The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem

Abstract

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+η$, $μ^2(η)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length, unlike previous versions.