A Point with 3 Rational Distances From The Corners of A Square

[2004.09.12]正方形の３頂点からの距離が有理数になる点

正方形の３頂点からの距離が有理数になる点を計算するプログラム(pari/GP)

■次の問題を考察する。

□ABCDを一辺の長さが正整数sである正方形とする。この正方形の内部のある点Pに対して、PA=c, PB=a, PC=bが全て正整数となるようなsを求めよ。

[2004.09.26追記]
この問題は、数論における未解決問題[3]の１つ(D19,Three-Distance Problem)である。

■Euclid座標で、A(0,s),B(0,0),C(0,s),D(s,s),P(x,y)とすると、Pythagorasの定理より、
    S: x²+y² = a²,     (s-x)²+y² = b²,     x²+(s-y)² = c²
となる。
点Pは□ABCDの内部にあるので、
    0 < x, y < s
である。
x,yは有理数であることが簡単に分かる。

以下では、Allan J. MacLeod[2]による解法を紹介する。

x,yの分母の最小公倍数をs,a,b,c,x,yに掛けることにより、全ての変数を整数として良い。
ある整数α,β,γ,m,n,p,q,r,tに対して、
    x = α2mn, y = α(m²-n²),
    s-x = β2pq, y = β(p²-q²),
    x = γ2rt, s-y = γ(r²-t²)
となる。よって、
    2αmn+2βpq=γ(r²-t²)+α(m²-n²), ------- (1)
    β = α(m²-n²)/(p²-q²), ----- (2)
    γ = {αmn}/{rt} ----- (3)
を得る。
(2),(3)を(1)に代入して、
    m²rt(p²-2pq-q²) +mn(r²-2rt-t²)(p²-q²) -n²rt(p²-2pq-q²) = 0
ここで、u=m/n,v=p/q,w=r/tとすると、
    u²w(v²-2v-1) +u(w²-2w-1)(v²-1) -w(v²-2v-1) = 0
つまり、
    u(v²-1)w²+(u²(v²-2v-1)-2u(v²-1)-(v²-2v-1))w-u(v²-1) = 0 ---- (4)
を得る。
0 < u,v,w < 1より、u(v²-1) != 0である。
wの２次方程式(4)が有理数解を持つので、ある有理数dに対して、
    d² = (u²(v²-2v-1)-2u(v²-1)-(v²-2v-1))²+4u(v²-1)u(v²-1)
つまり、
    C: d² = (u⁴-4u³+6u²+4u+1)v⁴-4(u²-1)(u²-2u-1)v³+ 2(u⁴+4u³-10u²-4u+1)v² +4(u²-1)(u²-2u-1)v+u⁴-4u³+6u²+4u+1 ----- (5)
となる。
Q(u)上で定義された４次曲線Cは、Q(u)-有理点v=1,d=2(u²-1)を持つので、Q(u)上で定義されたある楕円曲線EにQ(u)-isomorphicである。

４次曲線Cは、双有理変換φ:(v,d)→(g,t)
    g = {-(u+1)((u³-3u²-u-1)v²-2(u³-u²-3u-1)v-(3u³-5u²+u+5)}/{(v-1)²},
    t = -2{(u⁶-2u⁵-7u⁴+4u³+7u²-2u)v⁵-(5u⁶-6u⁵-35u⁴+12u³+35u²-6u)v⁴+(6u⁶-3u⁵-62u⁴+6u³+64u²-3u)v³
          +(2u⁶+7u⁵+42u⁴+14u³-48u²-7u-1)v²-(7u⁶-5u⁵+3u⁴+10u³-9u²-5u-2)v+(3u⁶-u⁵-5u⁴-2u³+u²+u+3)}/{(v-1)³}

[逆変換ψ:(g,t)→(v,d)は、
    v = {2(u²-u-1)g+t+4(u+1)(u²-2u-1)}/{-2ug+y-4u(u+1)(u²-2u-1)},
    d = (u²-1){2x³+2(u⁴-2u³+2u²-10u-5)x³+8(u²-2u-1)(u⁴-2u³-u²-4u-2)x-y²+8u²(u²-2u-1)y-8(u+1)²(u²+1)(u²-2u-1)²}/{8u²(u+1)²(u²-2u-1)}
]
によって、楕円曲線
    E: t² = g³+2(u⁴-2u³-u²-4u-2)g²-4(u⁶-3u⁴-4u³-5u²-4u-1)g
つまり、
    E: t² = g(g-2(u²-u-1))(g+2(u⁴-2u³-2u-1))

