We know that there is no highest prime number (trivial). We have also proved that, as we go up the number line, prime numbers get scarcer (decidedly non-trivial) and that there is no highest distance between two consecutive prime numbers (a moderately obvious corrolary thereof).

What I want to know is: is it true that all even distances between consecutive prime numbers occur?

Obviously it's not true for all positive distances, because there's only one even prime, so odd distances must always involve 2, and the only prime number consecutive to 2 is 3. Odd distances other than 1 are impossible. But it might be true for even distances.

Next question: if the above is not true, is it true that al even distances occur between prime numbers, not necessarily consecutive ones?

(This one is definitely not true for odd distances, either: 2+7 isn't prime.)

For the record: I have no idea whether either conjecture is true. I don't even know where to start solving it. But someone on this forum might. And if not, someone on this forum might win a Fields Medal for solving them!

Proving that is fields prize stuff most likely. Seems that prime gaps have captured the fascination of others too. The links from this place could prove helpful..

Originally posted by talzamir Proving that is fields prize stuff most likely. Seems that prime gaps have captured the fascination of others too. The links from this place could prove helpful..

http://oeis.org/A001223

Yeah, I'm afraid that's true. But it seems such a simple problem - it took me just two minutes of fiddling about with prime numbers to come up with it! And it's probably going to take someone a couple of decades to prove it, or even disprove it.

For the first conjecture is there any mileage in investigating how scarce prime numbers are getting and whether the gaps tend to be increasing at a rate too fast to accomodate all even numbers? (By any mileage I am meaning by someone far cleverer than me).

The second one I imagine would be a nightmare.

I do love these remarkably simple but hard to prove problems.

The differences between consecutive prime numbers can get arbitrarily large, as described (under "simple observations" ) here:

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

But this does not give a way to generate a arbitrary distance, so it does not prove that all even integers are possible gaps.

However a perusal of the above page and http://en.wikipedia.org/wiki/Prime_number_theorem
illustrates how tricky proving just about anything to do with prime numbers is.

One "proof" that might work in the hands of a mathematician is:

If there is a difference X, which is even, and is not a possible difference between two consecutive primes, then this, for every prime P would give two numbers P-X and P+X which cannot be primes, such a constraint on which numbers cannot be primes would violate the random distribution of primes.

Except for the gap between 3 and 5 a gap of 2 (mod 6) can only follow a prime with a value of 5 mod 6 and must precede a prime with a value of 1 mod 6.

a gap of 4 (mod 6) must follow a prime with a value of 1 mod 6, and precede a prime with a value of 5 mod 6.

These are provable from the fact that there are no primes that are 3, mod 6 (except for 3) as any such "primes" would be divisible by 3.

Originally posted by iamatiger One "proof" that might work in the hands of a mathematician is:

If there is a difference X, which is even, and is not a possible difference between two consecutive primes, then this, for every prime P would give two numbers P-X and P+X which cannot be primes, such a constraint on which numbers cannot be primes would violate the random distribution of primes.

That is a very common idea with conjectures about primes. We believe that we already know the patterns in primes. (Tao calls them "conspiracies".) We think there are no more patterns. We find their existence hard to believe. For example, it is hard to believe that there are only finitely many twin primes. If we've already found all the rigidity of primes' behavior, then there should be infinitely many twin primes. But it's one thing to think things like that, and it's another thing to prove them. The latter turns out extremely difficult, and might be impossible in many cases. But we (I mean the more skillful among us) have managed at times. The Green-Tao theorem is a beautiful example. I've never tried to understand the proof, but the fact that it has been proven makes me proud of being a human being.