SOLVED

Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$?

Erdős called this conjecture 'rather silly'. Using covering congruences Erdős

[Er50] proved that the set of such odd integers contains an infinite arithmetic progression.

Chen [Ch23] has proved the answer is no.

See also [9], [10], and [11].