A set of m distinct nonzero rationals <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1703_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\{a_1, a_2,\ldots , a_m\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1703_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$a_i a_j+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>a</mi> <mi>j</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a perfect square for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1703_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le i <j \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, is called a rational Diophantine m-tuple. If in addition, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1703_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$a_i^2+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>a</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a perfect square for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1703_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le i\le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, then we say the m-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.

Rational Diophantine sextuples with strong pair

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发表日期：

2025年1月27日

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Andrej Dujella,
Matija Kazalicki,
Vinko Petričević

摘要

A set of m distinct nonzero rationals $\{a_1, a_2,\ldots , a_m\}$ ${a_{1}, a_{2}, \dots, a_{m}}$ such that $a_i a_j+1$ $a_{i} a_{j} + 1$ is a perfect square for all $1\le i <j \le m$ $1 \leq i < j \leq m$ , is called a rational Diophantine m-tuple. If in addition, $a_i^2+1$ $a_{i}^{2} + 1$ is a perfect square for $1\le i\le m$ $1 \leq i \leq m$ , then we say the m-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.