is-there-a-general-solution-for-the-equation-x-x-x-x-n-times-x-m-with-x-gt-0-m-gt-0-

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

$${is}\:{there}\:{a}\:{general}\:{solution}\:{for} \\ $$$${the}\:{equation} \\ $$$$\underset{{n}\:{times}\:{x}} {\boldsymbol{{x}}\left[\boldsymbol{{x}}\left[\boldsymbol{{x}}\left[\boldsymbol{{x}}…\right]\right]\right]}=\boldsymbol{{m}} \\ $$$${with}\:{x}>\mathrm{0},\:{m}>\mathrm{0}. \\ $$

Commented by MJS last updated on 04/Jan/20

$$\mathrm{can}\:\mathrm{we}\:\mathrm{solve}\:\mathrm{for}\:\mathrm{all}\:{m},\:{n}?\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{so} \\ $$$$\mathrm{you}\:\mathrm{showed}\:{x}\left[{x}\left[{x}\left[{x}\right]\right]\right]=\mathrm{88}\:\Rightarrow\:{x}=\mathrm{22}/\mathrm{7} \\ $$$$\mathrm{but}\:{x}\left[{x}\left[{x}\left[{x}\right]\right]\right]=\mathrm{87}\:\mathrm{seems}\:\mathrm{to}\:\mathrm{have}\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{it}\:\mathrm{could}\:\mathrm{be}\:\mathrm{helpful}\:\mathrm{to}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{set}\:{S}_{{n}} \subset\mathbb{N} \\ $$$$\mathrm{with}\:{y}_{{n}} \in{S}_{{n}} ,\:{y}_{{n}} ={x}\left[{x}\right]_{{n}\:\mathrm{times}} \\ $$

Commented by mr W last updated on 04/Jan/20

$${you}\:{are}\:{right}\:{sir}.\:{for}\:{a}\:{given}\:{n}\:{the} \\ $$$${equation}\:{doesn}'{t}\:{have}\:{solution}\:{for} \\ $$$${every}\:{m}.\:{e}.{g}.\:{x}\left[{x}\right]=\mathrm{3}\:{has}\:{no}\:{solution}. \\ $$

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