# Necessary condition for x>0 being an integer

I was trying to solve a number theory problem and then I realized that I was needing to verify (prove or disprove) the following ”fact” about numbers. I would appreciate any help.

Q: Suppose $$x >0$$ is such that $$x^n in mathbb{Z}$$ for all $$n geq 2$$, then $$x$$ must be an integer? Furthermore, suppose that $$ngeq 3$$ is an odd integer, does the conclusion holds?

Mathematics Asked by Aeternal on November 18, 2021

Suppose that $$x^2,x^3inBbb Z$$. Then $$x^2(x-1)=x^3-x^2inBbb Z$$, so $$x-1$$ is rational. The square of a rational number is an integer if and only if the rational number itself is an integer, so $$x-1$$ is an integer, and therefore so is $$x$$.

Answered by Brian M. Scott on November 18, 2021

If $$x^2=kinmathbb{Z}$$, then $$x=sqrt{k}.$$ If $$x$$ is not an integer, then neither is $$x^3=ksqrt{k}$$.

A similar argument proves the second statement also.

Answered by saulspatz on November 18, 2021

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