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

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arxiv 2411.09446 v4 pith:W6YLRGQW submitted 2024-11-14 math.NT

Tianhan Dai , Yuchen Ding , Hui Wang This is my paper
classification math.NT
keywords alfonsconjectureprimeab-a-bconfirmsexistsformintegers
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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.

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  1. Primes of the form $ax+by$

    math.NT 2025-06 conditional novelty 6.0 of 10

    For coprime 3≤a<b, at least 0.005(ab-a-b)/log(ab-a-b) primes below the Frobenius number are representable as ax+by, and for fixed a the asymptotic fraction is 1/2-1/(2(a-1)).

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