Note on a conjecture of Ramírez Alfonsín and Skałba

TL;DR

This work resolves the Ramírez Alfonsín–Skałba conjecture by proving that for every pair of coprime integers $a,b$ with $2<a<b$ and $g=ab-a-b$, there exists at least one prime $p\le g$ expressible as $p=ax+by$ with $x,y\ge 0$. The authors combine effective prime-distribution results in arithmetic progressions (via explicit Siegel–Walfisz-type bounds) with Sylvester’s antisymmetry to count nonrepresentable integers and derive lower bounds for $\pi_{a,b}$. A lengthy case analysis (and supporting computer verifications) shows $\pi_{a,b}>0$ uniformly across parameter ranges, thereby confirming the conjecture. The result advances understanding of primes in Frobenius-type linear forms and provides effective bounds across various regimes of $(a,b)$, solidifying a 2020 conjecture and offering techniques applicable to related prime-representation questions.

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 Ramírez Alfonsín and Skałba.

Theorems & Definitions (10)

• Conjecture 1
• Remark 1
• Conjecture 2
• Theorem 1
• Lemma 1
• Remark 2
• Lemma 2
• Lemma 3
• Lemma 4
• proof : Proof of Theorem \ref{['thm1']}