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4階と5階のあいだ
1.
4階と5階のあいだ みうら 2017/4/1 ロマンティック数学ナイトボーイズ ~「有限」に思いを馳せて~
2.
・大学院(修士)で代数学を勉強中 ・素数大富豪世界1位(ってことになってる) ・最近の趣味は、パズルを作って 「みらいけん数学デー」に持っていくこと 自己紹介 楽しい
3.
今日の日付は… 1/4 4/1 = 4 =
0.25 有限小数! イギリス式英語での表記だと
4.
2以上の整数の逆数を小数で表す (1/1=1は小数とは言い難いので) 1/2=0.5 1/3=0.333333… 1/4=0.25 1/5=0.2 1/6=0.166666… 1/7=0.142857… 1/8=0.125 1/9=0.111111… 1/10=0.1 1/11=0.090909… 赤字:有限小数 隣接 している
5.
疑問 この先、このように有限小数が 連続することはあるのか?
6.
有理数𝑟について、 𝑟を既約分数で表したとき 分母の素因数が2と5のみ 1/4=1/(2∙2)=0.25 1/5=1/5=0.2 1/6=1/(2∙3)=0.1666… 1/7=1/7=0.142857… 𝑟が(十進法で) 有限小数で書ける 前提知識 ⇔ ↑有限小数 ↑無限小数
7.
これより、 1 𝑛 と 1 𝑛+1 がともに有限小数で 表せるための必要十分条件は 𝑛 + 𝑛
= 2 𝑎 ∙ 5 𝑏 𝑛 + 1 = 2 𝑐 ∙ 5 𝑑 なる非負整数 𝑎, 𝑏, 𝑐, 𝑑 が存在すること
8.
しかし、連続する二つの整数がともに偶数 だったり、ともに5の倍数ということはない 結局 𝑛, 𝑛 +
1 = {2 𝑒, 5 𝑓} (𝑒, 𝑓は1以上の整数) でなければならない
9.
ここで大道具
10.
カタラン予想
11.
〈カタラン予想〉 連続する「べき乗数」は8と9のみ べき乗数とは、 𝑢 𝑣(𝑢,
𝑣は2以上の整数) と書ける数のこと 2002年、P.ミハイレスクにより肯定的に解決
12.
(𝑛は2以上なので)カタラン予想より、 𝑛, 𝑛 +
1 = 2 𝑒 , 5 𝑓 ⇒ 𝑒 = 1 または 𝑓 = 1 結局、 𝑛, 𝑛 + 1 = 22 , 5 = 𝟒 , 𝟓 しか ありえない(証明完了!) ※カタラン予想を使わずに初等的に解く方法もあります
13.
系として、2以上の整数𝑛に対して 𝑛 + 1 𝑛 , 𝑛 𝑛
+ 1 がともに有限小数となるのは 𝑛 = 4 のとき のみであることがわかる ( 5 4 = 1.25 , 4 5 = 0.8 )
14.
15.
16.
踊り場の表記にはこういうのもある↓ 5|4 この書き方は特に、階段を下るときは 4を5で割りたくなる
17.
結論 「4階と5階のあいだ」は 上りも下りも有限小数となる 唯一の場所である 4階と5階を行き来するときは ぜひとも階段で!
18.
みなさん、 「有限」に思いを馳せてみませんか?
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