A remark on Fermat's last theorem

It is proved that the equation $x^p+y^p+z^p=0$, $(xyz,p)=1$ has no solutions in rational integers $x$, $y$, $z$ for all odd prime numbers $p$ for which $q=pk+1$ is a prime number, $k\leqslant82$, $k\not\equiv0$ $(\operatorname{mod}3)$.