Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations

Leandro Torres Di Gregorio

doi:doi:10.11648/j.pamj.20130205.11

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Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations

Leandro Torres Di Gregorio

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 5)

Received: 20 August 2013 Accepted: Published: 20 September 2013

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Abstract

The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n₁, n₂ and n₃ positive integers with n₁, n₂, n₃ > 2, if Xⁿ¹ +Yⁿ²=Zⁿ³ then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A² + B² = C² (here simply refereed as Pythagoras´ equation), δⁿ + γⁿ=αⁿ (here simply refereed as Fermat´s equation) and Xⁿ¹ +Yⁿ²=Zⁿ³ (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 5)
DOI	10.11648/j.pamj.20130205.11
Page(s)	149-155
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Beal Conjecture, Fermat´s Last Theorem, Diophantine Equations, Number Theory, Prime Numbers

References

[1]	D. R. Mauldin, "A generalization of Fermat´s last theorem: the Beal conjecture and prize problem", in Notices of the AMS, v. 44, n.11, 1997, pp. 1436-1437.
[2]	H. Darmon and A. Granville, "On the equations Z^m=F(x,y)and Ax^p+By^q=Cz^r", in Bull. London Math. Soc., 27, 1995, pp. 513-543.
[3]	K. Rubin and A. Silverberg, "A report on Wiles´ Cambridge lectures", in Bulletin (New Series) of the American Mathematical Society, v. 31, n. 1, 1994, pp. 15-38.
[4]	I. Stewart, Almanaque das curiosidades matemáticas, 1945, translationby Diego Alfaro, technicalreviewby Samuel Jurkiewicz, Rio de Janeiro: Zahar, 2009.
[5]	Al Shenk, Cálculo e geometria analítica, translationby Anna Amália Feijó Barroso,Rio de Janeiro: Campus, 1983-1984.
[6]	J. L. Boldrini, S. I. R. Costa, V. L. Figueiredo, H. G. Wetzler, Álgebra linear, 3 ed., São Paulo: Harper &Row do Brasil, 1980.
[7]	M. L. Crispino, Variedades lineares e hiperplanos, Rio de Janeiro: Editora Ciência Moderna Ltda, 2008.
[8]	K. Michio, Hiperespaço, translationby Maria Luiza X. de A. Borges, technicalreviewby Walter Maciel, Rio de Janeiro: Rocco, 2000.
[9]	Euclides, Os elementos, translationandintroductionby Irineu Bicudo, São Paulo: Unesp, 2009.
[10]	G. H. Edmund Landau, Teoria elementar dos números, Rio de Janeiro: Editora Ciência Moderna, 2002.

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Leandro Torres Di Gregorio. (2013). Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure and Applied Mathematics Journal, 2(5), 149-155. https://doi.org/10.11648/j.pamj.20130205.11

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Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure Appl. Math. J. 2013, 2(5), 149-155. doi: 10.11648/j.pamj.20130205.11

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AMA Style

Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure Appl Math J. 2013;2(5):149-155. doi: 10.11648/j.pamj.20130205.11

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@article{10.11648/j.pamj.20130205.11,
  author = {Leandro Torres Di Gregorio},
  title = {Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {5},
  pages = {149-155},
  doi = {10.11648/j.pamj.20130205.11},
  url = {https://doi.org/10.11648/j.pamj.20130205.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130205.11},
  abstract = {The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n1, n2 and n3 positive integers with n1, n2, n3 > 2, if Xn1 +Yn2=Zn3 then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A2 + B2 = C2 (here simply refereed as Pythagoras´ equation), δn + γn=αn (here simply refereed as Fermat´s equation) and Xn1 +Yn2=Zn3 (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.},
 year = {2013}
}

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TY  - JOUR
T1  - Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations
AU  - Leandro Torres Di Gregorio
Y1  - 2013/09/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130205.11
DO  - 10.11648/j.pamj.20130205.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 149
EP  - 155
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130205.11
AB  - The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n1, n2 and n3 positive integers with n1, n2, n3 > 2, if Xn1 +Yn2=Zn3 then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A2 + B2 = C2 (here simply refereed as Pythagoras´ equation), δn + γn=αn (here simply refereed as Fermat´s equation) and Xn1 +Yn2=Zn3 (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.
VL  - 2
IS  - 5
ER  -

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Author Information

Leandro Torres Di Gregorio

Souza Marques EngineeringCollege, Rio de Janeiro, Brazil

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Plain Text BibTeX RIS

APA Style

Leandro Torres Di Gregorio. (2013). Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure and Applied Mathematics Journal, 2(5), 149-155. https://doi.org/10.11648/j.pamj.20130205.11

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ACS Style

Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure Appl. Math. J. 2013, 2(5), 149-155. doi: 10.11648/j.pamj.20130205.11

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AMA Style

Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations. Pure Appl Math J. 2013;2(5):149-155. doi: 10.11648/j.pamj.20130205.11

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@article{10.11648/j.pamj.20130205.11,
  author = {Leandro Torres Di Gregorio},
  title = {Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {5},
  pages = {149-155},
  doi = {10.11648/j.pamj.20130205.11},
  url = {https://doi.org/10.11648/j.pamj.20130205.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130205.11},
  abstract = {The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n1, n2 and n3 positive integers with n1, n2, n3 > 2, if Xn1 +Yn2=Zn3 then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A2 + B2 = C2 (here simply refereed as Pythagoras´ equation), δn + γn=αn (here simply refereed as Fermat´s equation) and Xn1 +Yn2=Zn3 (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.},
 year = {2013}
}

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TY  - JOUR
T1  - Developments on Beal Conjecture from Pythagoras´ and Fermat´s Equations
AU  - Leandro Torres Di Gregorio
Y1  - 2013/09/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130205.11
DO  - 10.11648/j.pamj.20130205.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 149
EP  - 155
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130205.11
AB  - The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory´s field. It states that, for X, Y, Z, n1, n2 and n3 positive integers with n1, n2, n3 > 2, if Xn1 +Yn2=Zn3 then X, Y, Z must have a common prime factor. This article aims to present developments on Beal Conjecture, obtained from the correspondences between the real solutions of the equations in the forms A2 + B2 = C2 (here simply refereed as Pythagoras´ equation), δn + γn=αn (here simply refereed as Fermat´s equation) and Xn1 +Yn2=Zn3 (here simply referred as Beal´s equation). Starting from a bibliographical research on the Beal Conjecture, prime numbers and Fermat's Last Theorem, these equations were freely explored, searching for different aspects of their meanings. The developments on Beal Conjecture are divided into four parts: geometric illustrations; correspondence between the real solutions of Pythagoras´ equation and Fermat's equation; deduction of the transforms between the real solutions of Fermat's equation and the Beal´s equation; and analysis and discussion about the topic, including some examples. From the correspondent Pythagoras´ equation, if one of the terms A, B or C is taken as an integer reference basis, demonstrations enabled to show that the Beal Conjecture is correct if the remaining terms, when squared, are integers.
VL  - 2
IS  - 5
ER  -

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