Primes of the form $ax+by$ in certain intervals with small solutions
Yuchen Ding, Takao Komatsu, Honghu Liu
TL;DR
This paper studies primes relative to the $\ell$-numerical semigroup generated by $a$ and $b$, focusing on primes up to the $\ell$-Frobenius number $g_{\ell,a,b}=(\ell+1)ab-a-b$ that have at least $\ell+1$ representations and those with at most $\ell$. Using analytic methods for primes in residue classes (notably the Siegel–Walfisz theorem) and structural results on representations of $n=ax+by$, the authors derive an asymptotic density for $\pi_{\ell,a,b}$ as $b\to\infty$ for fixed $a$, namely $\pi_{\ell,a,b}=\left(\frac{a-2}{2(\ell a+a-1)}+o(1)\right)\pi(g_{\ell,a,b})$, and establish positive lower bounds for $\pi_{\ell,a,b}$ and strong lower bounds for $\pi^{*}_{\ell,a,b}$ across a variety of parameter regimes. These results generalize prior work of Chen and Zhu (2025) from $\ell=0$ to arbitrary $\ell\ge0$, with explicit constants and case analyses, highlighting the interaction between Frobenius-type representations and prime distribution in arithmetic progressions. The methods yield insights into primes in short intervals and residue classes within the Frobenius-structured context, contributing to the broader understanding of primes constrained by additive representation constraints.
Abstract
Let $1<a<b$ be two relatively prime integers and $\mathbb{Z}_{\ge 0}$ the set of non-negative integers. For any non-negative integer $\ell$, denote by $g_{\ell,a,b}$ the largest integer $n$ such that the equation $$n=ax+by,\quad (x,y)\in\mathbb{Z}_{\ge 0}^{2} \quad (1)$$ has at most $\ell$ solutions. Let $π_{\ell,a,b}$ be the number of primes $p\leq g_{\ell,a,b}$ having at least $\ell+1$ solutions for (1) and $π(x)$ the number of primes not exceeding $x$. In this article, we prove that for a fixed integer $a\ge 3$ with $\gcd(a,b)=1$, $$ π_{\ell,a,b}=\left(\frac{a-2}{2(\ell a+a-1)}+o(1)\right)π\bigl(g_{\ell,a,b}\bigr)\quad(\text{as}~ b\to\infty). $$ For any non-negative $\ell$ and relatively prime integers $a,b$, satisfying $e^{\ell+1}\leq a<b$, we show that \begin{equation*} π_{\ell,a,b}>0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} Let $π_{\ell,a,b}^{*}$ be the number of primes $p\leq g_{\ell,a,b}$ having at most $\ell$ solutions for (1). For an integer $a\ge 3$ and a large sufficiently integer $b$ with $\gcd(a,b)=1$, we also prove that $$ π^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$ Moreover, if $\ell<a<b$ with $\gcd(a,b)=1$, then we have \begin{equation*} π^{*}_{\ell,a,b}>\frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case $\ell=0$.