に写される。

[pari/gpによる計算]

gp>  read("sq3.gp")
time = 29 ms.
gp>  ph1([x,y])
time = 8 ms.
%1 = [((-u^4 + 2*u^3 + 4*u^2 + 2*u + 1)*x^2 + (2*u^4 - 8*u^2 - 8*u - 2)*x + (u^4 - 2*u^3 + y*u^2 + 6*u + (-y + 3)))/(x^2 - 2*x + 1), ((-2*u^5 + 8*u^4 + 4*u^3 - 8*u^2 - 2*u)*x^3 + (-2*u^6 + 10*u^5 - 2*u^4 - 20*u^3 + 2*u^2 + 10*u + 2)*x^2 + (4*u^6 - 2*u^5 - 20*u^4 + (2*y + 4)*u^3 + 20*u^2 + (-2*y - 2)*u - 4)*x + (2*u^6 - 6*u^5 + (2*y + 2)*u^4 + (-2*y + 12)*u^3 + (-4*y - 2)*u^2 + (2*y - 6)*u + (2*y - 2)))/(x^3 - 3*x^2 + 3*x - 1)]
gp>  ph2([x,y])
time = 66 ms.
%2 = [((2*u^2 - 2*u - 2)*x + (-4*u^3 + 4*u^2 + 12*u + (y + 4)))/(-2*u*x + (-4*u^4 + 4*u^3 + 12*u^2 + 4*u + y)), ((4*u^2 - 4)*x^3 + (4*u^6 - 8*u^5 + 4*u^4 - 32*u^3 - 28*u^2 + 40*u + 20)*x^2 + (16*u^8 - 64*u^7 + 16*u^6 + 64*u^5 + 80*u^4 + 128*u^3 - 80*u^2 - 128*u - 32)*x + (-16*u^10 + 32*u^9 + 80*u^8 - 64*u^7 + (16*y - 160)*u^6 + (-32*y - 128)*u^5 + (-32*y - 96)*u^4 + (32*y + 64)*u^3 + (-2*y^2 + 16*y + 176)*u^2 + 96*u + (2*y^2 + 16)))/(4*u^2*x^2 + (16*u^5 - 16*u^4 - 48*u^3 - 16*u^2 - 4*y*u)*x + (16*u^8 - 32*u^7 - 80*u^6 + 64*u^5 + (-8*y + 176)*u^4 + (8*y + 96)*u^3 + (24*y + 16)*u^2 + 8*y*u + y^2))]

EのQ(u)-有理点T₁(0,0),T₂(2(u²-u-1),0),T₃(-2(u⁴-2u³-2u-1),0)は、いずれも位数２のねじれ点である。
よって、Eのねじれ点群 E(Q(u))_torsは、{ O, T₁, T₂, T₃ } \cong Z/2Z×Z/2Zを部分群として持つことが分かる。

■楕円曲線E/Q(u)の判別式は、
Δ(E) = 1024u²⁰ - 4096u¹⁹ + 12288u¹⁷ - 4096u¹⁶ - 4096u¹⁵ - 20480u¹³ + 6144u¹² + 20480u¹¹ + 4096u⁹ - 4096u⁸ - 12288u⁷ + 4096u⁵ + 1024u⁴
= 2¹⁰u⁴(u+1)⁴(u-1)⁴(u²+1)²(u²-2u-1)²
である。

