Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression

Mohammad Amin Sharifi

doi:doi:10.11648/j.pamj.20221104.12

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Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression

Mohammad Amin Sharifi

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 4)

Received: 25 June 2022 Accepted: 18 July 2022 Published: 26 July 2022

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Abstract

The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: A^X+B^Y=C^Z where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Y^p=X^q+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a²+b²=c²), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.

Published in	Pure and Applied Mathematics Journal (Volume 11, Issue 4)
DOI	10.11648/j.pamj.20221104.12
Page(s)	51-69
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’s Conjecture, Catalan’s Theorem, Fermat’s Last Theorem, Pythagoras’ Equation, Arithmetic Progression, Prime Numbers, Integer Numbers

References

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[2]	Aithani, H, S. (218). The Mathematical Solution for the Beal Conjecture. AMIERJ. Volume- VII, Issues- IV, ISSN-2278-5655.
[3]	Buya, S, B. (2019). A Review of Buya’s Proof of Beal’s Conjecture and Simple Proof of Fermat’s Last Theorem. American Journal of Mathematical Analysis. Vol. 7, No. 1, 15-16. DOI: 10.12691/ajma-7-1-3.
[4]	Buya, S, B. (2021). A final of Buya’s Proof of Beal’s Conjecture. ResearchGate.
[5]	Nyambuya, G, G. (2014). A Simple and General Proof of Beal’s Conjecture. Advances in Pure Mathematics. 4, 518-521.
[6]	Buya, S, B. (2018). Proof of Beal’s conjecture. African Journal of Mathematics and Computer Science Research. Vol. 11 (8), PP. 109-113. ISSN: 2006-9731. DOI: 10.5897/AJMCSR2018.0764.
[7]	Porras Ferreira, J, W. (2014). The Solution for Fermat’s Last Theorem using Exponential Algebra Properties. Global Open Research Journal of Mathematics. Vol. 2, No. 1, PP. 01-11.
[8]	Stewart, I., & Tall, D. (2002). Algebraic Number Theory and Fermat’s Last Theorem. A K Peters, Ltd. 63 South Avenue. Natick, MA 01760. (3^th ed).
[9]	Marecek, L., Anthony-Smith, M, A,. & Mathis, A, H. (2020). Elementary Algebra 2e. Rice University. 6100 Main Street MS-375, Houston, Texas 77005.
[10]	Di Gregorio, L, T. (2018). A Deeper Analysis on a Generalization of Fermat’s Last Theorem. Canadian center of Science and Education. Journal of Mathematics Research. Vol. 10, No. 2. ISSN 1916-9795, E-ISSN 1916-9809.
[11]	Mazaris, A. (2018). A new Proof of Fermat’s Last theorem. Open Access Journal of Mathematical and Theoretical Physics. Volume 1, Issue 3.
[12]	Kumar, V. (2014). Proof of Fermat Last Theorem based on Odd Even Classification of integers. Int. J. Open Problems Compt. Math. Vol. 7, No. 4. ISSN 1998-6262.
[13]	Sghiar, M. (2020). The Fermat Classes and The Proof of Beal Conjecture. IOSR Journal of Mathematics (IOSR-JM). Volume 16. Issue 3. PP 09-13. e-ISSN: 2278-5728, p-ISSN: 2319-765X.
[14]	Frey, G. (2009). The Way to the Proof of Fermat’s Last Theorem. DE TOULOUSE Mathématiques. Vol. XVIII, PP. 5-23.
[15]	Metsankyla, T. (2003). Catalan’s Conjecture: Another Old Diophantine Problem Solved. Bulletin (New Series) of the American Mathematical Society. Volume 41, Number 1, Pages 43-57.
[16]	Koymans, P, H. (2017). The Catalan equation. Indagationes Mathematicae. 321-352.
[17]	Daems, J. (2003). A Cyclotomic Proof of Catalan’s Conjecture.
[18]	Darmon, H., & Granville, A. (1995). On The Equations z^m=F(x,y) And Ax^p+By^q=Cz^r. Bull. London Math. Soc. 27 (1995) 513-543.
[19]	Bird, J. (2017). Higher Engineering Mathematics. Routledge, 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN. 711 Third Avenue, New York, NY 10017. (8^th ed).
[20]	Lemma, M., Lambright, J., & Epps, B. (2015). The Mathematical Beauty of Triangular Numbers. Hawaii University International Conferences. ISSN 2333-4916 (CD-ROM), ISSN 2333-4908 (ONLINE).
[21]	Ellis, W., & Burzynski, D. (2012). Elementary Algebra. Rice University, Houston, Texas.

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Mohammad Amin Sharifi. (2022). Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure and Applied Mathematics Journal, 11(4), 51-69. https://doi.org/10.11648/j.pamj.20221104.12

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Mohammad Amin Sharifi. Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure Appl. Math. J. 2022, 11(4), 51-69. doi: 10.11648/j.pamj.20221104.12

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AMA Style

Mohammad Amin Sharifi. Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure Appl Math J. 2022;11(4):51-69. doi: 10.11648/j.pamj.20221104.12

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@article{10.11648/j.pamj.20221104.12,
  author = {Mohammad Amin Sharifi},
  title = {Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression},
  journal = {Pure and Applied Mathematics Journal},
  volume = {11},
  number = {4},
  pages = {51-69},
  doi = {10.11648/j.pamj.20221104.12},
  url = {https://doi.org/10.11648/j.pamj.20221104.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221104.12},
  abstract = {The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a2+b2=c2), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.},
 year = {2022}
}

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TY  - JOUR
T1  - Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression
AU  - Mohammad Amin Sharifi
Y1  - 2022/07/26
PY  - 2022
N1  - https://doi.org/10.11648/j.pamj.20221104.12
DO  - 10.11648/j.pamj.20221104.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 51
EP  - 69
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20221104.12
AB  - The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a2+b2=c2), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.
VL  - 11
IS  - 4
ER  -

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Author Information

Mohammad Amin Sharifi

Civil Department, Engineering Faculty, Herat University, Herat, Afghanistan

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Plain Text BibTeX RIS

APA Style

Mohammad Amin Sharifi. (2022). Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure and Applied Mathematics Journal, 11(4), 51-69. https://doi.org/10.11648/j.pamj.20221104.12

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Mohammad Amin Sharifi. Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure Appl. Math. J. 2022, 11(4), 51-69. doi: 10.11648/j.pamj.20221104.12

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AMA Style

Mohammad Amin Sharifi. Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression. Pure Appl Math J. 2022;11(4):51-69. doi: 10.11648/j.pamj.20221104.12

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@article{10.11648/j.pamj.20221104.12,
  author = {Mohammad Amin Sharifi},
  title = {Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression},
  journal = {Pure and Applied Mathematics Journal},
  volume = {11},
  number = {4},
  pages = {51-69},
  doi = {10.11648/j.pamj.20221104.12},
  url = {https://doi.org/10.11648/j.pamj.20221104.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221104.12},
  abstract = {The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a2+b2=c2), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression
AU  - Mohammad Amin Sharifi
Y1  - 2022/07/26
PY  - 2022
N1  - https://doi.org/10.11648/j.pamj.20221104.12
DO  - 10.11648/j.pamj.20221104.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 51
EP  - 69
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20221104.12
AB  - The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a2+b2=c2), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.
VL  - 11
IS  - 4
ER  -

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