I am submitting this proof that there are infinitely many primes because it has come to my attention that this simple proof is one key to understanding a lot of math.

The original document is available here . It is possibly the easiest proof I have come across and because of that, I am able to understand it ;-). I’ve made some additions to simplify it even further.

## Euler’s Proof that there are infinitely many primes

This is a proof by contradiction. That is, we shall prove that a STATEMENT is TRUE by assuming that the NEGATION of the STATEMENT is TRUE and reach a contradiction.

The STATEMENT to be proved is **There are infinitely many positive primes.**

**Proof:** Suppose there are NOT infinitely many primes — i.e., there are only finitely many positive primes, say K of them. List them all: p1, p2, p3, … , pK.

Now define some positive integer,** m**, by

m = ( p1 × p2 × p3 × … × pK ) + 1.

Since m > pj, for all j = 1, …, K, m is NOT a prime.

Thus, for some j, pj is a factor of m. Thus

m/pj = [( p1 × p2 × p3 × … × pK ) + 1 ] /pj is greater than, > 1

by multiplying & rearranging the equation we get

m/pj = ( p1 × p2 × p3 × … × pK )/pj + 1/pj

thus,

1/pj = m/pj – ( p1 × p2 × p3 × … × pK )/pj

now we know that m/pj > 1 and also that ( p1 × p2 × p3 × … × pK )/pj >1

thus 1/pj = [ m/pj – ( p1 × p2 × p3 × … × pK )/pj ] must also be > 1

But we know that 1/(pj) must satisfy 0 < 1/(pj) < 1.

This is a contradiction (to our above TRUE statement?).

thus the original statement must be wrong and There are infinitely many positive primes is TRUE.