I mean, according to my knowledge, Bertrand's postulate has already been proved, I've already read and understood one, but Paul Erdos, and a few other mathematicians have proved it using various methods. I just finished reading Ramanujan's proof. Its amazingly advanced, and really short. The porblem of the inequality that Ramanujan uses to prove Bertrand's postulate is amazingly similar to an alternate formulation of Legendre's conjecture. Well, I played a little bit with other representations of the Prime Counting Function and Ramanujan's inequality of the alternate formulation of Legendre's, and just assuming Bertrand's postulate to be true (which it has already been proved as I mentioned), I found a rather elementary (and obvious) proof of Legendre's. Can anyone explain why they say that Bertrand's postulate doesn't imply Legendre's conjecture? I read about the supposed error ratio of the Prime Number Theorem, so if anyone knows about this would they be so kind to elaborate why BT doesnt imply LC?