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Question Number 100622 by MJS last updated on 27/Jun/20

Commented by MJS last updated on 27/Jun/20

$$\mathrm{building}\mathrm{on}\mathrm{my}\mathrm{work},\mathrm{can}\mathrm{someone}\mathrm{try}\mathrm{to}$$$$\mathrm{find}\mathrm{the}\mathrm{possible}\mathrm{values}\mathrm{for}x\in \mathbb{Q}\mathrm{and}$$$$x\left[x\left[x\left[x\right]\right]\right]=n\in \mathbb{N}?$$$$\mathrm{in}\mathrm{the}\mathrm{picture},i\mathrm{and}f\mathrm{are}\mathrm{the}\mathrm{integer}\mathrm{and}$$$$\mathrm{the}\mathrm{fractional}\mathrm{part}\mathrm{of}x$$

Commented by prakash jain last updated on 03/Jul/20

$$\mathrm{Q}96329\mathrm{has}\mathrm{a}\mathrm{special}\mathrm{case}.$$$$x\lfloor x\lfloor x\lfloor x\rfloor \rfloor \rfloor =n$$$$\mathrm{Trying}\mathrm{for}\mathrm{general}\mathrm{case}$$$$x⩽\lfloor n^{1/4}\rfloor$$$$x=i+f\mathrm{where}i⩽\lfloor n^{1/4}\rfloor$$$$x⩽n^{1/4}\Rightarrow x\lfloor x\lfloor x\rfloor \rfloor \rfloor ⩾n^{3/4}$$$$\mathrm{induction}\mathrm{may}\mathrm{help}.$$

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