Infinite Primes

The Fundamental theorem of Arithmetic.why short is beautiful

I have always believed that brevity is the key to effective communication and that great authors—whether in history, mathematics, engineering, or everyday interactions—excel at getting straight to the point. We should always aim at cutting away redundant information and only communicating the essential.

An example of this is one of my favorite mathematical proofs - Euclid’s demonstration that there are infinitely many prime numbers (numbers that can only be divided by one and themselves).

• Assume there are finitely many primes: p1,p2,p3…,pn-1,pn

• Let S=p1 x p2 x p3 ⋅……⋅pn-1x pn​, the product of all these primes.

• the fact that N=S+1, cannot be divided by any primes in S, leads to a contradiction, because S is suppose to contain all the finite primes.

Therefore, there are infinitely many primes.