20150403, 02:07  #45 
Oct 2007
London, UK
2×653 Posts 
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 GreenTao 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? 
20150427, 02:10  #46 
"Matthew Anderson"
Dec 2010
Oregon, USA
1001010100_{2} Posts 
Hi Math People,
The ktuple conjecture states that there for any admissible ktuple, 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 4tuples. 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 ktuples for k up to 12. My webpage is here. If I had to guess, I would say that the ktuple conjecture is probably true. I wouldn't even know where to look for a counterexample. Regards, Matt 
20151117, 09:22  #47  
"Gang aft agley"
Sep 2002
2×1,877 Posts 
Quote:
https://terrytao.wordpress.com/2015/...etweenprimes/ Quote:


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