Is There Evidence Supporting Multiple Twin Primes in Specified Intervals?

[jnorris]

Bear with me. I'm new to forum and don't yet know all protocol. My question concerns twin primes. The previous thread on this topic seems to be closed. My question is this: When considering the Twin Primes Conjecture, has anyone researched the idea that (heuristically speaking) there is evidence that if p and p+2 are twin primes, there should be at least one pair of twin primes q and q+2 such that p^2 < q < (p+2)^2-2? From a probability standpoint, it seems that we should expect 2 or more pairs of twin primes in each such interval. The reasons for this involve simple modular arithmetic. Proving this is obviously much harder, but it seems a reasonable way to search for twins (i.e. choose largest known twin prime pair p and p+2 and search the interval (p^2, (p+2)^2-2)). Anyway, I'm probably off base, but would welcome comments.

 

[shmoe]

You might want to look at the heuristic argument for the asymptotic in the twin primes conjecture. Details are in Hardy & Wright's Intro to Number Theory book (and elsewhere of course).

You'd expect twin primes in any interval of the form x^2 to (x+1)^2 when x is large enough. Their (conjectured) density is something like a constant times 1/\log^2{x} near x, so an interval of length 2x should have about (a constant times) x/2\log^2{x} twin primes. That is you expect quite a few as x gets large.

They are quite frequent and finding twin primes in a "reasonable" range of a few hundred digits can be done quickly on a home computer, just like finding primes can be. Breaking the record for the largest known is another story. The largest pair has over 50000 digits last I checked and was of a special form (similar idea to mersenne prime search) so a random grab and test algorithm isn't going to work well in this range.

 

[jnorris]

Thanks for your reply. Clearly for a number between p_n^2 and (p_n+2)^2-2 to be a twin prime (first of a pair) it must not be congruent to either 0 or p_i - 2 mod any prime p_i for i =1 to n. My thinking was that the probability of a given q in this interval to be a twin prime is greater than or equal to
(1/2)(1/3)(3/5)(5/7)...[(p_n - 2)/p_n] > 1/(2p_n). Since there are 4p_n + 1 numbers on this interval, we'd expect > 2 pairs of twin primes on any such interval. I've just read that there is a conjecture that says there is at least one prime between any 2 perfect squares so I suppose my idea is akin to this one. Obviously, provability is the problem, but I just thought it was interesting that the problem really could be viewed in terms of congruences (since they're periodic over the integers).

 

[shmoe]

You should really look into the Hardy-Littlewood asymptotic I mentioned. It's essentially based on the same idea (and the usual prime number theorem).

A problem is you aren't usually dealing with intervals whose length is divisible by the set of primes needed, so the events that a given number is divisible by different primes will not be independent. e.g. the event that a random number between 1 and 6 is divisible by 3 is independent from the event that it's divisible by 2. This is not the case for random numbers picked between 1 and 5. Of course it still seems reasonable that everything works out like you'd hope, like you say proving this is the case is still a problem.

The conjecture of a prime between consecutive squares is a special case of the primes in short intevals problem, you might want to take a look at some of the research in this area.