Finding the Real Solution for Fractional Part Equations with Given Values of k

[juantheron]

Calculation of Real $x$ in $\{x\} = \{x^2\} = \{x^3\}$, Where $\{x\} =$ fractional part of $x$

My Try:: We Know that $\{x\}$ and $\{x^2\}$ and $\{x^3\}$ are are $\in \left[0,1\right)$

So Let we take $\{x\} = \{x^2\} = \{x^3\} = k$, where $k\in \left[0,1\right)$

So If $\{x\} = k$ , Then $x-\lfloor x \rfloor = k\Rightarrow x = \lfloor x \rfloor +k$

where $\lfloor x \rfloor =$ floor function of $x$

Similarly If $\{x^2\} = k$ , Then $x^2-\lfloor x^2 \rfloor = k\Rightarrow x^2 = \lfloor x^2 \rfloor +k$

Sililarly If $\{x^3\} = k$ , Then $x^3-\lfloor x \rfloor = k\Rightarrow x^3 = \lfloor x^3 \rfloor +k$

Now How Can I proceed after that,

please Help me

Thanks

 

[verty]

Hint: test some values for k.