That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$?

In this case, the [Hamming weight][1] of a number is the number of $1$s in its binary expansion.

Many problems of this sort have been considered, but perhaps not in such language. For instance, the question "Are there infinitely many [Fermat primes][2]?" corresponds to asking, "Are there infinitely many distinct primes with Hamming weight exactly $2$?" Also related is the question of whether there are infinitely many [Mersenne primes][3].

These examples suggest a class of such problems, "Do there exist infinitely many primes which are the sum of exactly $n$ distinct powers of two?"

Since this question is open even for the $n=2$ case, I pose a much weaker question here.

What is known is that for every $n \leq 1024$ there is such a prime.

The smallest such prime is listed in the Online Encyclopedia of Integer Sequences [A061712][4].

The number of zeros in the smallest such primes are listed in [A110700][5]. The number of zeros in a number with a given Hamming weight is a reasonable measure of how large that number is. The conjecture at OEIS is quite a bit stronger than the question I pose.

Is there a theorem ensuring such primes for every $n \in \mathbb{Z}_{>0}$?


  [1]: http://en.wikipedia.org/wiki/Hamming_weight
  [2]: http://en.wikipedia.org/wiki/Fermat_number
  [3]: http://en.wikipedia.org/wiki/Mersenne_prime
  [4]: https://oeis.org/A061712
  [5]: https://oeis.org/A110700