[pari/gpによる計算]

gp>  e2 
time = 0 ms.
%6 = [0, 2*u^4 - 4*u^3 - 2*u^2 - 8*u - 4, 0, -4*u^6 + 12*u^4 + 16*u^3 + 20*u^2 + 16*u + 4, 0, 8*u^4 - 16*u^3 - 8*u^2 - 32*u - 16, -8*u^6 + 24*u^4 + 32*u^3 + 40*u^2 + 32*u + 8, 0, -16*u^12 + 96*u^10 + 128*u^9 + 16*u^8 - 256*u^7 - 704*u^6 - 1024*u^5 - 1008*u^4 - 768*u^3 - 416*u^2 - 128*u - 16, 64*u^8 - 256*u^7 + 320*u^6 - 256*u^5 + 256*u^4 + 256*u^3 + 320*u^2 + 256*u + 64, -512*u^12 + 3072*u^11 - 6912*u^10 + 8704*u^9 - 7680*u^8 + 7680*u^7 - 2560*u^6 - 7680*u^5 - 7680*u^4 - 8704*u^3 - 6912*u^2 - 3072*u - 512, 1024*u^20 - 4096*u^19 + 12288*u^17 - 4096*u^16 - 4096*u^15 - 20480*u^13 + 6144*u^12 + 20480*u^11 + 4096*u^9 - 4096*u^8 - 12288*u^7 + 4096*u^5 + 1024*u^4, (256*u^24 - 3072*u^23 + 16128*u^22 - 50176*u^21 + 108288*u^20 - 178176*u^19 + 226304*u^18 - 221184*u^17 + 177408*u^16 - 99328*u^15 + 23808*u^14 - 9216*u^13 - 55808*u^12 + 9216*u^11 + 23808*u^10 + 99328*u^9 + 177408*u^8 + 221184*u^7 + 226304*u^6 + 178176*u^5 + 108288*u^4 + 50176*u^3 + 16128*u^2 + 3072*u + 256)/(u^20 - 4*u^19 + 12*u^17 - 4*u^16 - 4*u^15 - 20*u^13 + 6*u^12 + 20*u^11 + 4*u^9 - 4*u^8 - 12*u^7 + 4*u^5 + u^4), 0, 0, 0, 0, 0, 0]
gp>  e2.disc
time = 0 ms.
%7 = 1024*u^20 - 4096*u^19 + 12288*u^17 - 4096*u^16 - 4096*u^15 - 20480*u^13 + 6144*u^12 + 20480*u^11 + 4096*u^9 - 4096*u^8 - 12288*u^7 + 4096*u^5 + 1024*u^4
gp>  factor(e2.disc)
time = 115 ms.
%8 = 
[u - 1 4]

[u 4]

[u + 1 4]

[u^2 - 2*u - 1 2]

[u^2 + 1 2]

■楕円曲線Eは、Q(u)-有理点P₁(4(u²+1),4(u-1)²(u²+1))を持つ。

[pari/gpによる計算]

gp>  P1
time = 0 ms.
%9 = [4*u^2 + 4, 4*u^4 - 8*u^3 + 8*u^2 - 8*u + 4]
gp>  ellisoncurve(e2,P1)
time = 0 ms.
%10 = 1
gp>  factor(P1[1])
time = 2 ms.
%11 = 
[u^2 + 1 1]

gp>  factor(P1[2])
time = 4 ms.
%12 = 
[u - 1 2]

[u^2 + 1 1]

よって、任意の整数nに対して、nP₁,nP₁+T₁,nP₁+T₂,nP₁+T₃も、楕円曲線EのQ(u)-有理点である。

これらのQ(u)-有理点を双有理変換ψで写すことにより、４次曲線CのQ(u)-有理点を求めることができる。

...

