Three summation criteria for Fermat’s last theorem

Abstract:

This paper extends the search for solutions of the congruences \[ \sum \limits _1^{[p/6]} {\frac {1}{i} \equiv 0,} \quad \sum \limits _1^{[p/6]} {\frac {1}{{{i^2}}} \equiv 0} \quad {\text {and}}\quad \sum \limits _{[p/6] + 1}^{[p/5]} {\frac {1}{i} \equiv 0\;\pmod p} \] to the limit $p < 600000$. The only solutions found were $p = 61$ in the first case, in the second $p = 205129$, and in the third case $p = 109$ and $p = 491$.