TITLE: Is the $n$th triangular number greater than or equal to the $n$th prime?
QUESTION [6 upvotes]: Let $p_n$ be the $n$th prime. For which values of $p_n$ would the following inequality be satisfied?
$$\frac {n^2+n}{2}\ge p_n$$
I have no programming skills, so checking this one number at a time is rather difficult. So far I have verified it for every number less than $100$. I would guess that a way to prove it would be Bertrand's Postulate or the PNT, but I am not sure. Is this a corollary of another theorem or just an obvious statement or is it plain wrong?

REPLY [2 votes]: According to Wikipedia, $p_n<n(\log n+\log\log n)$ for $n>6$. So all you have to do is prove $\log n+\log\log n\le(n+1)/2$ for $n$ sufficiently large, and that's a standard intro Calculus type problem.