山田 智宏の別館


343と239


投稿日: 2017/12/25 00:00 GMT

本ページは前回の続きの記事で、インテジャーズ Advent Calendar 2017の25日目(最終日)の記事です。

前回、Fermatの2つの問題を紹介し、それぞれ \begin{equation}\label{eq1} \sigma(x^3)=y^2 \end{equation} および \begin{equation}\label{eq2} \sigma(x^2)=y^3 \end{equation} となる整数 $x, y$ を求める問題として定式化できることと、 $x=p$ が素数の場合に限れば問題は \begin{equation}\label{eq3} p^3+p^2+p+1=y^2 \end{equation} および \begin{equation}\label{eq4} p^2+p+1=y^3 \end{equation} と多項式を使ってあらわすことができ、解は \[\sigma(7^3)=1+7+7^2+7^3=20^2\] しかないことを証明しました。

そこで、次に多項式であらわされた方程式 $(\ref{eq3})$ および $(\ref{eq4})$ のほうに着目し、 $p$ が素数という条件を外すとどうなるでしょうか。つまり \begin{equation}\label{eq5} 1+x+x^2+x^3=y^2 \end{equation} および \begin{equation}\label{eq6} 1+x+x^2=y^3 \end{equation} で $x$ が素数でなくてもよいとして、整数解を考えるわけです。

まず、 $x$ が素数でないときにはこれはFermatのもとの問題とは全く別の問題であることに 注意しなければいけません。たとえば $(\ref{eq6})$ には自明と思える $(x, y)=(-1, 1), (0, 1)$ のほか \[ 1+18+18^2=7^3 \] という解が存在します。驚くべきことに、ここで再び $7^3=343$ が登場します! 一方で、 \[ \sigma(18^2)=\sigma(2^2\times 3^4)=7\times 11^2 \] は立方数ではありません(もちろんその他の累乗数でもありません)。 また $(\ref{eq5})$ には前回取り上げました $(x, y)=(7, 20)$ のほかに (どちらかといえば自明ながら)$(-1, 0), (0, 1), (1, 2)$ といった解が存在します。

ところで、$(\ref{eq5})$ に関しては、 $y$ を $-y$ に変えても解になりますから、 $y\geq 0$ の場合のみ考えても問題ありません。また $x\geq -2$ ならば $1+x+x^2+x^3=(1+x)(1+x^2)<0$ となって、当然負の数は平方数ではありえませんから ( $(-1, 0)$ を別にすれば) $x$ も $0$ 以上の場合のみ考えて問題ありません。 また $(\ref{eq6})$ に関しては、なお、左辺は常に正で、かつ \[1+(-x)+(-x)^2=x^2-x+1=(x-1)^2+(x-1)+1\] となりますので、 この方程式でも $x, y$ は $0$ 以上として問題はありません。

$(\ref{eq5}), (\ref{eq6})$ のどちらの方程式も楕円曲線をあらわしているので、 Magma や Sage を使って、整数解をすべて求めることはできますがこれらのソフトは楕円曲線の整数点を求めるために、 楕円対数というかなり高度な道具を使っています。

実は19世紀末から20世紀初頭にかけて、すでにこれらの2つの方程式は解かれています。

$(\ref{eq5})$ の解は $(x, y)=(-1, 0), (0, \pm 1), (1, \pm 2), (7, \pm 20)$ のみで、 $(\ref{eq6})$ の解は $(x, y)=(-19, 7), (-1, 1), (0, 1), (18, 7)$ のみである。

$(\ref{eq5})$ は1876年にGerenoが初等的な方法で解き、その後1883年にGennochiが より簡単な解き方を発見しました。 一方 $(\ref{eq6})$は1921年にNagellが解きましたが、これは二次体 $\mathbb{Q} (\sqrt{-3})$ での分解を使う、幾分複雑なものでした。なお、Nagellは一般に \[1+x+x^2=y^n\] および \[1+x+x^2=3y^n\] の解もそれぞれ $(x, y^n)=(0, 1), (18, 7^3)$ および $(x, y^n)=(1, 1)$ しかないことを示しています。

