2, 3, 5, 7, 11, 13, 17, 19...

Prime-Numbers.net

Where Prime Numbers mysteries are solved.

What is the Biggest Prime Number?

The biggest known prime to date is the Mersenne prime 243,112,609 - 1. It has 12,978,189 digits! However, we know there are larger primes out there, because there are an infinite number of prime numbers! How do we know that? Euclid actually proved this over 2,000 years ago!

Proof that there are an Infinite Number of Primes

This proof is a proof by contradiction. First, we'll make an assumption, and then we'll show that the assumption must be false. Follow along:

  1. Suppose there are a finite number of prime numbers, up to N of them. We'll call this set P. So P = {p1, p2, p3, ..., pN-1, pN}.
  2. Now, take all the numbers in P and multiply them together. We'll call this q. So q = p1 x p2 x p3 x ... x pN-1 x pN.
  3. Now look at q - 1. This number will not be divisible by any of the numbers in P. If you take any number in P, say Pm, since Pm is greater than 1, and since q is divisible by Pm, q - 1 will not be divisible by Pm. The next lower number than q to be divisible by Pm would be q - Pm.
  4. Since q - 1 is not divisible by any of the prime numbers, it is by definition prime! This means our supposition in step 1 was false, and therefore, there must be an infinite number of prime numbers.