Assume the number of primes is finite. Then, for any positive K, K! = \prod_{p}{p^{\phi(p, K)}}, where \phi is the function from Legendre's Formula and the product is taken over all primes.

\phi(p, K) = \sum_{s=1}\bigg\lfloor\frac{K}{p^{s}}\bigg\rfloor \lt \sum_{s=1}\frac{K}{p^{s}} = \frac{K}{p-1} \le K.

Therefore, K! \lt (\prod_{p}p)^{K}. But \lim_{K\rightarrow\infty}(\prod_{p}p)^{K}/{K!} = 0, which is impossible.

References

  • J. P. Whang, Another Proof of the Infinitude of the Prime Numbers, The American Mathematical Monthly, v 117, n 2 (2010), p 181