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Question Number 77204 by mr W last updated on 04/Jan/20

$$isthereageneralsolutionfor$$ $$theequation$$ $$\underset{ntimesx}{\mathit{x}\left[\mathit{x}\left[\mathit{x}[\mathit{x}...]\right]\right]}=\mathit{m}$$ $$withx>0,m>0.$$

Commented byMJS last updated on 04/Jan/20

$$\mathrm{can}\mathrm{we}\mathrm{solve}\mathrm{for}\mathrm{all}m,n?\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{think}\mathrm{so}$$ $$\mathrm{you}\mathrm{showed}x\left[x\left[x\left[x\right]\right]\right]=88\Rightarrow x=22/7$$ $$\mathrm{but}x\left[x\left[x\left[x\right]\right]\right]=87\mathrm{seems}\mathrm{to}\mathrm{have}\mathrm{no}\mathrm{solution}$$ $$\mathrm{it}\mathrm{could}\mathrm{be}\mathrm{helpful}\mathrm{to}\mathrm{determine}\mathrm{the}\mathrm{set}{S}_n\subset \mathbb{N}$$ $$\mathrm{with}{y}_n\in S_n,y_n=x{\left[x\right]}_{n\mathrm{times}}$$

Commented bymr W last updated on 04/Jan/20

$$youarerightsir.foragivennthe$$ $$equation{doesn}^{\prime }thavesolutionfor$$ $$everym.e.g.x\left[x\right]=3hasnosolution.$$

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