wayne 发表于 2025-5-2 08:52
由 $m^4=\sum _{k=0}^{n-1} (k+p)^2 , 2 p + n - 1 = q_1$,得到$12 m^4-3 n q_1^2=n^3-n$,这个可以稍微变换一下,由 $M=m^4=\sum _{k=0}^{n-1} (k+p)^2 , 2 p + n - 1 = q_1$,得到$12 M-3 n q_1^2=n^3-n$,其中$n|6M$
也就是$4 M n-\frac{n^4}{3}+\frac{n^2}{3} = (nq_1)^2=q^2$,这个四次曲线双有理等价于一个椭圆曲线. $(x,y)\to(-\frac{36M + n}{n}, \frac{108Mq}{n^2})$,或者$(n,q)\to(-\frac{36M}{x+1},\frac{12 M y}{(x+1)^2})$
对应的椭圆曲线$x^3 + 3888*M^2 + y^2 - 3*x - 2=0$,四次方就是$M=m^4$,三次方就是$M=m^3$,平方就是$M=m^2$
首先$n|M$,其次根据表达式$x=-\frac{36M + n}{n}$ 得知,x是整数, 所以, 我们就是要求椭圆曲线 $x^3 + 3888*M^2 + y^2 - 3*x - 2=0$ 的整数解. 这就开始变得有趣了.
也就是求方程$Y^2 = X^3+ 3X^2- 3888M^2$的整数解.其中$X=\frac{36M}{n}, Y=\frac{108Mq_1}{n}$,
或者求方程$Y^2 = X^3+ \frac{X^2 }{12}- \frac{M^2 }{12}$的有理解,其中$X=\frac{M}{n}, Y=\frac{Mq_1}{2n}$, 因为$n|6M$,所以需要$6X$是整数
Mathematica验证均满足恒等关系.
Factor[-3888 M^2 + 3 X^2 + X^3 - y^2 /. Thread[{X, y} -> {36 M/n, 108 M q/n}]]
Factor[-(M^2/12)+x^2/12+x^3-y^2/.Thread[{x,y}->{M/n,1/2M q/n}]]
比如计算m=143^2的情况
m=143^2;
E=ellinit();
ellratpoints(E,)
得到这么多有理解
格式是n,a,b, 这里的n,a,b都成了有理数. :)
{33,-39,-7}
{33,7,39}
{11,-48,-38}
{11,38,48}
{121/2,-(71/2),24}
{121/2,-24,71/2}
{1,-143,-143}
{1,143,143}
{1014/19,-(727/19),268/19}
{1014/19,-(268/19),727/19}
{99/26,-(5825/78),-(2803/39)}
{99/26,2803/39,5825/78}
{676/109,-(13079/218),-(11945/218)}
{676/109,11945/218,13079/218}
{338/181,-(19019/181),-(18862/181)}
{338/181,18862/181,19019/181}
{3718/427,-(22311/427),-(19020/427)}
{3718/427,19020/427,22311/427}
{81796/1321,-(89169/2642),71781/2642}
{81796/1321,-(71781/2642),89169/2642}
{122694/8263,-(361814/8263),-(247383/8263)}
{122694/8263,247383/8263,361814/8263}
{363/2702,-(1053001/2702),-(527670/1351)}
{363/2702,527670/1351,1053001/2702}
再比如 109^3,得到
{{218, -153, 64}, {218, -64, 153}}
而 13^4 得到的解
{52,-(87/2),15/2}
{52,-(15/2),87/2}
{2,-120,-119}
{2,119,120}
{1,-169,-169}
{1,169,169}
{13182/367,-(16032/367),-(3217/367)}
{13182/367,3217/367,16032/367}
{85683/14702,-(532206/7351),-(993431/14702)}
{85683/14702,993431/14702,532206/7351}
{85683/499394,-(101773177/249697),-(203960065/499394)}
{85683/499394,203960065/499394,101773177/249697}
只可惜m不能太大, 因为四次方就是m^8.这蹭蹭的往上涨.
m=2026的时候,
{3/2,-37,-(73/2)}
{3/2,73/2,37}
{4052/211,-(3746/211),95/211}
{4052/211,-(95/211),3746/211}
{961/91,-(51609/2821),-(24639/2821)}
{961/91,24639/2821,51609/2821}
{58081/2149,-(8689644/517909),4789968/517909}
{58081/2149,-(4789968/517909),8689644/517909}
cases="[, , , , , , , ]";
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