P ∈ E(Q(u))		ψ(P) ∈ C(Q(u))	v ∈ Q(u)	w ∈ Q(u)
O	[0:1:0]	[1,2(u²-1)]	1	-
T₁	[0,0]	[1/u,(-u⁴+1)/u²]	1/u	1/u, u
T₂	[2(u+1)²,0]	[-1,2(u²-1)]	-1	-
T₃=T₁+T₂	[-(u²+1)(u²-2u-1),0]	[-u,-(u⁴-1)]	-u	-u, 1/u
P₁	[4(u²+1), 4(u-1)²(u²+1)]	[-u,(u⁴-1)]	-u	1/u, -u
-P₁	[4(u²+1), -4(u-1)²(u²+1)]	[-(u³-u+2)/(2u³+u²-1), (u-1)(u+1)(u²+1)(u⁴+4u³+18u²-4u+1)/(2u³+u²-1)²]	{-(u³-u+2)}/{2u³+u²-1}	{-(3u²-2u+1)}/{u(u²+2u+3)}, {u(u²+2u+3)}/{3u²-2u+1}
P₁+T₁	[-(u+1)²(u²-2u-1), (u-1)²(u+1)²(u²-2u-1)]	[-1,-2(u²-1)]	-1	-
-P₁+T₁	[-(u+1)²(u²-2u-1), -(u-1)²(u+1)²(u²-2u-1)]	[(3u²+2u-3)/(u²-2u-1), -2(u-1)(u-1)(u⁴+4u³+18u²-4u+1)/(u²-2u-1)²]	{3u²+2u-3}/{u²-2u-1}	{-(u²-2u-1)}/{4u}, {4u}/{(u²+2u-1)}
P₁+T₂	[2(u+1)²(u²+1), -4u²(u+1)²(u²+1)]	[1/u,(u⁴-1)/u²]	1/u	-u, 1/u
-P₁+T₂	[2(u+1)²(u²+1), 4u²(u+1)²(u²+1)]	[(2u³+u²-1)/(u³-u+2), (u-1)(u+1)(u²+1)(u⁴+4u³+18u²-4u+1)/(u³-u+2)²]	{2u³+u²-1}/{u³-u+2}	{u(u²+2u+3)}/{3u²-2u+1}, {-3(3u²-2u+1)}/{u(u²+2u+3)}
P₁+T₃	[2(u²-2u-1), -4u²(u²-2u-1)]	[1,-2(u²-1)]	1	-
-P₁+T₃	[2(u²-2u-1), 4u²(u²-2u-1)]	-	-	-
2P₁	[(u²+2u+3)²/4, -(u²+2u-1)(u²+2u+3)(3u²-2u+1)/8]	[-(u²-2u-1)/(3u²+2u-3), 2(u-1)(u+1)(u⁴+4u³+18u²-4u+1)/{3(3u²+2u-3)²}]	{-(u²-2u-1)}/{3u²+2u-3}	{-(u²+2u-1)}/{27u}, {27u}/{u²+2u-1}
-2P₁	[(u²+2u+3)²/4, (u²+2u-1)(u²+2u+3)(3u²-2u+1)/8]	[{7u⁶+22u⁵+31u⁴-36u³-31u²+22u-7}/{(u²-2u-1)(3u⁴+12u³-10u²-12u+3)}, {2(u-1)(u+1)(u¹²+12u¹¹+486u¹⁰+996u⁹+207u⁸-296u⁷+468u⁶+296u⁵+207u⁴-996u³+486u²-12u+1)}/{(u²-2u-1)²(3u⁴+12u³-10u²-12u+3)²}	{7u⁶+22u⁵+31u⁴-36u³-31u²+22u-7}/{(u²-2u-1)(3u⁴+12u³-10u²-12u+3)}	{(u²+2u-1)(u⁴+4u³+18u²-4u+1)}/{4u(5u⁴+4u³-6u²-4u+5)}, {-4u(5u⁴+4u³-6u²-4u+5)}/{(u²+2u-1)(u⁴+4u³+18u²-4u+1)}
2P₁+T₁	[-16(u+1)²(u²+1)(u²-2u-1)/(u²+2u+3)², -8(u+1)²(u²+1)(u²+2u-1)(3u²-2u+1)/(u²+2u+3)³]	[(2u³+u²-1)/(u³-u+2), -(u-1)(u+1)(u²+1)(u⁴+4u³+18u²-4u+1)/(u³-u+2)²]	{-(u²-2u-1)}/{3u²+2u-3}	{-(3u²+2u-1)}/{u(u²+2u+3)}, {u(u²+2u+3)}/{3u²-2u+1}