そこで、初等的な解き方が知られている $(\ref{eq5})$ について解説したいと思います。

$1+x+x^2+x^3=y^2$

まず2つの不定方程式(これらも楕円曲線をあらわしています)を解くことからはじめます。

補題1

\[2Y^2=X^4-1\] の整数解は (X, Y)=(\pm 1, 0) のみである。

右辺を因数分解して \[2Y^2=(X^2-1)(X^2+1)\] とあらわすことで(さらに $(X-1)(X+1)(X^2+1)$ と因数分解できますが、)、 $X^2-1$ と $X^2+1$ の公約数は $1$ または $2$ しかありえない ことがわかりますので、 \[X^2-1=Y_1^2, X^2+1=2Y_2^2\] または \[X^2-1=2Y_1^2, X^2+1=Y_2^2\] のいずれかのでなければなりません。 したがって $X^2-1$ または $X^2+1$ のどちらかは平方数でなければならず、 そのようなものは $X=0, \pm 1$ のいずれかしかありません。 $X^4-1=2Y^2\geq 0$ なので $X=\pm 1, Y=0$ でなければならないことがわかります。

補題2

\[Y^2=8X^4+1\] の解は $(X, Y)=(0, \pm 1), (\pm 1, \pm 3)$ のみである。

$(X, Y)$ が解ならば、 $X$ や $Y$ の符号を変えても解ですから $X, Y\geq 0$ の場合のみ解けば、符号を変えればすべての解が得られます。 $Y$ は奇数でないといけないので $Y=2Z+1$ とおきます。すると $Y\geq 0$ より $Z\geq 0$ かつ \[8X^4+1=(2Z+1)^2=4Z^2+4Z+1\] つまり \[2X^4=Z(Z+1)\] となることがわかります。

よって \[Z=s^4, Z+1=2t^4 \textrm{または} Z=2s^4, Z+1=t^4\] のどちらかが成り立ちます。 つまり \[2t^4=s^4+1 \textrm{または} t^4=2s^4+1\] のどちらかが成り立ちます。
$t^4=2s^4+1$ ならば補題1の特殊な場合になりますので $(s, t)=(0, \pm 1)$ より $Z=X=0, Y=1$ です。 もう一つの場合、 $2t^4=s^4+1$ を変形すると \[(t^2)^4-s^4=t^8-2t^4+1=(t^4-1)^2\] となります。

実はFermatは \[A^4-B^4=C^2\] の整数解は、 $ABC=0$ となるものしか存在しないことを証明しています(ここから最終定理の4次の場合が導かれます)。 詳しい解説は 拙サイト にあります。

よって、$t, s, t^4-1$ のいずれかは $0$ でなければならず、 $s=0, t=-1, t=0, t=1$ のいずれかであることがわかります。 $s=0$ のとき $2t^4=1$ となってしまい $t=0$ のとき $s^4+1=0$ となってしまいますから、 これらの場合は不適当です。したがって、$t=\pm 1, s=\pm 1$ しかありえないことがわかります。 そうすると $Z=1, X=1$ より $(X, Y)=(1, 3)$ となることがわかります。

以上より、 $X, Y\geq 0$ の場合、解は $(X, Y)=(0, 1), (1, 3)$ のみであることがわかります。 一般の場合には、$(X, Y)=(0, \pm 1), (\pm 1, \pm 3)$ がすべての解であることがわかります。


