In the first part of this paper, we give a generalization of Kummer congruences and Mirimanoff congruences for the first case of Fermat's Last Theorem based upon a result of Eichler which asserts that if the first case of Fermat's Last Theorem fails for the prime $p>2$, then $p^[\sqrt{p}-1]$ divides the first factor $h^*$ of the class number $h$ of the $p-$cyclotomic number field $\Q(\xi)$.

With this generalization, we give more than $p^2$ new summation criteria $mod (p)$ and depending only on $p$ for the first case and also for the second case of Fermat's Last Theorem when $x^p+y^p+z^p=0$, $z\equiv 0 mod(p)$ and $z\not\equiv 0 mod(p^2)$.

This paper rests on classical algebraic number theory without any connection with Wiles proof.

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Roland Queme <106104.1447@compuserve.com>