Fermat's Equation Has No Solution with Some Prime Components

Abstract

Within the scope of elementary number theory, we prove that, as the main result, if $1\le x<y<z$ are integers such that at least one of $y,z,x+y$ is prime then $x^n+y^n\ne z^n$ for every odd integer $n\ge 3$. This result covers a special case of a conjecture of Abel, and furnishes a definite way to construct infinitely many setwise coprime integers that do not satisfy the Fermat's equation uniformly in $n$.

arXiv comment3 pages