Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation?
$2$ is the only prime with $1$ one, the Fermat primes are the only primes with $2$ ones, for $3$ ones we want primes of the form $2^a+2^b+1$ with $a>b$, so
and so on.
This would be a start if one hoped for the existence of infinitely many Fermat primes.
Edit: This paper proves that for all sufficiently large $t$ we can find a prime with exactly $t$ ones in their binary representation.