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On irrationals with Lagrange value exactly 3.

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Title: On irrationals with Lagrange value exactly 3.
Authors: Cao, Zhe1 (AUTHOR) zhecao@mail.nankai.edu.cn, Erazo, Harold2 (AUTHOR) harold.erazo@impa.br, Moreira, Carlos Gustavo2,3 (AUTHOR) gugu@impa.br
Source: Journal of Number Theory. May2026, Vol. 282, p147-173. 27p.
Subjects: Irrational numbers, Lagrange problem, Real numbers, Cardinal numbers, Mathematics, Number theory
Abstract: For c > 0 , let X c denote the set of x ∈ R ﹨ Q such that | x − p q | < 1 c q 2 has only finitely many rational solutions p q. It is a classical fact, known since the 1950s, that X c is uncountable for c > 3 and countable for c < 3. However, the cardinality of X 3 does not appear to be present in the literature. We prove that X 3 is uncountable. More generally, we show that for any n ∈ N ∪ { ∞ } , the set of x ∈ R ﹨ Q with Lagrange value exactly 3 and such that | x − p q | < 1 3 q 2 has exactly n rational solutions p q is also uncountable. For a video summary of this paper, please visit https://youtu.be/VyKB99-kVeY. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: On irrationals with Lagrange value exactly 3.
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  Data: &lt;searchLink fieldCode=&quot;JN&quot; term=&quot;%22Journal+of+Number+Theory%22&quot;&gt;Journal of Number Theory&lt;/searchLink&gt;. May2026, Vol. 282, p147-173. 27p.
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  Data: For c &gt; 0 , let X c denote the set of x ∈ R ﹨ Q such that | x − p q | &lt; 1 c q 2 has only finitely many rational solutions p q. It is a classical fact, known since the 1950s, that X c is uncountable for c &gt; 3 and countable for c &lt; 3. However, the cardinality of X 3 does not appear to be present in the literature. We prove that X 3 is uncountable. More generally, we show that for any n ∈ N ∪ { ∞ } , the set of x ∈ R ﹨ Q with Lagrange value exactly 3 and such that | x − p q | &lt; 1 3 q 2 has exactly n rational solutions p q is also uncountable. For a video summary of this paper, please visit https://youtu.be/VyKB99-kVeY. [ABSTRACT FROM AUTHOR]
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  Data: &lt;i&gt;Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.jnt.2025.11.006
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      – Code: eng
        Text: English
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        PageCount: 27
        StartPage: 147
    Subjects:
      – SubjectFull: Irrational numbers
        Type: general
      – SubjectFull: Lagrange problem
        Type: general
      – SubjectFull: Real numbers
        Type: general
      – SubjectFull: Cardinal numbers
        Type: general
      – SubjectFull: Mathematics
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      – SubjectFull: Number theory
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      – TitleFull: On irrationals with Lagrange value exactly 3.
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            NameFull: Cao, Zhe
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            NameFull: Erazo, Harold
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            – D: 01
              M: 05
              Text: May2026
              Type: published
              Y: 2026
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              Value: 282
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