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  −1(mod 6). With Pollaczek’s result [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  2437 and Conjecture2 for primes  less than 100003.