Romanov type problems

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form [Formula: see text] and [Formula: see text] where [Formula: see text] and p, q are primes. In the opposite direction, Erdős constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove a much better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form [Formula: see text] is larger than [Formula: see text] . (2) The proportion of positive integers not of the form [Formula: see text] is at least [Formula: see text] .