これらを踏まえて、元の方程式 $(\ref{eq5})$ に戻ります。 ただし $x\geq 0$ の場合に限定して考えます。

\[y^2=(x+1)(x^2+1)\] より \[x+1=z^2, x^2+1=w^2, y=zw\] または \[x+1=2z^2, x^2+1=2w^2, y=2zw\] のいずれかが成り立つことがわかります。それぞれ第一の場合、第二の場合と呼ぶことにします。 といっても第一の場合、 $x^2+1=w^2$ の解は $(x, w)=(0, \pm 1)$ しかありえませんから この場合にあたるのは $(x, y)=(0, \pm 1)$ しかありません。

第二の場合、 $x\geq 0$ の場合に限定していますから、 $z>0$ となることに注意します。 さて $x=2z^2-1$ を $x^2+1=2w^2$ に代入すると、 \[2w^2=(2z^2-1)^2+1=4z^4-4z^2+2\] つまりピタゴラス型の関係 \[w^2=2z^4-2z^2+1=z^4+(z^2-1)^2\] が成り立つことがわかります。よってピタゴラス方程式の解の公式(これも拙サイトに解説があります)から \[z^2=m^2-n^2, z^2-1=2mn\] または \[z^2=2mn, z^2-1=m^2-n^2\] のいずれかが成り立つ、互いに素な整数 $m, n$ がとれます。改めて、これらをそれぞれ 第一の場合、第二の場合と呼ぶことにします。

第一の場合、$z^2=2mn+1$ より $z$ は奇数で、 \[z^2=(m+n)(m-n)\] となります。$m, n$ は互いに素なので $\gcd(m+n, m-n)$ は $1$ または $2$ ですが $z$ は奇数なので結局 $m+n, m-n$ も互いに素でなければならず、これら2つの数 $m+n, m-n$ は共に平方数です。

したがって \[m+n=r^2, m-n=s^2\] とおくと \[2m^2-(m+n)^2=m^2-n^2-2mn=z^2-(z^2-1)=1\] より \begin{equation}\label{eq7} 2m^2-1=(m+n)^2=r^4 \end{equation} となりますが \[m^4-r^4=m^4-2m^2+1=(m^2-1)^2\] より、先程も触れましたFermatの結果から $r=0, m=-1, m=0, m=1$ のいずれかでなければなりません。 しかし $r=0$ ならば $2m^2=r^4+1=1$、 $m=0$ ならば $r^4=2m^2-1=-1$ となりますから $m=\pm 1, r=\pm 1$ の可能性しかありません。このとき $n=0, z=\pm 1, x=1, y=\pm 2$ となります。

第二の場合、 $z^2=2mn$ より $z$ は偶数ですから \[m^2-n^2=z^2-1\equiv 3\pmod{4}\] より $m$ は偶数、 $n$ は奇数でなければならず、かつ前にも述べたように互いに素ですから $z^2=2mn$ より $m=2r^2, n=s^2$ とおくことができます。 よって \[4r^4-s^4=z^2-1=2mn-1=4r^2s^2-1\] つまり \[-4r^4+4r^2s^2+s^4=1\] です。よって \[8r^4+1=(2r^2+s^2)^2\] となりますから、補題2より $(r, 2r^2+s^2)=(0, \pm 1), (\pm 1, \pm 3)$ のいずれかです。$r=0$ ならば $z=0$ となってしまいます。 $r=\pm 1$ の場合、 $s^2=\pm 1$ より $s=\pm 1$ となります。 したがって $m=2, n=1$ でなければならず、 $z^2=2mn=4$ より $z=2, x=2z^2-1=7$ でなければならないことがわかります。

以上より、 $x\geq 0$ のときの $(\ref{eq5})$ の解は $(x, y)=(0, \pm 1), (1, \pm 2), (7, \pm 20)$ で、 $(x, y)=(-1, 0)$ とあわせると $(\ref{eq5})$ の解は $(x, y)=(-1, 0), (0, \pm 1), (1, \pm 2), (7, \pm 20)$ のみであることがわかります。


