TITLE: Is there a prime between every pair of consecutive triangular numbers?
QUESTION [10 upvotes]: Between two triangular numbers there is at least one prime number. Is there a mathematical proof for this statement?

REPLY [2 votes]: Note that if $m \geq 2$ and $T_{n-1} < m^2 $ then
$$(n-1)n < 2 m^2 $$
From here is easy to deduce that $n < m$. 
Also note that $T_{n+1}-T_n=n+1$.
Therefore, for each positive integer $m \geq 2$ if $n$ is the largest integer so that $T_{n-1} <m^2$, then we have by the above we have
$$T_{n+1} = T_n+n+1 < T_{n-1}+2n+1< m^2+2m+1$$
This shows that
$$m^2  \leq T_n <T_{n+1}< (m+1)^2$$
This shows that the statement you ask for, if true, would imply the Legendre conjecture.