Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?

Solution 1:

Your conjectured statement isn't true. Take $x=\frac{36}{100}$ and $n=2$. Then

$$x^{1-\frac1n}+y^{1-\frac1n} =\left(\tfrac{36}{100}\right)^{\tfrac12}+\left(\tfrac{64}{100}\right)^{\tfrac12}=\tfrac75 $$

Solution 2:

Let $u=x^{n-1}$ and $v=y^{n-1}$ By Boreico’s theorem (see section "Higher powers" on pages 91-92 at http://www.thehcmr.org/issue2_1/mfp.pdf), $u^{\frac{1}{n}}+v^{\frac{1}{n}}$ is rational iff $u^{\frac{1}{n}}$ and $v^{\frac{1}{n}}$ are both rational. Since $n-1$ and $n$ are coprime, this is equivalent to $x^{\frac{1}{n}}$ and $y^{\frac{1}{n}}$ being both rational. As Jack M remarked, your conjecture is then equivalent to Fermat’s last theorem.