 197 ; ( 321/377 , 2337/1508 ), ( 197/69 , -619/92 )
**u= 80/19 , ht= 71 ; ( -71/43 , -191/172 ), ( 653/121 , -4805/484 )
**u= 80/27 , ht= 133 ; ( 79/133 , -991/532 ), ( 79/667 , 5197/8004 )
**u= 80/39 , ht= 3443 ; ( -3443/899 , -19025/3596 ), ( 2267/4139 , -27543/16556 )
**u= 80/47 , ht= 7579 ; ( 8163/2899 , -55557/11596 ), ( -7579/4923 , 7597/2188 )
**u= 80/67 , ht= 20139 ; ( -22313/4219 , 182809/16876 ), ( -20139/6527 , 191045/26108 )
**u= 80/91 , ht= 1931 ; ( 1931/657 , -19831/2628 ), ( -21431/773 , -221947/3092 )
**u= 80/99 , ht= 3019 ; ( -743/3019 , 33287/12076 ), ( 10793/10771 , -159131/43084 )
**u= 81/4 , ht= 11 ; ( 9/11 , -193/99 ), ( 207/107 , 5191/963 )
**u= 81/11 , ht= 1 ; ( -1 , 11/9 ), ( -1 , -11/9 )
**u= 81/16 , ht= 27 ; ( -23/27 , -779/486 ), ( -139/135 , -3757/2430 )
**u= 81/35 , ht= 383 ; ( -571/31 , -5699/279 ), ( -383/317 , 7205/2853 )
**u= 81/43 , ht= 955 ; ( 555/1079 , 245/117 ), ( 249/955 , -2573/1719 )
**u= 81/44 , ht= 529 ; ( 1/529 , 9412/4761 ), ( 529 , 9412/9 )
**u= 81/80 , ht= 127 ; ( -127/77 , -6095/1386 ), ( 77/127 , -6095/2286 )
**u= 81/100 , ht= 303 ; ( 303/49 , -4994/315 ), ( 189/337 , 44597/15165 )
**u= 85/2 , ht= 329 ; ( -163/329 , -1324/329 ), ( 511/13 , -2362/13 )
**u= 85/6 , ht= 11 ; ( -943/2939 , -7654/2939 ), ( -11/7 , 24/7 )
**u= 85/18 , ht= 291 ; ( 184/913 , -3405/1826 ), ( -247/291 , 1012/873 )
**u= 85/26 , ht= 79 ; ( 1026/1013 , -3717/2026 ), ( 79/27 , -16/3 )
**u= 85/46 , ht= 32 ; ( 8631/10747 , -24564/10747 ), ( -32/9 , -43/6 )
**u= 85/54 , ht= 2189 ; ( 647/2189 , 13958/6567 ), ( 1178/4621 , -47905/27726 )
**u= 85/62 , ht= 379 ; ( -1357/396 , -5239/792 ), ( 379/303 , 330/101 )
**u= 85/78 , ht= 1067 ; ( -3377/2356 , 18153/4712 ), ( 1/1067 , -2264/1067 )
**u= 85/86 , ht= 3093 ; ( -89/3093 , 7616/3093 ), ( 22482/9539 , -118723/19078 )
**u= 85/98 , ht= 159 ; ( 733/111 , 12634/777 ), ( 159/112 , 6935/1568 )
**u= 88/45 , ht= 7 ; ( -5934/1651 , -103879/19812 ), ( 2/7 , 41/28 )
**u= 88/53 , ht= 338 ; ( -338/33 , 2147/132 ), ( 1137/130 , -9293/520 )
**u= 89/4 , ht= 4019 ; ( -4969/4919 , 2566/4919 ), ( -4019/1811 , 12166/1811 )
**u= 89/11 , ht= 705 ; ( 705/683 , -729/683 ), ( -2179/483 , 1551/161 )
**u= 89/16 , ht= 185 ; ( -185/149 , -299/298 ), ( -6223/3233 , -11281/3233 )
**u= 89/19 , ht= 831 ; ( 