The irrational-exponent conjecture for Fermat-type equations

Let $a,b,c$ be positive integers and let $n$ be an exponent such that

$$a^n+b^n=c^n.$$

Irrational-exponent conjecture. Then $n$ is irrational. Equivalently, no such equation has a rational exponent $n$ under these hypotheses. This statement is presented after the paper's claimed proof of Fermat's Last Theorem, so its mathematical status is not established by the supplied text.