A prime circle of order 2n is a circular permutation of the numbers from 1 to 2n with each adjacent pair summing to a prime. Filz (1982) asked whether there exists a prime cycle for all even 2n. In 2021, Chen et al. proved that there exist prime circles for infinitely many of even numbers. In this note, we consider an infinite version of prime circles, that is, a two-way rearrangement of positive integers, say <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2976_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$(a_{i})_{i=-\infty }^{+\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mo>-</mo> <mi>∞</mi> </mrow> <mrow> <mo>+</mo> <mi>∞</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2976_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$a_{i}+a_{i+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is a prime for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2976_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$i\in {{\mathbb {Z}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>. We call such a permutation an infinite prime circle of the positive integers set <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2976_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {N}}^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>. With a theorem due to Zhang on bounded gaps between primes, we show that there exists an infinite prime circle of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2976_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {N}}^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>.

An infinite version of prime circles

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发表日期：

2025年9月28日

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Wangyi Shang,
Binlong Li,
Shenggui Zhang

摘要

A prime circle of order 2n is a circular permutation of the numbers from 1 to 2n with each adjacent pair summing to a prime. Filz (1982) asked whether there exists a prime cycle for all even 2n. In 2021, Chen et al. proved that there exist prime circles for infinitely many of even numbers. In this note, we consider an infinite version of prime circles, that is, a two-way rearrangement of positive integers, say $(a_{i})_{i=-\infty }^{+\infty }$ ${(a_{i})}_{i = - \infty}^{+ \infty}$ , such that $a_{i}+a_{i+1}$ $a_{i} + a_{i + 1}$ is a prime for all $i\in {{\mathbb {Z}}}$ $i \in Z$ . We call such a permutation an infinite prime circle of the positive integers set ${\mathbb {N}}^*$ $N^{*}$ . With a theorem due to Zhang on bounded gaps between primes, we show that there exists an infinite prime circle of ${\mathbb {N}}^*$ $N^{*}$ .