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### The n-th prime is less than $n^2$? [duplicate]

Let $p_n$ be the n-th prime number, e.g. $p_1=2,p_2=3,p_3=5$. How do I show that for all $n>1$, $p_n<n^2$?
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### $n$-th prime is less than a polynomial of $n$ [duplicate]

Let $p_n$ be the $n$-th prime. Is there an elementary way to prove the statement that there exists a polynomial function $P$ such that $$p_n \leq P(n),$$ for all $n \in \mathbb{N}$? (Or, alternatively,...
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### n-th prime number is less than $4^n$

Prove that $\forall n \in \mathbb{N}^*, P_n \lt 4^n$ where $P_n$ is the $n$th prime number. I'm searching for a proof that doesn't use induction and uses only the elementary concepts of number theory. ...
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### some properties of consecutive primes [closed]

If $p_1$ and $p_2$ are two odd consecutive primes and n is their midpoint, then $p_1p_2$ is the largest odd multiple of $p_1$ not exceeding $n^2$. This sounds obvious, but I still have problem to ...
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### Show that $p_n^{1-\epsilon}\le n$ using PNT

Assuming PNT $$\pi(x)\sim \frac{x}{\log{x}}$$ How can we show that given any $\epsilon>0$ $$p_n^{1-\epsilon}< n,$$ for all sufficiently large $n$ ($p_n$ denotes the $n^{th}$ prime.) My work: ...
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