A Result on Odd Powers in Fermat’s Last Theorem

Kelvin Muzundu

doi:doi:10.11648/j.ml.20241001.11

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A Result on Odd Powers in Fermat’s Last Theorem

Kelvin Muzundu

Published in Mathematics Letters (Volume 10, Issue 1)

Received: 5 December 2023 Accepted: 9 January 2024 Published: 23 January 2024

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Abstract

In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.

Published in	Mathematics Letters (Volume 10, Issue 1)
DOI	10.11648/j.ml.20241001.11
Page(s)	1-6
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

Fermat’s Last Theorem, Odd Powers, Integers

References

[1]	Albert, A. A., Modern Higher Algebra, Cambridge University Press, 1938.
[2]	Z. I. Borevich, Z. I., Shafarevich I. R., Number Theory, Academic Press; New York, 1966.
[3]	Faltings, G. The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles, Notices of the American Mathematical Society 1995, 42(7), 743-746.
[4]	Grosswald, E., Topics from the Theory of Numbers, Macmillan, New York, 1965.
[5]	Ireland K., Rosen M., A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, New York, 1990.
[6]	Lenstra, H. W., Jr., Euclidean Number Fields 1, Math. Intelligencer 1979, 2, 6-15.
[7]	Long, C. T., Elementary Introduction to Number Theory, D. C. Heath and Company, 1972.
[8]	Maeda, Y., Modularity of special cycles on unitary Shimura varieties over CM-fields, Acta Arithmetica 2022, 204(1), 1-18.
[9]	Mahoney, M. S., The Mathematical Career of Pierre de Fermat, 1601-1665, (2^nd ed.), Princeton University Press, 1994.
[10]	McLarty, C., The large structures of Grothendieck founded on finite order arithmetic, Rev. Symb. Log. 2020, 13(2), 296-325.
[11]	Ribet, K., Galois Representations and Modular Forms, Bulletin of AMS 1995, 32, 375-402.
[12]	Ribenboim, P., 13 Lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979.
[13]	Singh, S. Fermat’s Last Theorem. Notting Hill: Fourth Estate; 1997.
[14]	Taylor, R., Wiles, A., Ring-Theoretic Properties of Certain Hecke Algebras, Ann. Of Math. 1995, 141, 553-572.
[15]	van der Poorten, A., Notes on Fermat's Last Theorem, J. Wiley & Sons, New York, 1996.
[16]	Washington, L., An Introduction to Cyclotomic Fields, 2nd ed., Springer-Verlag, New York, 1997.
[17]	Wiles, A. Modular elliptic curves and Fermat’s Last Theorem. Ann. Of Math. 1995, 141, 443-551.
[18]	Zhang, S., On a comparison of Cassels pairings of different elliptic curves, Acta Arithmetica 2023, 211(1), 1-23.

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Muzundu, K. (2024). A Result on Odd Powers in Fermat’s Last Theorem. Mathematics Letters, 10(1), 1-6. https://doi.org/10.11648/j.ml.20241001.11

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ACS Style

Muzundu, K. A Result on Odd Powers in Fermat’s Last Theorem. Math. Lett. 2024, 10(1), 1-6. doi: 10.11648/j.ml.20241001.11

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AMA Style

Muzundu K. A Result on Odd Powers in Fermat’s Last Theorem. Math Lett. 2024;10(1):1-6. doi: 10.11648/j.ml.20241001.11

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@article{10.11648/j.ml.20241001.11,
  author = {Kelvin Muzundu},
  title = {A Result on Odd Powers in Fermat’s Last Theorem},
  journal = {Mathematics Letters},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ml.20241001.11},
  url = {https://doi.org/10.11648/j.ml.20241001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241001.11},
  abstract = {In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.
},
 year = {2024}
}

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TY  - JOUR
T1  - A Result on Odd Powers in Fermat’s Last Theorem
AU  - Kelvin Muzundu
Y1  - 2024/01/23
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241001.11
DO  - 10.11648/j.ml.20241001.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241001.11
AB  - In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.

VL  - 10
IS  - 1
ER  -

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Author Information

Kelvin Muzundu

Department of Mathematics and Statistics, University of Zambia, Lusaka, Zambia

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Muzundu, K. (2024). A Result on Odd Powers in Fermat’s Last Theorem. Mathematics Letters, 10(1), 1-6. https://doi.org/10.11648/j.ml.20241001.11

Copy | Download

ACS Style

Muzundu, K. A Result on Odd Powers in Fermat’s Last Theorem. Math. Lett. 2024, 10(1), 1-6. doi: 10.11648/j.ml.20241001.11

Copy | Download

AMA Style

Muzundu K. A Result on Odd Powers in Fermat’s Last Theorem. Math Lett. 2024;10(1):1-6. doi: 10.11648/j.ml.20241001.11

Copy | Download

@article{10.11648/j.ml.20241001.11,
  author = {Kelvin Muzundu},
  title = {A Result on Odd Powers in Fermat’s Last Theorem},
  journal = {Mathematics Letters},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ml.20241001.11},
  url = {https://doi.org/10.11648/j.ml.20241001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241001.11},
  abstract = {In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - A Result on Odd Powers in Fermat’s Last Theorem
AU  - Kelvin Muzundu
Y1  - 2024/01/23
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241001.11
DO  - 10.11648/j.ml.20241001.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241001.11
AB  - In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.

VL  - 10
IS  - 1
ER  -

Copy | Download