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

TL;DR

This paper proves a conjecture stating that for two coprime integers a and b greater than 2, there exists at least one prime of a specific linear form less than or equal to g, confirming a 2020 conjecture.

Contribution

The paper confirms a 2020 conjecture by proving the existence of a prime of a certain linear form within a bound for coprime integers a and b.

Findings

Existence of a prime p of the form p=ax+by within p ≤ g.

Confirmation of the 2020 conjecture by Ramírez Alfonsín and Skałba.

Advancement in understanding primes in linear forms.

Abstract

Let $2< a<b$ be two relatively prime integers and $g=ab-a-b$. It is proved that there exists at least one prime $p\le g$ of the form $p=ax+by~(x,y\in \mathbb{Z}_{\ge 0})$, which confirms a 2020 conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba.