Note for the Beal&apos;s Conjecture

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 the ongoing challenge that a short proof for the Beal's Conjecture presents to mathematicians.

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Author and article information

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Title: ScienceOpen Preprints

Publisher: ScienceOpen

Publication date (Electronic preprint): 11 May 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.v1

SO-VID: 52ef21d2-949b-4776-a2e2-351e259b44d4

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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