Monday, 14 May 2012

Dirichlet's Theorem and composite numbers seen as the difference of two squares

Back in January I made a series of posts where I explored the patterns in composite numbers adjacent to numbers of the form 3*2^n (= 6, 12, 24, 48, etc).

What I've recently realised is that what I was seeing there was a specific, narrow example of Dirichlet's Theorem on arithmetic progressions, which states that "for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0."

For instance, the series of numbers adjacent to (eg 1 more or 1 less than) 48, can be expressed as the two series 1+ 48n and 47 + 48n. The patterns I was seeing show that each time we go to a higher multiple of 3*2^n, half the composite numbers remain, so we have a very similar pattern - and because the composites are distributed in this way it also seems reasonable to expect that primes are also being divided in more or less the same manner. The first diagram in the post linked to above, shows how composite numbers divide equally between numbers that can be expressed as 12n+1, 12n+5, 12n+7 and 12n+11 - I tend to use the pigeonhole analogy to picture the process which leads to equal numbers of composites being put into each of these modulo residues.

So then when you look at composite numbers adjacent to 24, half of the 12n+1 numbers will go into the 24n+1 pigeonhole and half of them will go into the 24n+13 pigeonhole, while half of the 12n+11 composites go to 24n+23 and half go to 24n+11.

Going back to Wikipedia, I see that "stronger forms of Dirichlet's theorem state that, for any arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges, and that different arithmetic progressions with the same modulus have approximately the same proportions of primes." Which is kind of what I was just trying to say (though obviously seeing a pattern is not the same as proving a theorem).

Of course this applies to all values of d, not just multiples of 3*2n - for other values (for instance 4n+3 and 4n+1, or 10n+1, 10n+3, 10n+7, 10n+9) there will also be an ongoing process whereby the composites are evenly spread among the available pigeonholes, which will lead to each having a similar proportion of primes.

I have looked at a few proofs of Dirchlet's theorem and haven't got my head round them yet. Will try a bit harder but I think they might be too technical for me to follow.

The other question which I am interested in is whether this has any significance for twin primes and other Polignac primes - in the later posts on composite numbers expressed as the difference of two squares, I was also exploring the fact that (for instance) the patterns for 48n + 1 and 48n + 47 can be seen as the same progression, going "through zero" (if you ignore the minus signs) -95, -47, 1, 49, 97.

When it comes to composite numbers as the difference of two squares, it works like this. Take x^2 - y^2 where the difference between x and y is 7.

24^2 - 17^2 =287, 12^2 - 5^2 = 119, 0^2 - 7^2 = -49, 12^2-19^2 = -217

Which can also be expressed as
7*41, 7* 17, -7*7, -31*7

So this is a continuous arithmetic progression with a gap of 7*24 = 168 - if we extend it a bit further we get this series

-791 -623 -455 -287 -119 49 217 385 553 721 889 1057 1225 1393

If we look at this in modulo 96, the same series produces these remainders

25, 49, 73, 97, 25, 49, 73, 97, 25, 49, 73, 97, 25, 49...

Which is a constant loop, meaning that half of these numbers are next to a multiple of 48, and half of those are next to a multiple of 96 and son.

Of course, if we ignore the minus signs, the first few would instead be 23, 47, 71, 95... (eg 6n-1 numbers instead of 6n+1 numbers). But with regard to which multiples of 24 they are adjacent to the pattern is a continuous one.

I'll try to come back to look at this again soon to see if I have anything useful to say about how this relates to twins.

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