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  1. It has been proved that the set of irregular primes is infinite (in fact, it has been proven that even the smaller subset of irregular primes of the form $4n+3$ is infinite); whether the set of regular primes is infinite is still an open question (or, put another way, it has not yet been proven that the set of regular primes is finite).
  2. It is not known whether the set of Mersenne primes is finite or infinite.
  3. It has been conjectured that infinitely many Wilson primes exist.

etc.

Ignoring any sets defined by trivial limitations (e.g., “primes less than a billion” or similarly defined by some magnitude-based criteria), is there any subset of the integer primes that has been proven to be finite?

EDIT: The distinction between “trivial” and “nontrivial” is clearly critical in this question. I’m not sure I can think of a better wording. I don’t accept the AP theorem given in the first answer (below) as a counterexample, because it necessarily includes a magnitude-based criteria: the number of primes in an AP of given [fixed] difference $d$ starting with a given prime $q$ is finite, but the number of primes which are in any AP of given [fixed] distance $d$ is apparently infinite.

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  • $\begingroup$ I wouldn't count this as trivial, so I think this answers your question. $\endgroup$ – B. Mehta May 20 '17 at 13:48
  • $\begingroup$ @B.Mehta: You wouldn't count what as trivial? $\endgroup$ – Kieren MacMillan May 20 '17 at 13:50
  • $\begingroup$ The progressions shown in that article $\endgroup$ – B. Mehta May 20 '17 at 13:51
  • $\begingroup$ @B.Mehta How is that a finite subset of primes? $\endgroup$ – Blitzkrieg May 20 '17 at 13:53
  • $\begingroup$ Ah, misread the original post. Apologies! $\endgroup$ – B. Mehta May 20 '17 at 13:53
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$\textbf{Szemerédi's theorem :}$ Any subset of natural numbers with a positive natural density has a $k$-term arithmetic progression for every natural number $k$,

later extended to the $\textbf{Green-Tao Theorem}$ showed that the set of prime numbers has arbitrarily long arithmetic progressions.

From this theorem it can be asserted that any arithmetic progression of prime is numbers finite or in other words any tuple of primes which differ by the same number is finite.

I think it has also been shown that there are infinite such tuples of the same length.

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  • 2
    $\begingroup$ As you point out in your comment (to the main post), this doesn’t exactly answer my question. That is, if my official conjecture was that “every non-trivially-defined subset of the primes is infinite”, I’d be able to find a definition of “non-trivially-defined” that got around APs of primes. But it is helpful information! Thanks. $\endgroup$ – Kieren MacMillan May 20 '17 at 15:09

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