So a new largest prime number has been found.  It is

2^74,207,281 −  1

Of course we know that there is an infinitude of primes, so the number above is just the largest known.

A standard proof is via contradiction. Assume there is a largest prime p_n . Then it can be shown that the number p₁p₂p₃ … p_n + 1 is also a prime, contradicting our assumption that there is a largest prime.

Yesterday I found another proof from Wikipedia which is fascinating. There is a formula:

π/4 = 3/4 × 5/4  × 7/8 × 11/12 × 13/12 × 17/16 × 19/20 × 23/24 × 29/28 × 31/32 × …

The numerator is all primes (except 2), one after the other. The denominator is the nearest multiple of 4 of the numerator.

We know  π is irrational. If there are a finite number of primes, the right hand side is a rational number, which doesn’t makes sense since the left hand side is irrational. Hence the right hand side is an infinite product. Hence there is an infinitude of primes!