TITLE: Consequences of Legendre's conjecture
QUESTION [10 upvotes]: I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.

REPLY [20 votes]: I am given credit here for a conjecture that I did not make (on the maximal gaps between consecutive primes).  This is also wrong in Wikipedia (can someone please correct that?). 
What I noted, on page 24 of my paper, "Harold Cramér and the distribution of prime numbers, Scandanavian Actuarial J., 1 (1995) 12- 28" is that if one includes in Cramér's model the fact that every pth integer is divisible by $p$, for small primes $p$, then
$\limsup_{n\to \infty}  (p_{n+1}-p_n)/ (\log p_n)^2  \geq  2e^{-\gamma} .$
Cramér's conjecture is that the limsup equals 1 (which is smaller than $2e^{-\gamma}$) and so is likely to be false (even if there is not enough computational evidence yet to say that). I have not made a conjecture as to the correct value of the limsup.