ところで証明の過程で不定方程式 \[2Y^2=X^4-1, Y^2=8X^4+1\] を解き、また $(\ref{eq7})$ に関する考察から \[2Y^2=X^4+1\] の解は $(X, Y)=(\pm 1, \pm 1)$ のみであることもわかります。 そこで類似した不定方程式についても考えてみます。すなわち \[Y^2=2Y^4+1, Y^2=2Y^4-1, Y^2=8Y^4-1\] のような方程式の解をすべて求めたいわけです。

3つめの $Y^2=8Y^4-1$ は $8$ を法として考えると、すぐに不可能であることがわかります。 1つめの $Y^2=2X^4+1$ は補題1から比較的容易に解くことができます。実際 $Y>0$ のとき $Y-1=2s^4, Y+1=t^4$ または $Y-1=s^4, Y+1=2t^4$ ですが前者は $t^4=2s^4+2\equiv 2, 4 \pmod{8}$ より不可能で、後者は $s$ が偶数なので $s=2u$ とおくと $t^4-1=8u^4=2(2u^2)^2$ ですから、ここで補題1を使うわけです。 すると $(t, u)=(\pm 1, 0)$ より $Y=1$ でなければいけません。

2つめの $Y^2=2X^4-1$ はどうでしょう。 $(X, Y)=(\pm 1, \pm 1)$ はすぐにわかりますが、 他に解はあるでしょうか。実はさらに $(X, Y)=(\pm 13, \pm 239)$ という解をもちます。 そして解はこれ以外にないことがLjunggrenによって1942年に証明されましたが、その証明は容易ではありません。
その後、1991年にSteinerとTzanakisが対数一次形式を用いる、比較的簡単な証明を与えましたが、 完全に初等的な証明は実は未だに知られていません。もちろんこの方程式は楕円曲線をあらわしていますから MagmaやSageで容易に解くことができますが(実際 $x=2X^2, y=2XY$ とおくとWeierstrassの標準形 $y^2=x^3-2x$ に変換できます)、上記の通り、内部処理には楕円対数というかなり高度な道具が使われていますから、 初等的な証明とは言い難いものがあります。

ところで、$Y^2=2X^4-1$ の解として現れた $239$ ですが、円周率に詳しい方ならば、 Machinの公式 \[\frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}\] や、その他の円周率に関する複数の公式 (これについては、インテジャーズ本家ブログで触れられています) に出現することでお馴染みの数かと思います。 このことは決して偶然ではなく $239^2+1=2\times 13^4$ と小さな素数の累乗の$2$倍という形に 素因数分解される事実が円周率の公式に出現する背景となっているのです(さらに、$239$ は $n^2+1$ が $13$ 以下の素数の積で あらわされる最大の $n$ でもあります (Størmer, 1897)。この事実も円周率の公式と関連があります)。これについては、 詳しい話は来年の関西日曜数学友の会でお話しする予定です。

参考文献

あとがき

はじめに書いたとおり、本ページは前回の続きの記事で、またインテジャーズ Advent Calendar 2017の25日目にして最後の記事です。

今回は、純粋な多項式不定方程式の話題でしたがいかがでしょうか?

実は今回の不定方程式 $y^2=x^3+x^2+x+1$ は、以前にtwitterで何度か話題になっていて (関連するtweetが消えていますが これとかこれ)、 私も3月に一度Genocchiの解を参考とした解説をtweetしたのですが(よく見ると、 $Y^2=2X^4+1$ は必要ではないことがわかります)、 今回、インテジャーズ Advent Calenderでこれについてお話ししようと考え、まとめ直しました。

で、過去のtweetを読み直していましたところ、前回触れました \[y^2=p^3+p^2+p+1\] について、もっと単純な解き方が存在することを思い出しましたので、前回の記事も修正いたします。

最後に、インテジャーズ Advent Calendar 2017の企画をされた、インテジャーズ管理人のせきゅーん氏(twitterはこちら)と、前回記事および本記事を読んでくださった読者の皆様方に感謝申し上げます。

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