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What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger problem with prime numbers where this might be needed.

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    $\begingroup$ How about primes in arithmetic progressions? $\endgroup$ – lhf Jul 2 '14 at 18:53
  • $\begingroup$ The set of all prime numbers is one interesting infinite family of prime numbers. The set of all twin primes is another if in fact it's infinite, but nobody's proved that yet. It has been proved that the sum of the reciprocals of the members of the latter set is finite. $\endgroup$ – Michael Hardy Jul 2 '14 at 18:54
  • $\begingroup$ What is the larger problem with primes ? $\endgroup$ – Dietrich Burde Jul 2 '14 at 18:58
  • $\begingroup$ @DietrichBurde I can't tell you, because I need to do the problem independently. $\endgroup$ – anon Jul 2 '14 at 19:01
  • $\begingroup$ I don't want to do your problem, do not worry ! But your question is then very vague and hard to answer, because we don't know what you need. $\endgroup$ – Dietrich Burde Jul 2 '14 at 19:04
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One list of families of prime numbers is

https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes

which I have collected over several years. Most of these families are infinite though knowledge of their size is varied.

Perhaps worth noting is that, for any $n$, there are infinitely many primes with more than $n$ 1s in binary, since otherwise there could be only at most $k\choose n$ $k$-bit primes, in contradiction to the Prime Number Theorem.

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  • $\begingroup$ One comment on your list of families: in the "Prime constellations" section, the sieve (Selberg sieve or an elaboration of Brun's sieve) has proved big-$O$ upper bounds of the same order of magniture as the conjectured asymptotic densities. $\endgroup$ – Greg Martin Jul 11 '14 at 7:28
  • $\begingroup$ @GregMartin: Thanks -- I thought that was the case but hadn't found the time to check. $\endgroup$ – Charles Jul 11 '14 at 14:01

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