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oskar
2026-05-01 04:47:38 +02:00
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@@ -274,34 +274,20 @@ header-auto-numbering:
> > - $\underbrace{(\neq 2)222}_{\hspace{-13ex}\mathrlap{\text{Par le thm. du jour 2}}}1]$
> > - $3211]$
> > - $31221]$
31221
> > - $3112211]$
3112211
> > - $3212221]$
3212221
> > - $312113211]$
312113211
> > - $3111221131221]$
3111221131221
> > - $\underbrace{(\neq 3)33}_{\hspace{-5em}\mathrlap{\text{Par le thm. du jour 2}}}1222113112211]$
331222113112211
> > - $2\cdot \underbrace{\overbracket{\color{#FCD600}311}^{\mathclap{[3^{1}X^{\neq 3}}}\color{#FCD600}322113212221]}_{\text{cycle }(1)}$
2311322113212221
> > - $2\cdot \color{#FCD600}13211322211312113211]$
213211322211312113211
> > - $2\cdot \color{#FCD600}1113122113322113111221131221]$
21113122113322113111221131221
> > - $2 \cdot 311311222\cdot \overbracket{\color{#FCD600}12}^{\mathclap{[1^{1}X^{1}}}\color{#FCD600}322211331222113112211]$
231131122212322211331222113112211
> > - $2\cdot 1321132132\cdot \overbracket{111}^{[1^{3}}2133\cdot \overbracket{2212}^{\mathclap{[2^{2}1^{1}X^{1}}}\cdot \underbrace{\overbracket{\color{#FCD600}311}^{\mathclap{[3^{1}X^{\neq 3}}}\color{#FCD600}322113212221]}_{\text{cycle } (1)}$
2132113213211121332212311322113212221
> > - $2\cdot 111312211312111312\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}2112\cdot \overbracket{3\cdot \underbracket{22}_{\mathclap{[2^{2}1^{3}]}}}^{\mathclap{[3^{1}X^{\neq 3}}} 1112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}211322211312113211$
> > - $2\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}3112221131112\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}31112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}21122112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}22\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq_3}}}2\cdot \overbracket{\color{#FCD600}111}^{[1^{3}}\color{#FCD600}3122113322113111221131221$
> > - 23113112221131112311311121321122112132231121113122113322113111221131221
> > - $2\cdot 1321132\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}221133112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}21133112\cdot \overbracket{111}^{[1^{3}}31221222112\cdot \overbracket{111}^{[1^{3}}3\cdot \overbracket{2213}^{\mathclap{[2^{2}1^{1}X^{1}}}2112\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}311222\cdot \overbracket{12}^{[1^{1}X^{1}}322211331222113112211$
> > - 213211321322113311 213211331121113122122211211132213211231131122212322211331222113112211
> > - 2111312211312111322212322121113122123211231131122113221123113221113122112132113213211121332212311322113212221
> > - 2111312211312111322212321121113122123211231131122113221123113221113122112132113213211121332212311322113212221
> > - $2\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}2112\cdot \overbracket{3\cdot \underbracket{22}_{\mathclap{[2^{2}1^{3}]}}}^{\mathclap{[3^{1}X^{\neq 3}}} 1112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}211322211312113211$
> > - $2\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}21122112\cdot \overbracket{13}^{\mathclap{[1^{1}X^{1}}}22\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq_3}}}2\cdot \overbracket{\color{#FCD600}111}^{[1^{3}}\color{#FCD600}3122113322113111221131221$
> > - $2\cdot \overbracket{2213}^{\mathclap{[2^{2}1^{1}X^{1}}}2112\cdot \overbracket{311}^{\mathclap{[3^{1}X^{\neq 3}}}311222\cdot \overbracket{12}^{\mathclap{[1^{1}X^{1}}}322211331222113112211$
> > - $2\cdot 1321132132\cdot \overbracket{111}^{[1^{3}}2133\cdot22\cdot \overbracket{\color{firebrick}\underset{\ce{Ca}}{12}}^{\mathclap{[1^{1} X^{1}}}\cdot \overbracket{\color{#FCD600}311}^{\mathrlap{\hspace{-3ex}[3^{1}X^{\neq 3}}\hspace{-3ex}}\color{#FCD600}322113212221$
> >
> >
> > Ce qui montre bien que toute chaîne qui termine par $1$ finit par atteindre le cycle $(1)$.
@@ -478,7 +464,8 @@ Conway leur donne des noms d'éléments (de l'hydrogène à l'uranium, ce qui fa
- Tous les $C_{\geq t_0 + 400}$ contiendront simultanément tous les éléments
- $\ce{U} \in C_{t_0+300}$ donc $\ce{Pa} \in C_{t_0+301}$, et même $\ce{U\&}\ce{Pa} \in C_{t_0 + 301}$ par la propriété précédente. Pour les mêmes raison : $\ce{U\&Pa\&Th} \in C_{t_0+302},\quad$ $\ce{U\&Pa\&Th\&Ac} \in C_{t_0+303},\quad$ $\ce{U\&Pa\&Th\&Ac\&Ra} \in C_{t_0+303},\dots$ et ainsi de suite.
3. Soit $L$ une chaîne différente de $[\;]$ ou $[22]$.
Si $L$ est de la forme $L'2^{2}]$, la propriété suit du fait que Calcium (qui correspond à $12$)
Si $L$ est de la forme $L'2^{2}]$, on considère $L'$ à la place de $L$ (on ignore le $2^{2}]$).
Ainsi, on peut affirmer que $L$ correspond soit au cycle $(1)$, soit au cycle $(2)$ dans le théorème de la fin. En observant la preuve du théorème de la fin, on remarque l'apparition du Calcium $(\ce{Ca})$ dans les deux cycles, ce qui montre qu'un descendant assez avancé de $L$ contient du Calcium,