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On the subgroup generated by solutions of Pell's equation.

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Title: On the subgroup generated by solutions of Pell's equation.
Authors: Covill, Elena C.1 ec20covi@siena.edu, Javaheri, Mohammad1 mjavaheri@siena.edu, Krylov, Nikolai A.1 nkrylov@siena.edu
Source: Journal of Number Theory. Jan2019, Vol. 194, p356-371. 16p.
Subjects: Mathematical equivalence, Diophantine equations, Integers, Pell's equation, Mathematical models
Abstract: Abstract Equivalence classes of solutions of the Diophantine equation a 2 + m b 2 = c 2 form an infinitely generated abelian group G m , where m is a fixed square-free positive integer. Solutions of Pell's equation x 2 − m y 2 = 1 generate a subgroup P m of G m. We prove that P m and G m / P m have infinite rank for all m > 1. We also give several examples of m for which G m / P m has nontrivial torsion. [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.)
Database: Engineering Source
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Header DbId: egs
DbLabel: Engineering Source
An: 132105930
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Data: On the subgroup generated by solutions of Pell's equation.
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  Data: <searchLink fieldCode="AR" term="%22Covill%2C+Elena+C%2E%22">Covill, Elena C.</searchLink><relatesTo>1</relatesTo><i> ec20covi@siena.edu</i><br /><searchLink fieldCode="AR" term="%22Javaheri%2C+Mohammad%22">Javaheri, Mohammad</searchLink><relatesTo>1</relatesTo><i> mjavaheri@siena.edu</i><br /><searchLink fieldCode="AR" term="%22Krylov%2C+Nikolai+A%2E%22">Krylov, Nikolai A.</searchLink><relatesTo>1</relatesTo><i> nkrylov@siena.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Jan2019, Vol. 194, p356-371. 16p.
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  Data: <searchLink fieldCode="DE" term="%22Mathematical+equivalence%22">Mathematical equivalence</searchLink><br /><searchLink fieldCode="DE" term="%22Diophantine+equations%22">Diophantine equations</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink><br /><searchLink fieldCode="DE" term="%22Pell's+equation%22">Pell's equation</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: Abstract Equivalence classes of solutions of the Diophantine equation a 2 + m b 2 = c 2 form an infinitely generated abelian group G m , where m is a fixed square-free positive integer. Solutions of Pell's equation x 2 − m y 2 = 1 generate a subgroup P m of G m. We prove that P m and G m / P m have infinite rank for all m > 1. We also give several examples of m for which G m / P m has nontrivial torsion. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>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.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.jnt.2018.07.001
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      – Code: eng
        Text: English
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        PageCount: 16
        StartPage: 356
    Subjects:
      – SubjectFull: Mathematical equivalence
        Type: general
      – SubjectFull: Diophantine equations
        Type: general
      – SubjectFull: Integers
        Type: general
      – SubjectFull: Pell's equation
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      – SubjectFull: Mathematical models
        Type: general
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      – TitleFull: On the subgroup generated by solutions of Pell's equation.
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            NameFull: Covill, Elena C.
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            NameFull: Javaheri, Mohammad
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            NameFull: Krylov, Nikolai A.
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              Text: Jan2019
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              Y: 2019
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              Value: 194
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            – TitleFull: Journal of Number Theory
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