Article: First proof that infinitely many prime numbers come in pairs

[lavalamp]

It seems that everyone here pretty much expects there to be an infinite number of twin primes. However, What do people think about the possible infinitude of cousin primes, sexy primes etc.? Might there be an infinite number of primes p and p+n where n is any even number?

Along similar lines perhaps, the Green-Tao theorem states that there are arbitrarily long arithmetic progressions of primes. I could pick "Graham's number" as the length and surely one of that length must exist, but does this extend to arithmetic progressions of infinite length?

Also, does this theorem or any other give an indication to the number of sequences of length k? Might there be an infinite number of each such sequence?

[MattcAnderson]

Hi Math People,

The k-tuple conjecture states that there for any admissible k-tuple, there is an infinite sequence of primes. It also implies that for k=2 and any even number, there is an infinite number of twin prime pairs, cousin primes, ect. However it is just a conjecture. A conjecture is something that a mathematician thinks to probably be true. For k=2 and a separation of 246, there is a theorem about pairs of primes. A summary about prime pairs is here.

Puzzle Peter and others have been working on 3 and 4-tuples. k=3,4. But an enumeration of primes is not a theorem. It is just something else.

The Online Encyclopedia of Integer Sequences has a enumeration in A1359. 100,000 of the smallest twins are enumerated.

I have been working on prime constellations that is to say k-tuples for k up to 12. My webpage is here.

If I had to guess, I would say that the k-tuple conjecture is probably true. I wouldn't even know where to look for a counter-example.

Regards,
Matt

[only_human]

Originally Posted by lavalamp

It seems that everyone here pretty much expects there to be an infinite number of twin primes. However, What do people think about the possible infinitude of cousin primes, sexy primes etc.? Might there be an infinite number of primes p and p+n where n is any even number?

Along similar lines perhaps, the Green-Tao theorem states that there are arbitrarily long arithmetic progressions of primes. I could pick "Graham's number" as the length and surely one of that length must exist, but does this extend to arithmetic progressions of infinite length?

Also, does this theorem or any other give an indication to the number of sequences of length k? Might there be an infinite number of each such sequence?

Terry Tao mentioned a new paper:
https://terrytao.wordpress.com/2015/...etween-primes/

Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “Chains of large gaps between primes“...

I'm not sure if this is relevant to any of the places on this forum that look at prime gaps but it might be.