I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:


Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of factorials it is not difficult to prove that
$$p^{v\_p{n\choose a}}\leq n$$
(with $v\_p{n\choose a}$ denoting the multiplicity of the prime $p$ in the
prime-factorization of the binomial coefficient ${n\choose a}$)
for every prime number $p$ and integers $0\leq a\leq n$.


Since ${2n\choose n}$ involves only primes up to $2n$, we get
therefore
$${2n\choose n}\leq (2n)^{\pi(2n)}$$
with $\pi(2n)$ denoting the number of primes $\leq 2n$.


Using Stirling's approximation ${2n\choose n}\sim\frac{2^{2n}}{\sqrt{\pi n}}$
and taking the logarithm we get now
$$2n\log 2\leq \frac{\log \pi+\log n}{2}+\pi(2n)\log(2n)\ .$$


This implies asymptotically
$$\liminf \frac{\log(2n)}{2n}\pi(2n)\geq \log 2$$
which is just short by a factor $\log 2$ of the prime-number Theorem
and implies that there are infinitely many primes.


Remark: Assuming $\lim\frac{\pi(\lambda x)}{\pi(x)}=\lambda$
for all $\lambda>0$, these arguments can be refined to a proof of the
prime-number Theorem.



Reference for a proof of Euclid's Theorem for the infinitude of primes
As Ofir says in the comments, this is very similar to but somewhat simpler than [Erdős' proof of Bertrand's postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate). As in that argument, the appeal to Stirling's approximation can be replaced by the simpler estimate ${2n \choose n} \ge \frac{4^n}{2n+1}$ which just comes from observing that ${2n \choose n} \ge {2n \choose k}, 0 \le k \le 2n$.


