Solution to:

Prove that there has to be a prime number bigger than 500.

There is a number that is (2*3*4*...*499*500)+1

If you try to divide that number by 2 or 3 or 4 or anything up to 500, you will find that there is a remainder of one, so the big number isn't divisible by any of the numbers that are less than or equal to 500.

That big number has to have a prime factor, because every counting number bigger than 1 has prime factors (the prime factor might be the number itself, or it might be something less than that number). That prime factor can't be any of the prime numbers less than (or equal to) 500, so the prime factor has to be bigger than 500.

So there has to be a prime number bigger than 500.

Note: this proof is the same proof that there are infinitely many prime numbers that is in the book in chapter 2.3