Let 1 < c < d be two relatively prime integers, gc,d = cd − c − d, and let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9660_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb{P}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation> be the set of primes. For any given integer k ⩾ 1, we prove that<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9660_Article_IEq2.gif" Format="GIF" Height="30" Rendition="HTML" Resolution="72" Type="Linedraw" Width="479" /> </InlineMediaObject> <EquationSource Format="TEX">$\#\left\{{p}^{k}\leqslant{g}_{c,d}:p\in {\mathbb{P}},{p}^{k}=cx+dy,x,y\in {\mathbb{Z}}_{\geqslant0}\right\}\sim \frac{k}{K+1}\frac{{g}^{1/k}}{\mathrm{log}g} \mathrm{as } c\to \infty ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <mfenced close="}" open="{"> <msup> <mrow> <mi>p</mi> </mrow> <mi>k</mi> </msup> <mo>⩽</mo> <msub> <mi>g</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">P</mi> <mo>,</mo> <msup> <mrow> <mi>p</mi> </mrow> <mi>k</mi> </msup> <mo>=</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </mfenced> <mo>∼</mo> <mfrac> <mi>k</mi> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mfrac> <msup> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">log</mi> <mi>g</mi> </mrow> </mfrac> <mi mathvariant="normal">as</mi> <mi>c</mi> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation>which gives an extension of a recent result of Ding, Zhai, and Zhao.

The distribution of powers of primes related to the Frobenius problem

期刊论文
发表日期：

2025年1月1日

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Enxun Huang,
Tengyou Zhu

摘要

Let 1 < c < d be two relatively prime integers, g_c,d = cd − c − d, and let ${\mathbb{P}}$ $P$ be the set of primes. For any given integer k ⩾ 1, we prove that

$\#\left\{{p}^{k}\leqslant{g}_{c,d}:p\in {\mathbb{P}},{p}^{k}=cx+dy,x,y\in {\mathbb{Z}}_{\geqslant0}\right\}\sim \frac{k}{K+1}\frac{{g}^{1/k}}{\mathrm{log}g} \mathrm{as } c\to \infty ,$ $# \{p^{k} ⩽ g_{c, d} : p \in P, p^{k} = c x + d y, x, y \in Z_{⩾ 0}\} \sim \frac{k}{K + 1} \frac{g^{1 / k}}{\log g} as c \to \infty,$

which gives an extension of a recent result of Ding, Zhai, and Zhao.