What are all of the prime number?

Answer 1

For lists of prime numbers, see the link.

There are an infinite amount of prime numbers. The first to have proven this was Euclid. Here is an outline of his proof and as you can see it is a proof by contradiction.

We begin by assuming the contrary; we assume the number of primes is finite, say there are n of them.

1. Suppose there are n prime numbers overall. 2. Let N be a common multiple of all these primes.

3. Consider N + 1, is it prime? If the answer is yes, then we have found a new prime and we are done.

4. So suppose to the contrary that N + 1 is not prime. Thus, there exists a Prime number p which divides N + 1 evenly. If p is one of the primes dividing N, it also divides 1 evenly, which is impossible. Thus, p is not one of the n primes, and we found a new prime. We conclude N + 1is not prime and we further conclude the number of primes is infinite.

The proof is elegant because it is so simple. However many people without a strong math background may still have some trouble following the logic. A good technique for any math problem or proof when this happens is to get rid of some of the abstraction ( beautiful as it may be) and replace it with numbers. This allows you to have a concrete understanding of what is behind the proof and then the abstraction makes sense.

So for example between 1 and 10 we have the prime numbers 2, 3, 5, and 7. That is all of them. Say you want to prove there are more than that. Using the proof above suppose that 2, 3, 5, and 7 are all the prime numbers, there are 4 of them and n = 4. Let N=2 * 3 * 5 * 7 = 210. Now consider N + 1. Is 210 + 1 which is 211 prime? Is so then there are more than 4 primes, since N + 1 is the fifth prime and we are done because our hypothesis was there are only 4 of them. Our proof would be over, so assume the contrary and we say 210 + 1 is not prime. That means it is composite and every composite number can be broken down into the product of primes and this product is unique up to the order of the multiplication. So 210 + 1 can be broken down into primes and we pick an arbitrary prime call it p which must divide N + 1. Now we assumed that there are only 4 primes, 2, 3, 5, and 7. so one of them must divide 210 + 1, that is to say either 2, 3, 5, or 7 must divide 211. We try all 4 of them:

211/2 = 105r1

211/3 = 70r1

211/5 = 42r1

211/7 = 30r1

Now if a number, such at N+1 cannot be divided by any prime, we must conclude it is a prime, but we said it was not. Therefore we must conclude there are more than 4 primes.

If this is still hard to see use the idea of remainders to write N + 1 = 2 * 3 * 5 * 7 + 1 which is the same as 2 * 3 * 5 * 7 remainder 1. This tells us again that N + 1 is not divisible by 2, 3, 5, or 7, since there is a remainder of 1. Once again we conclude N + 1 must be a prime since no prime divides it.

Suppose there are a finite number of primes.

We can multiply them all together and we have a number that is divisible by every single prime.

Now add 1. We now have a number that, when you divide it by any prime, will have a remainder of 1.

This number leads us to a contradiction since it can't be prime, and it isn't divisible by any prime.

What type of number is not divisible by any prime and is not prime? It could be the number 1, but that is not possible here, so we must conclude this number

does not exist and our original assumption was there is a finite number of primes was wrong.

Answer 2

That's an infinite list.