gaps between primes

SteveL27

You're right, but that doesn't give any new information.

We know 2 is prime and no other prime is divisible by 2.

We know 3 is prime and no other prime is divisible by 3.

We don't need to consider 4 because if a number is divisible by 4, it's already divisible by 2, which we checked earlier.

We know 5 is prime and no other prime is divisible by 5.

Continuing like this, we see that we could figure out the distribution of primes ... if we already knew the distribution of primes. We haven't gotten any more insight.

However, your idea is actually the basis of the famous Sieve of Eratosthenes. You start with a list of all the counting numbers from 2 onward. You draw a circle around two; then you cross out 4,6,8, and all the other multiples of 2.

Then you put a circle around 3; and cross out all the multiples of 3. Continuing like this, you end up with all the primes circled. You can use this algorithm to find all the primes below any given number. The algorithm's about 2300 years old -- and still as good as ever.

http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

There's a very cool animation on that page showing the algorithm in action.

coolul007

It's akin to proving a negative proposition. "It's not one of these!" That applies to a lot of things. The closest we will come is proving the Goldbach Conjecture, that places two primes equidistant from a fixed point.