Romanov type problems

Abstract

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 $$n=p+2^{2^k}+m!$$ and $$n=p+2^{2^k}+2^q$$ where $$m,k \in \mathbb {N}$$ 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 $$p+2^{2^k}+m!$$ is larger than $$\frac{3}{4}$$ . (2) The proportion of positive integers not of the form $$p+2^{2^k}+2^q$$ is at least $$\frac{2}{3}$$ .

Locations

• The Ramanujan Journal - View - PDF
• PubMed Central - View
• PubMed - View