I'm stuck on this question its from my math class. I appreciate your help.
Turn this into a skeptic-proof proof.
Suppose that there were only finitely many primes. Then I'll use that assumption to create a compression scheme for arbitrary numbers. Indeed, given $n$, I can express it as a product of primes, so $$n = 2^{v_2(n)} \cdot 3^{v_3(n)} \cdot 5^{v_5(n)} \cdots P^{v_P(n)}$$
where $P$ is the largest prime. Now my compression for n is the array $$(v_2(n), v_3(n),...v_P(n))$$ Crudely, let me convince that the compression uses fewer bits than just communicating or storing $n$.
The worst case scenario for the size of $v_2(n)$ would be $n$ having no other prime factors, so $v_2(n)$ could be as large as $\log_2 (n)$ and I might need $\log_2 \log_2 (n)$ bits to store it. No other $v_p(n)$ could be even that big. So the total number of bits I'll need is bounded by $N \log_2 \log_2 (n)$ where $N$ is the total number of primes. But it usually requires $\log_2 (n)$ bits to store a number, and if $n$ is large enough $\log_2 n$ is much larger that $N \log_2 \log_2 (n)$.