Twin Primes

[Chronos]

Quoth mla3:

I know how that N! function works and it doesn’t describe the number of potential differences accurately at all.

Actually, it does. The statement “There are a finite number of possible differences between prime pairs” is false, as can be proven using that construction. Thus:

Suppose that there are a finite number of possible difference between prime pairs.
Then there must be some largest element of that set, that is, a largest possible difference between prime pairs. Call this largest difference d.
Let f = d+2. Then f!+2, f!+3, … f!+f are all composites. Let p be the largest prime smaller than f!+2, and let q be the smallest prime larger than f!+f.
Then p and q are a prime pair, and there is a difference of at least f-1 between p and q.
But this difference is larger than d, contrary to our definition that d was the largest possible difference.
This is a contradiction, and therefore our initial assumption must be wrong. QED.

[Great_Antibob]

@Chronos

And Great Antibob, one can easily use the formula for digits of pi in hexidecimal to construct a similar formula for digits of pi in any base which is a power of 2.

True, but the basic point stands. Just because something isn’t random doesn’t always mean we have a simple, straightforward calculation for it.

I just didn’t really want to get too bogged down in details.

I think the real hangup here is the reliance on the 6n +/- 1 relationship.

Just because all prime differences must be constrained by this relationship doesn’t mean prime differences can’t get arbitrarily large. I reasonably sure this is where mla3 is still getting caught.

[deltasigma]

For anyone interested, it seems the maximum difference between “twins” has now been reduced to 5414.

A major advance came this spring, when University of New Hampshire mathematician Yitang Zhang showed that there are infinitely many primes separated by some number smaller than 70 million. That may be a lot bigger than their eventual goal of 2, but by the end of July mathematicians had already whittled that limit down to 5,414.

[Hari_Seldon]

As for probability, consider the Goldbach conjecture that every even number > 2 is a sum of two primes. The evidence is that the larger you do the easier it is to write an even number as a sum of two primes. Probably someone has even formulated a hypothesis as to how many ways 2n can be decomposed as a sum of two primes. But none of this amounts to a proof. In this case, numerical evidence seems to make it overwhelmingly likely.

In the case of twin primes I think there is even a conjecture as to their density (I think it is 1/(log n)^2) but no one has anything like a proof.

[deltasigma]

Record is now down to 600.

Using the [Selberg] sieve, Maynard found a list of 105 numbers (0, 10, 12, 24, … 594, 598, 600) that serves as a template for prime pairs. This means that infinitely many numbers, when added to the numbers in the template, produce at least two primes. (For example, 3 works because 3+0 and 3+10 are both prime.) The template produces pairs of primes that are separated by 600 at most — a major improvement over the record of 4,680. Even more importantly, Maynard showed that longer templates exist for prime triplets, quadruplets and higher order prime multiples, thereby establishing world records that had not even existed. (Tao independently used the same ideas to reach a similar, but slightly weaker result.)

[ultrafilter]

For what it’s worth, the application of probabilistic methods in number theory dates back almost as far as the modern theory of probability. Rényi’s survey states some of the major ideas and results in a relatively accessible manner, and might be worth at least glancing through.