-2P₁+T₁	[-16(u+1)²(u²+1)(u²-2u-1)/(u²+2u+3)², 8(u+1)²(u²+1)(u²+2u-1)(3u²-2u+1)/(u²+2u+3)³]	[{2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5}/{5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2}, {(u-1)(u+1)(u²+1)(u¹²+12u¹¹+486u¹⁰+996u⁹+207u⁸-296u⁷+468u⁶+296u⁵+207u⁴-996u³+486u²-12u+1)}/{(5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2)²}	{2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5}/{5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2}	{(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}/{u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}, {-u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}/{(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}
2P₁+T₂	[2(u+1)²(3u²-2u+1)²/(u²+2u-1)², -8u²(u-1)²(u+1)²(u²+2u+3)(3u²-2u+1)/(u²+2u-1)³]	[(3u²+2u-3)/(u²-2u-1), 2(u-1)(u+1)(u⁴+4u³+18u²-4u+1)/(u²-2u-1)²]	{3u²+2u-3}/{u²-2u-1}	{4u}/{u²+2u-1}, {-(u²+2u-1)}/{4u}
-2P₁+T₂	[2(u+1)²(3u²-2u+1)²/(u²+2u-1)², 8u²(u-1)²(u+1)²(u²+2u+3)(3u²-2u+1)/(u²+2u-1)³]	[-(u²-2u+1)(3u⁴+12u³-10u²-12u+3)/(7u⁶+22u⁵+31u⁴-36u³-31u²+22u-7), 2(u-1)(u+1)(u¹²+12u¹¹+486u¹⁰+996u⁹+207u⁸-296u⁷+468u⁶+296u⁵+207u⁴-996u³+486u²-12u+1)/(7u⁶+22u⁵+31u⁴-36u³-31u²+22u-7)²]	{-(u²-2u+1)(3u⁴+12u³-10u²-12u+3)}/{7u⁶+22u⁵+31u⁴-36u³-31u²+22u-7}	{-4u(5u⁴+4u³-6u²-4u+5)}/{(u²+2u-1)(u⁴+4u³+18u²-4u+1)}, {(u²+2u-1)(u⁴+4u³+18u²-4u+1)}/{4u(5u⁴+4u³-6u²-4u+5)}
2P₁+T₃	[-2(u²-2u-1)(u²+1)(u²+2u-1)²/(3u²-2u+1)², -8u²(u-1)²(u²+1)(u²+2u-1)(u²+2u+3)/(3u²-2u+1)³]	[-(u³-u+3)/(2u³+u²-1), (u-1)(u+1)(u²+1)(u⁴+4u³+18u²-4u+1)/(2u³+u²-1)²]	{-(u³-u+3)}/{2u³+u²-1}	{u(u²+2u+3)}/{3u²-2u+1}, {-(3u²-2u+1)}/{u(u²+2u+3)}
-2P₁+T₃	[-2(u²-2u-1)(u²+1)(u²+2u-1)²/(3u²-2u+1)², 8u²(u-1)²(u²+1)(u²+2u-1)(u²+2u+3)/(3u²-2u+1)³]	[(5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2)/(2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5), (u-1)(u+1)(u²+1)(u¹²+12u¹¹+486u¹⁰+996u⁹+207u⁸-296u⁷+468u⁶+296u⁵+207u⁴-996u³+486u²-12u+1)/(2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5)²]	{5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2}/{2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5}	{-u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}/{(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}, {(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}/{u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}
