Let N be a large enough natural number and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">B</mi> </math></EquationSource> </InlineEquation> be subsets of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="131" /> </InlineMediaObject> <EquationSource Format="TEX">$\{N+1, \cdots , 2N\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>2</mn> <mi>N</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that there exist integers a, b with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$a\in \mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="fraktur">A</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$b\in \mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi mathvariant="fraktur">B</mi> </mrow> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ55"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_Equ55.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} ab=P_k^2 + O(P_k^{1-\delta }) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msubsup> <mi>P</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>P</mi> <mi>k</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for some <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1167_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$P_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> denotes an almost-prime with at most k prime factors, counted with multiplicity.

On a multiplicative hybrid problem over almost-primes

期刊论文
发表日期：

2025年7月21日

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Yuetong Zhao,
Wenguang Zhai

摘要

Let N be a large enough natural number and $\mathfrak {A}$ $A$ and $\mathfrak {B}$ $B$ be subsets of $\{N+1, \cdots , 2N\}$ ${N + 1, \dots, 2 N}$ . In this paper, we prove that there exist integers a, b with $a\in \mathfrak {A}$ $a \in A$ , $b\in \mathfrak {B}$ $b \in B$ such that $\begin{aligned} ab=P_k^2 + O(P_k^{1-\delta }) \end{aligned}$ $\begin{matrix} a b = P_{k}^{2} + O (P_{k}^{1 - δ}) \end{matrix}$ for some $\delta >0$ $δ > 0$ , where $P_k$ $P_{k}$ denotes an almost-prime with at most k prime factors, counted with multiplicity.