What is the highest prime number?

Answer 1

There is no highest Prime number. Given any set of prime numbers, you can prove that there is at least one prime number that is not in that set. Here's how.

First, recall that every natural number is either a prime number, or is a composite number and thus the product of some series of prime numbers.

Now, suppose you have a set of n prime numbers, P1 .. Pn. Multiply them all together and add one. Call this number Q.

Q is either prime or composite.

If Q is prime, then it is obviously not one of P1 .. Pn, and thus is a new prime number not in that set.

If Q is composite, then there must be a list of primes that evenly divide into it. Because Q is one greater than the product of P1 .. Pn, dividing by any of those primes will have a remainder of 1. So there must be some new prime, call it R, that divides evenly into Q.

In either case, Q or R is a new prime.

Now suppose that you have some potential highest prime. Enumerate all of the primes lower than it, and follow the above procedure with that set of primes. You will end up with a new prime not in that list. Since you have listed all primes less than your purported highest prime, any new prime number must be greater than your highest prime.

Thus, there is no highest prime.