# Proof of $p_n<n^2$ by Elementary Means

Is there any proof of the inequality $p_n<n^2$ (for all sufficiently large $n$) by elementary means and without using Prime Number Theorem?

I searched in google but in vain. The results that I have found (and from which the inequality can be proved) are mostly results that arises as a consequence of PNT.

• You can prove by elementary means $p_n=O(n\log n)$. – karvens Oct 21 '14 at 5:09
• The statement is not true for $p_1=2$, so you need an exception for that case. – Esteban Crespi Oct 21 '14 at 5:50
• @karvens: But how does it conclude that $p_n<n^2$, can you elaborate? – user 170039 Oct 21 '14 at 6:10
• If $p_n=O(n\log n)$, we have $p_n<Cn\log n<n^2$ for all sufficiently large $n$. – karvens Oct 21 '14 at 6:14
• @user170039 Chebyshev already proved something of the strength $p_n \le (1.2 + o(1)) n \log n$; this was nearly 50 years before PNT was established. The proof that karvens refers to is probably the simpler one due to Erdős. – Erick Wong Oct 21 '14 at 8:52

As said in the comments, Chebyshev already proved by elementary means that $p_n<C n \log(n)$ for all $n\ge n_0$. As there was explicitly asked for a reference, I include one (of many): Theorem $5.1$ here.