4471/9525 , 3439/1905 ), ( -831/575 , -1429/575 )
**u= 89/20 , ht= 13 ; ( 3/13 , -24/13 ), ( 727/777 , -10/7 )
**u= 89/24 , ht= 31 ; ( 31/14 , -69/56 ), ( 2242/6275 , 5887/25100 )
**u= 89/40 , ht= 1241 ; ( -7789/51 , 9082/51 ), ( 1241/526 , -9713/2104 )
**u= 89/51 , ht= 361 ; ( 259/361 , 823/361 ), ( -541/683 , 1497/683 )
**u= 89/67 , ht= 8215 ; ( 8215/3443 , -17075/3443 ), ( 3443/8215 , -3415/1643 )
**u= 89/76 , ht= 2099 ; ( -8161/3087 , -2572/441 ), ( -1169/2099 , -5033/2099 )
**u= 89/91 , ht= 5679 ; ( -11071/8221 , -32287/8221 ), ( 5679/4405 , -17227/4405 )
**u= 89/96 , ht= 8167 ; ( -8167/977 , -38517/1954 ), ( -23222/6793 , 486723/54344 )
**u= 89/100 , ht= 1687 ; ( 487/1687 , 4496/1687 ), ( 91373/29411 , 1227508/147055 )
**u= 90/7 , ht= 313 ; ( 54/313 , 4853/1878 ), ( -613/654 , -593/3924 )
**u= 90/19 , ht= 708 ; ( -71/708 , 7967/4248 ), ( 1692/191 , 19031/1146 )
**u= 90/23 , ht= 103 ; ( 34/103 , -1123/618 ), ( 338/361 , 3353/2166 )
**u= 90/31 , ht= 57 ; ( 17061/2522 , 73529/15132 ), ( 26/57 , -365/342 )
**u= 90/43 , ht= 524 ; ( -524/207 , -4615/1242 ), ( -207/524 , 4615/3144 )
**u= 90/59 , ht= 5569 ; ( 5392/5569 , 88615/33414 ), ( 43156/34837 , -645823/209022 )
**u= 90/83 , ht= 52 ; ( 19309/3788 , 252395/22728 ), ( 31/52 , 797/312 )
**u= 97/3 , ht= 217 ; ( 653/1057 , 483/151 ), ( 217/173 , -539/173 )
**u= 97/8 , ht= 471 ; ( -308/471 , 3823/1884 ), ( 4923/1789 , -11831/1789 )
**u= 97/12 , ht= 685 ; ( -547/685 , 220/137 ), ( -1525/667 , -3080/667 )
**u= 97/20 , ht= 47 ; ( 47/33 , 32/33 ), ( -33/47 , 32/47 )
**u= 97/27 , ht= 157 ; ( 509/151 , 253/453 ), ( 59/157 , 71/157 )
**u= 97/40 , ht= 4398 ; ( -4398/3733 , -33207/14932 ), ( 3733/4398 , -11069/5864 )
**u= 97/48 , ht= 34591 ; ( 34591/1903 , 45399/1903 ), ( -47267/25351 , -194297/50702 )
**u= 97/59 , ht= 141 ; ( -1185/1529 , -3645/1529 ), ( -141/95 , -65/19 )
**u= 97/68 , ht= 3413 ; ( 753451/105071 , 1354387/105071 ), ( -3413/3209 , 9251/3209 )
**u= 97/80 , ht= 12539 ; ( 12539/2629 , -25562/2629 ), ( -118859/14231 , -542011/28462 )
**u= 97/83 , ht= 2867 ; ( 1525/3019 , -7603/3019 ), ( -2867/121 , -6599/121 )
**u= 97/84 , ht= 275 ; ( -3245/3587 , 10620/3587 ), ( -61/275 , 116/55 )
**u= 98/3 , ht= 41 ; ( 215/218 , 2693/3052 ), ( 41/2 , 2329/28 )
**u= 98/15 , ht= 32 ; ( -244/301 , -6683/4214 ), ( 32/17 , -829/238 )
**u= 98/19 , ht= 1270 ; ( -1270/839 , -4975/11746 ), ( -2466/365 , 13123/1022 )
**u= 98/27 , ht= 74 ; ( -74/23 , 463/966 ), ( -97/38 , 2461/532 )
**u= 98/31 , ht= 535 ; ( 535/8 , 2995/112 ), ( -368/2515 , -3379/7042 )
**u= 98/47 , ht= 191 ; ( 3135/268 , 55899/3752 ), ( 191/144 , -1909/672 )
**u= 98/55 , ht= 428 ; ( -1312/393 , 29347/5502 ), ( 428/373 , -14237/5222 )
**u= 98/75 , ht= 586 ; ( -167/586 , -93101/41020 ), ( 1138/187 , -35477/2618 )
**u= 98/83 , ht= 418 ; ( 123/790 , 5115/2212 ), ( -418/3 , 4469/14 )
**u= 98/87 , ht= 1091 ; ( -40/1091 , 35663/15274 ), ( -1091/40 , -35663/560 )
**u= 99 , ht= 1 ; ( -221/223 , -667/669 ), ( -1 , 1/3 )
**u= 99/4 , ht= 1 ; ( 1 , -2/3 ), ( 85 , 905/3 )
**u= 99/16 , ht= 1271 ; ( -1205/1271 , -10787/7626 ), ( -2359/2293 , -19507/13758 )
**u= 99/20 , ht= 41 ; ( -21/41 , -221/123 ), ( 299/399 , -965/1197 )
**u= 99/25 , ht= 91 ; ( -91/67 , -1517/1005 ), ( -809/757 , -20491/11355 )
**u= 99/49 , ht= 1 ; ( -127/73 , -4577/1533 ), ( 1 , 7/3 )
**u= 99/50 , ht= 12 ; ( -5/12 , -727/360 ), ( -12/5 , 727/150 )
**u= 99/56 , ht= 10 ; ( -4782/1021 , 89071/12252 ), ( -3/10 , -193/120 )
**u= 99/64 , ht= 1 ; ( -1 , -8/3 ), ( -1157/127 , 28999/1524 )
**u= 99/74 , ht= 63 ; ( -63/11 , 358/33 ), ( -11/63 , -358/189 )
**u= 99/97 , ht= 5299 ; ( 10595/4193 , -76565/12579 ), ( -5299/4135 , 9469/2481 )
**u= 99/100 , ht= 1 ; ( -1 , -10/3 ), ( -1 , 10/3 )
**u= 100/3 , ht= 1807 ; ( 1015/1807 , -61611/18070 ), ( 3887/2009 , 27401/4018 )
**u= 100/7 , ht= 125 ; ( -57/125 , -3093/1250 ), ( -4555/2553 , -35299/8510 )
**u= 100/11 , ht= 1061 ; ( 1061/973 , -1327/1946 ), ( -83581/83091 , 312243/276970 )
**u= 100/43 , ht= 135 ; ( 135/49 , -249/70 ), ( -1297/2745 , 12907/9150 )
**u= 100/79 , ht= 4941 ; ( -4941/2801 , 113877/28010 ), ( 13859/8487 , -7773/1886 )
**u= 100/99 , ht= 7813 ; ( 199989/195611 , -19563289/5868330 ), ( 7813/1059 , -115217/6354 )
352
>

■これらのuについて、(2),(3a),(3b)を満たす有理数解(x,y,t)を持たないものもあれば、有理数解(x,y,t)を持つものもある。
これらのuを順に調べれば良い。

ここからは、A^4+B^4=C^4+D^4と同様なので、最終的に得られた整点のみ記述する。
ここで、対応する整点が見つかった各有理数uについて、0 < A < B, 0 < C < D, 0 < A < Cを満たすように、
A,B,C,Dの符号を変更したり、A,B,C,Dを交換して、Dの小さい順に並び替えると、以下のようになる。

[参考文献]

Last Update: 2026.06.16
H.Nakao

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