An arbitrary question
- I think this is my first post here after several years lurking, so hi!
The following thoughts have been nagging at me for a few weeks now,
and I'd be grateful for suggestions or pointers.
According to Wikipedia, Green & Tao's theorem states that there
are 'arbitrarily long' arithmetic progressions of prime numbers. I
take this to mean that if you give me a number then I can find an AP
of primes longer than the number you nominate. Assuming my
interpretation is in the ballpark, then is the following very sketchy
reasoning saying anything useful?
Let p + kd (k = 0, 1, ..., (arbitrarily large) n) be an arbitrarily
long AP of primes.
Let g be such that gcd(p + g, d) = 1.
Let q = p + g.
Consider the finite AP q + kd, terminating at the above (arbitrarily
large) n. This AP is arbitrarily long, and so perhaps by some density
argument or other it contains arbitrarily many primes. So there are
arbitrarily many prime pairs (p + kd, q + kd) with difference g.
Thus if you give me a number m, I can find a g such that there are
more than m prime pairs (p, q) satisfying q - p = g.
There are thus numbers g for which there are arbitrarily many prime
pairs (p, q) satisfying q - p = g.
Can I pass from 'arbitrarily many' to 'infinitely many'? Can I
conclude that there is at least one finite number g for which there
are infintely many prime pairs satisfying q - p = g?