  In this paper two conjectures are proposed based on which we can prove the
first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1
(\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first
case of FLT can be proved for all primes greater than 3. With a computer
Conjecture1 was verified to be true for primes $\leq 2437$ and Conjecture2 for
primes $\leq 100003$.
