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ܼ

[̿]x3|x2ʤ顢3|xǤ롣

[]3ˡȤx0,1,2Τɤ줫˹ƱǤ롣

x1,2ʤx21 (mod 3)ȤʤΤǡx23dzڤʤ

äơ3|x2Ȥʤ뤿ˤϡx0 (mod 3)ǤʤƤϤʤʤĤޤꡢ3|xΩġ

ߤͭ

[]X2+Y2=3ˤͭϤʤȤ衣

[]ˡǹԤΤˡͭ(X,Y)¸ߤȲꤹ롣

X2+Y2=3(*)

XYͭʤΤǡx,y,zˤäơΤ褦ɽ롣

X=x/z,Y=y/z(**)

ΤȤx,y,zˤ3dzڤʤΤȲꤷƤ褤ʤʤС3ĤȤ3dzڤ줿顢x,y,zx/3,y/3,z/3֤Ƥ⡢(**)ΩĤǤ롣

X,Y(*)Τǡx,y,zϼ

x2+y2=3z2(***)

ǡ⤷x3dzڤȤȡ(***)x23z23dzڤΤǡ3|y2ǤʤƤϤʤʤȡ3|yȤʤʢ[̿]x3|x2ʤ顢3|xǤפˡǡx,yξȤ3dzڤ뤳ȤˤʤäΤǡ(***)κդ32dzڤ롣32|3z2ǤΤǡ3|z2Ǥ롣ȡ(***)ˤäơ3|zȤȤ狼롣ɡx,y,z3dzڤ뤳ȤˤʤäƤޤx,y,zˤ3dzڤʤΤȤȿ롣

äơx3dzڤʤ

x,yؤƾ嵭Ʊ򤹤ȡy3dzڤʤȤˤʤ롣

椨ˡx,y϶3dzڤʤ

ǡ(***)3ˡȤƹͤ롣x,y϶3dzڤʤΤǡx,y1,2 (mod 3)Ǥ롣

äơx2,y21 (mod 3)ȤʤΤǡx2+y22 (mod 3)Ǥ롣

(***)ȡ3z22 (mod 3)Ȥʤ롣3z20 (mod 3)Ǥ뤿ᡢ̷⤹롣

äơX2+Y2=3Ȥ߾ͭȤ̷꤫⤬ƳΤǡ꤬̿줿

Y2=X3-2X

ͭˡ

̵ͭ¸ĵˡˤĤƹͻ롣

Y2=X3-2X(*)

ζˤϡɸǤȯǤ롣ͭǤ롣Ūˤϡ(X,Y)=(-1,1),(2,2)ʤɤǤ롣

1Ĥͭ(X0,Y0)̤ͭ(X1,Y1)򸫤Ĥˡͤ롣⤷ˡΩС̵¸Ĥͭ򸫤Ĥ뤳ȤǤ롣ͭ򤹤٤ƵȤϸ¤ʤ

ޤ(X0,Y0)϶(*)ʤΤǡY_0^2=X_0^3-2X_0Ωġ

(*)ξդXʬȡ2Y~\frac{dY}{dX}~=~3X^2~-2ȤʤΤǡY0ΤȤ˼Ωġ

\frac{dY}{dX}~=~\frac{3X^2~-2}{2Y}(**)

(**)ˤäơ(X0,Y0)Ǥζ(*)ηϡ\frac{3X_0^2-2}{2Y_0}ʤΤǡ(X0,Y0)Ǥ(*)ϼΤ褦ˤʤ롣

Y=Y_0~+~(\frac{3X_0^2-2}{2Y_0})(X-X_0)(***)

(X1,Y1)Ͼ嵭ľ(***)뤿ᡢΩġ

Y_1=Y_0~+~(\frac{3X_0^2-2}{2Y_0})(X_1-X_0)

(X1,Y1)϶(*)(***)ȤθʤΤǡ(X1,Y1)뤿(***)(*)ȡΤ褦ˤʤ롣

(Y_0~+~(\frac{3X_0^2-2}{2Y_0})(X-X_0))^2=X^3-2X
{X_0}^3~-2X_0~+(3{X_0}^2-2)(X-X_0)+\frac{(3{X_0}^2-2)^2(X-X_0)^2}{4({X_0}^3-2X_0)}~=~X^3-2X
\frac{(3{X_0}^2-2)^2(X-X_0)^2}{4({X_0}^3-2X_0)}=X^3-2X~-~\{~{X_0}^3~-2X_0~+(3{X_0}^2-2)(X-X_0)~\}ʢ嵭κդ3ޤǤܹԤ׻
\frac{(3{X_0}^2-2)^2(X-X_0)^2}{4({X_0}^3-2X_0)}=(X-X_0)^2(X+2X_0)
(X-X_0)^2(X+2X_0~-~\frac{(3{X_0}^2-2)^2(X-X_0)^2}{4({X_0}^3-2X_0)})=0
X_1=-2X_0~+~\frac{(3{X_0}^2-2)^2(X-X_0)^2}{4({X_0}^3-2X_0)}ʢX1X=X0ʳβǤ뤫
X_1=\frac{({X_0}^2+2)^2}{4X_0({X_0}^2-2)}

X1X0ȤäɽȤϡY1ͤ롣

ʾˤꡢ(X0,Y0)Ȥäơ(X1,Y1)ɽ줿

ʸ

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