Note for the Beal&apos;s Conjecture

We changed the abstract and the content. Indeed, we create a new Lemma that supports the Lemma 2 (which is the Lemma 1 in the previous version). This new version is complete since the previous version contains gaps and flaws that were filled and fixed in this current version.

Abstract

This work explores two famous conjectures in number theory: Fermat's Last Theorem and Beal's Conjecture. Fermat's Last Theorem, posed by Pierre de Fermat in the 17th century, states that there are no positive integer solutions for the equation $a^{n} + b^{n} = c^{n}$, where $n$ is greater than $2$. This theorem remained unproven for centuries until Andrew Wiles published a proof in 1994. Beal's Conjecture, formulated in 1997 by Andrew Beal, generalizes Fermat's Last Theorem. It states that for positive integers $A$, $B$, $C$, $x$, $y$, and $z$, if $A^{x} + B^{y} = C^{z}$ (where $x$, $y$, and $z$ are all greater than $2$), then $A$, $B$, and $C$ must share a common prime factor. Beal's Conjecture remains unproven, and a significant prize is offered for a solution. This paper provides a concise introduction to both conjectures, highlighting their connection and presenting a short proof of the Beal's Conjecture.

Content

Author and article information

Journal

Title: ScienceOpen Preprints

Publisher: ScienceOpen

Publication date (Electronic preprint): 4 June 2024

Affiliations

[1 ] GROUPS PLUS TOURS INC., 9611 Fontainebleau Blvd, Miami, FL, 33172, USA;

Author notes

[* ]Email: vega.frank@ 123456gmail.com .

Author information

Frank Vega https://orcid.org/0000-0001-8210-4126

Article

DOI: 10.14293/PR2199.000861.v2

SO-VID: 18685049-1f3d-40fa-bfad-13e2d92d4d58

License:

This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

History

Date received : 11 May 2024

References

Irwin Klee. Toward the Unification of Physics and Number Theory. Reports in Advances of Physical Sciences. Vol. 03(01)2019. World Scientific Pub Co Pte Lt. [Cross Ref]
Mihailescu P.. Primary cyclotomic units and a proof of Catalans conjecture. Journal für die reine und angewandte Mathematik (Crelles Journal). Vol. 2004(572)2004. Walter de Gruyter GmbH. [Cross Ref]
Anni Samuele, Siksek Samir. Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp. Algebra & Number Theory. Vol. 10(6):1147–1172. 2016. Mathematical Sciences Publishers. [Cross Ref]
Rahimi Amir M.. An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass. Missouri Journal of Mathematical Sciences. Vol. 29(2)2017. University of Central Missouri, Department of Mathematics and Computer Science. [Cross Ref]
Freitas Nuno, Naskręcki Bartosz, Stoll Michael. The generalized Fermat equation with exponents 2, 3,. Compositio Mathematica. Vol. 156(1):77–113. 2020. Wiley. [Cross Ref]
Ribet Kenneth A.. Galois representations and modular forms. Bulletin of the American Mathematical Society. Vol. 32(4):375–402. 1995. American Mathematical Society (AMS). [Cross Ref]
Wiles Andrew. Modular Elliptic Curves and Fermat's Last Theorem. The Annals of Mathematics. Vol. 141(3)1995. JSTOR. [Cross Ref]

Comments

Comment on this article

[1] Irwin Klee. Toward the Unification of Physics and Number Theory. Reports in Advances of Physical Sciences. Vol. 03(01)2019. World Scientific Pub Co Pte Lt. [Cross Ref]

[2] Mihailescu P.. Primary cyclotomic units and a proof of Catalans conjecture. Journal für die reine und angewandte Mathematik (Crelles Journal). Vol. 2004(572)2004. Walter de Gruyter GmbH. [Cross Ref]

[3] Anni Samuele, Siksek Samir. Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp. Algebra & Number Theory. Vol. 10(6):1147–1172. 2016. Mathematical Sciences Publishers. [Cross Ref]

[4] Rahimi Amir M.. An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass. Missouri Journal of Mathematical Sciences. Vol. 29(2)2017. University of Central Missouri, Department of Mathematics and Computer Science. [Cross Ref]

[5] Freitas Nuno, Naskręcki Bartosz, Stoll Michael. The generalized Fermat equation with exponents 2, 3,. Compositio Mathematica. Vol. 156(1):77–113. 2020. Wiley. [Cross Ref]

[6] Ribet Kenneth A.. Galois representations and modular forms. Bulletin of the American Mathematical Society. Vol. 32(4):375–402. 1995. American Mathematical Society (AMS). [Cross Ref]

[7] Wiles Andrew. Modular Elliptic Curves and Fermat's Last Theorem. The Annals of Mathematics. Vol. 141(3)1995. JSTOR. [Cross Ref]

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Note for the Beal's Conjecture