3P₁	[4(u²+1)(5u⁴+4u³-6u²-4u+5)²/(u⁴+4u³-6u²+12u-7)², 4(u-1)²(u²+1)(u⁴+4u³+18u²-4u+1)(5u⁴+4u³-6u²-4u+5)(7u⁴+12u³+6u²+4u-1)/(u⁴+4u³-6u²+12u-7)³]	[(5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2)/(2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5), (u-1)(u+1)(u²+1)(u¹²+12u¹¹+486u¹⁰+996u⁹+207u⁸-296u⁷+468u⁶+296u⁵+207u⁴-996u³+486u²-12u+1)/(2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5)²]	{5u⁷+16u⁶+5u⁵+14u⁴-13u³-20u²+3u-2}/{2u⁷+3u⁶+20u⁵-13u⁴-14u³+5u²-16u+5}	{(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}/{u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}, {-u(u²+2u+3)(u⁴+4u³-6u²+12u-7)}/{(3u²-2u+1)(7u⁴+12u³+6u²+4u-1)}
-3P₁	[4(u²+1)(5u⁴+4u³-6u²-4u+5)²/(u⁴+4u³-6u²+12u-7)², -4(u-1)²(u²+1)(u⁴+4u³+18u²-4u+1)(5u⁴+4u³-6u²-4u+5)(7u⁴+12u³+6u²+4u-1)/(u⁴+4u³-6u²+12u-7)³]	[-(5u¹³+32u¹²-282u¹¹-596u¹⁰+3u⁹+292u⁸-268u⁷+792u⁶-69u⁵-56u⁴+582u³-452u²+93u-12)/(12u¹³+93u¹²+452u¹¹+582u¹⁰+56u⁹-69u⁸-792u⁷-268u⁶-292u⁵+3u⁴+596u³-282u²-32u+5), (u-1)(u+1)(u²+1)(u²⁴+24u²³+14556u²²+119128u²¹+391842u²⁰+580296u¹⁹+79724u¹⁸-1047288u¹⁷-687505u¹⁶-91536u¹⁵+406456u¹⁴+2989808u¹³+282076u¹²-2989808u¹¹+406456u¹⁰+91536u⁹-687505u⁸+1047288u⁷+79724u⁶-580296u⁵+391842u⁴-119128u³+14556u²-24u+1)/(12u¹³+93u¹²+452u¹¹+582u¹⁰+56u⁹-69u⁸-792u⁷-268u⁶-292u⁵+3u⁴+596u³-282u²-32u+5)²]	{-(5u¹³+32u¹²-282u¹¹-596u¹⁰+3u⁹+292u⁸-268u⁷+792u⁶-69u⁵-56u⁴+582u³-452u²+93u-12)}/{12u¹³+93u¹²+452u¹¹+582u¹⁰+56u⁹-69u⁸-792u⁷-268u⁶-292u⁵+3u⁴+596u³-282u²-32u+5}	{-(7u⁴+12u³+6u²+4u-1)(17u^{8+40u⁷+4u⁶-56u⁵+22u⁴-104u³+100u²-8u+1)}/{u(u⁴+4u³-6u²+12u-7)(u⁸+8u⁷+100u⁶+104u⁵+22u⁴+56u³+4u²-40u+17)}, {u(u⁴+4u³-6u²+12u-7)(u⁸+8u⁷+100u⁶+104u⁵+22u⁴+56u³+4u²-40u+17)}/{(7u⁴+12u³+6u²+4u-1)(17u^{8+40u⁷+4u⁶-56u⁵+22u⁴-104u³+100u²-8u+1)}}}
...	...	...

