On a conjecture of Ramírez Alfonsín and Skałba II
Yuchen Ding1 ; Wenguang Zhai2 ; Lilu Zhao3
1 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China
2 China University of Mining and Technology, Beijing 100083, People’s Republic of China
3 School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 357-371

Let $\pi (t)$ be the number of primes not exceeding $t$. Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by combining the Hardy–Littlewood method with the Siegel–Walfisz theorem, a 2020 conjecture of Ramírez Alfonsín and Skałba which states that

\[ \#\left\lbrace p\le g_{c,d}:p\in \mathcal{P},~p=cx+dy,~x,y\in \mathbb{Z}_{\geqslant 0}\right\rbrace \sim \frac{1}{2}\pi \left(g_{c,d}\right) \]

as $c\rightarrow \infty ,$ where $\mathcal{P}$ and $\mathbb{Z}_{\geqslant 0}$ denote the sets of primes and nonnegative integers, respectively.

Soit $\pi (t)$ le nombre de nombres premiers inférieurs ou égaux à $t.$ Soient $1<c<d$ deux entiers premiers entre eux et $g_{c,d}=cd-c-d$. En combinant la méthode de Hardy–Littlewood avec le théorème de Siegel–Walfisz, nous démontrons la conjecture, énoncée en 2020 par Ramírez Alfonsín et Skałba, qui affirme que

\[ \#\left\lbrace p\le g_{c,d}:p\in \mathcal{P},~p=cx+dy,~x,y\in \mathbb{Z}_{\geqslant 0}\right\rbrace \sim \frac{1}{2}\pi \left(g_{c,d}\right) \]

quand $c\rightarrow \infty ,$ où $\mathcal{P}$ et $\mathbb{Z}_{\geqslant 0}$ désignent l’ensemble des nombres premiers et l’ensemble des entiers non négatifs respectivement.

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DOI : 10.5802/jtnb.1324
Classification : 11N05, 11P55
Keywords: Frobenius–type problems, Hardy–Littlewood method, primes, Siegel–Walfisz theorem

Yuchen Ding 1 ; Wenguang Zhai 2 ; Lilu Zhao 3

1 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China
2 China University of Mining and Technology, Beijing 100083, People’s Republic of China
3 School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Yuchen Ding; Wenguang Zhai; Lilu Zhao. On a conjecture of Ramírez Alfonsín and Skałba II. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 357-371. doi: 10.5802/jtnb.1324

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