  In this paper, we make some conjectures on prime numbers that are sharper
than those found in the current literature. First we describe our studies on
Legendre's Conjecture which is still unsolved. Next, we show that Brocard's
Conjecture can be proved assuming our improved version of Legendre's
Conjecture. Finally, we sharpen the Bertrand's Postulate for prime numbers. Our
results are backed by extensive empirical investigation.