■有理数u!=0,±1に対して、Q上の楕円曲線
E_u: y² = x(x-2(u²-u-1))(x+2(u⁴-2u³-2u-1))
および
C_u: y² = (u⁴-4u³+6u²+4u+1)x⁴-4(u²-1)(u²-2u-1)x³+ 2(u⁴+4u³-10u²-4u+1)x² +4(u²-1)(u²-2u-1)x+u⁴-4u³+6u²+4u+1
を定義する。

u=2,3,...,20に対して、E_u,C_uの有理点をいくつか求めると、以下のようになる。

u	E_u:[a₁,a₂,a₃,a₄,a₆]	E_u(Q)_tors E_u(Q)_torsの生成元	rank(E_u(Q))	E_u(Q)/E_u(Q)_torsの生成元	E_u(Q)/E_u(Q)_tors の生成元の高さ	C_u: y²=ax⁴+bx³+cx²+dx+e (a,b,c,d,e)	C_u(Q)	u
2	[0, -28, 0, 180, 0]	Z/2Z×Z/2Z [0, 0], [10, 0]	1	[2 : 16 : 1]	1.1462288522241	(17, 12, 2, -12, 17)	[1, -6], ...	2
3	[0, 8, 0, -1280, 0]	Z/2Z×Z/2Z [0, 0], [32, 0]	1	[-32 : 128 : 1]	1.1462288522241	(40, -64, 176, 64, 40)	[-1, -16], ...	3
4	[0, 188, 0, -11900, 0]	Z/2Z×Z/2Z [0, 0], [50, 0]	2	[-175 : 1575 : 1], [338 : -7488 : 1]	1.74076491514234, 2.87222544209511	(113, -420, 674, 420, 113)	[-1,30], [1/56, 34455/3136], ...	4
5	[0, 656, 0, -52416, 0]	Z/2Z×Z/2Z [0, 0], [72, 0]	1	[-28 : 1400 : 1]	1.58898526713929	(296, -1344, 1712, 1344, 296)	[47/95, -319344/9025], ...	5
6	[0, 1604, 0, -166796, 0]	Z/2Z×Z/2Z [0, 0], [98, 0]	2	[-1127 : 28175 : 1], [-1694 : 4928 : 1]	2.09652859736981, 4.61886316103567	(673, -3220, 3554, 3220, 673)	[-1, -70], [-839/191, -27749530/36481], ...	6
7	[0, 3272, 0, -435200, 0]	Z/2Z×Z/2Z [0, 0], [128, 0]	1	[200 : -7200 : 1]	1.88735304910278	(1352, -6528, 6512, 6528, 1352)	[-169/367, 2776800/134689], ...	7
8	[0, 5948, 0, -989820, 0]	Z/2Z×Z/2Z [0, 0], [162, 0]	2	[10530 : 1347840 : 1], [1058 : 82432 : 1]	2.35389724714885, 3.50417269289867	(2465, -11844, 10946, 11844, 2465)	[1087/506, 29750175/256036], [110/29, 7605/29], ...	8
9	[0, 9968, 0, -2033600, 0]	Z/2Z×Z/2Z [0, 0], [200, 0]	2	[-124 : 20088 : 1], [21043064 : -5618336256 : 343]	2.11385872299762, 9.29897481113499	(4168, -19840, 17264, 19840, 4168)	[319/639, -52036000/408321], [56393/128703, 2005620010400/16564462209], ...	9
10	[0, 15716, 0, -3861836, 0]	Z/2Z×Z/2Z [0, 0], [242, 0]	1	[-9559 : 774279 : 1]	2.55627951586215	(6641, -31284, 25922, 31284, 6641)	[-1, -198], ...	10
11	[0, 23624, 0, -6886656, 0]	Z/2Z×Z/2Z [0, 0], [288, 0]	1	[-14112 : 1411200 : 1]	2.29684079614009	(10088, -47040, 37424, 47040, 10088)	[-1, -240], ...	11
12	[0, 34172, 0, -11664380, 0]	Z/2Z×Z/2Z [0, 0], [338, 0]	2	[-1183 : 244881 : 1], [57154 : 11385495 : 8]	4.00722977176705, 5.96548872344736	(14737, -68068, 52322, 68068, 14737)	[-29/81, -137566/6561], [81/29, 137566/841], ...	12
13	[0, 47888, 0, -18925760, 0]	Z/2Z×Z/2Z [0, 0], [392, 0]	1	[680 : 97920 : 1]	2.45050692698816	(20840, -95424, 71216, 95424, 20840)	[-13, 28560], ...	13
14	[0, 65348, 0, -29609100, 0]	Z/2Z×Z/2Z [0, 0], [450, 0]	2	[-37575 : 6350175 : 1], [-859321222020 : 240272491990356 : 430368875]	2.8656153445757, 13.9669383677014	(28673, -130260, 94754, 130260, 28673)	[-1, -390], [-1809592/4913683, -11030391941205/455552464613], ...	14
15	[0, 87176, 0, -44896256, 0]	Z/2Z×Z/2Z [0, 0], [512, 0]	2	[2312 : -612000 : 1], [-62856 : 9945216 : 1]	4.51685317534012, 6.7743966730988	(38536, -173824, 123632, 173824, 38536)	[-95/241, 1647296/58081], [-10529/7503, -46006299584/56295009], ...	15
16	[0, 114044, 0, -66251516, 0]	Z/2Z×Z/2Z [0, 0], [578, 0]	1	[-64447 : 14500575 : 1]	2.98964082567156	(50753, -227460, 158594, 227460, 50753)	[-1, -510], ...	16
17	[0, 146672, 0, -95463360, 0]	Z/2Z×Z/2Z [0, 0], [648, 0]	2	[1160 : 296960 : 1], [705672 : 651442176 : 1]	2.69959392001499, 5.03326438313988	(65672, -292608, 200432, 292608, 65672)	[-17, 83520], [1703/1033, 551221056/1067089], ...	17
18	[0, 185828, 0, -134689100, 0]	Z/2Z×Z/2Z [0, 0], [722, 0]	1	[234650 : 152053200 : 1]	3.09958168655876	(83665, -370804, 249986, 370804, 83665)	[11987/5816, 14073473375/33825856], ...	18
19	[0, 232328, 0, -186502400, 0]	Z/2Z×Z/2Z [0, 0], [800, 0]	1	[-644 : 464968 : 1]	2.8036370798768	(105128, -463680, 308144, 463680, 105128)	[3239/6479, -25290248400/41977441], ...	19
20	[0, 287036, 0, -253943676, 0]	Z/2Z×Z/2Z [0, 0], [882, 0]	1..2	[-158319 : 57153159 : 1], ???	3.19833249499868, ???	(130481, -572964, 375842, 572964, 130481)	[-1, -798], ...	20

