On a conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba II

TL;DR

This paper confirms a 2020 conjecture by employing the Hardy--Littlewood method, showing that the count of primes of a specific form up to a certain bound asymptotically equals half the prime count, as one parameter grows large.

Contribution

The paper proves a conjecture relating to the distribution of primes in a linear form, using advanced analytic number theory techniques, specifically the Hardy--Littlewood method.

Findings

Confirmed the conjecture for large c values

Established asymptotic equivalence for prime counts in the specified form

Applied Hardy--Littlewood method to a new class of prime distribution problems

Abstract

Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by employing the Hardy--Littlewood method, a 2020 conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba which states that $$#\left\{p\le g_{c,d}:p\in \mathcal{P}, ~p=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}\right\}\sim \frac{1}{2}\pi\left(g_{c,d}\right) \quad (\text{as}~c\rightarrow\infty),$$ where $\mathcal{P}$ is the set of primes, $\mathbb{Z}_{\geqslant0}$ is the set of nonnegative integers and $\pi(t)$ denotes the number of primes not exceeding $t$.