■A.J.MacLeod[2]は、楕円曲線EのQ(u)-有理点-P₁(4(u²+1),-4(u-1)²(u²+1)), P₁+T₁(-(u+1)²(u²-2u-1),(u-1)²(u+1)²(u²-2u-1))から、Diophantus方程式系Sの2変数m,nによるパラメータ解を求めている。

    s = 4(m⁶+2m⁵n+m⁴n²+4m³n³-m²n⁴+2mn⁵-n⁶),
    x = 2mn(m²+2mn+3n²)(3m²-2mn+n²),
    y = (m²-n²)(m²+2mn+3n²)(3m²-2mn+n²),
    a = (m²+n²)(m²+2mn+3n²)(3m²-2mn+n²),
    b = (m²+n²)(5m⁴+4m³n-6m²n²-4mn³+5n⁴),
    c = (m²+n²)(m⁴+4m³n+18m²n²-4mn³+n⁴);

    s = 2(3m⁴+4m³n+6m²n²-4mn³n+3n⁴),
    x = 16mn(m²+2mn-n²),
    y = 8(m²-n²)(m²+2mn-n²),
    a = 8(m²+n²)(m²+2mn-n²),
    b = 2(5m⁴+4m³n-6m²n²-4mn³+5n⁴),
    c = 2(m⁴+4m³n+18m²n²-4mn³+n⁴).

[pari/gpによる検証]

gp>  X=2*m*n*(m^2+2*m*n+3*n^2)*(3*m^2-2*m*n+n^2);
time = 0 ms.
gp>  Y=(m^2-n^2)*(m^2+2*m*n+3*n^2)*(3*m^2-2*m*n+n^2);
time = 2 ms.
gp>  S=4*(m^6+2*m^5*n+m^4*n^2+4*m^3*n^3-m^2*n^4+2*m*n^5-n^6);
time = 1 ms.
gp>  A=(m^2+n^2)*(m^2+2*m*n+3*n^2)*(3*m^2-2*m*n+n^2);
time = 0 ms.
gp>  B=(m^2+n^2)*(5*m^4+4*m^3*n-6*m^2*n^2-4*m*n^3+5*n^4);
time = 0 ms.
gp>  C=(m^2+n^2)*(m^4+4*m^3*n+18*m^2*n^2-4*m*n^3+n^4);
time = 1 ms.
gp>  X^2+Y^2-A^2
time = 6 ms.
%10 = 0
gp>  (S-X)^2+Y^2-B^2
time = 1 ms.
%11 = 0
gp>  X^2+(S-Y)^2-C^2
time = 0 ms.
%12 = 0

[参考文献]

[1]Allan J. MacLeod, "Parametric Expressions for Integer Equilateral Triangles with a Point at ineteger distance from Each vertex (I)", May 21, 2002, p1-8.
[2]Allan J. MacLeod, "Parametric Expressions For 3 Rational Distances From The Corners of A Square", p1-4.
[3]Richard K. Guy, "Unsolved Problems in Number Theory Third Edtion", Springer-Verlag, 2004, p283-287, ISBN0-387-2060-7.

Last Update: 2007.02.22

H.Nakao