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Journal articles on the topic 'Beal's conjecture'

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Published: 3 June 2025
Last updated: 31 July 2025

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1

JAMES, E. JOSEPH, and M. P. NAYAR BHAMINI. "A PROOF OF BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 14, no. 1 (2018): 2324–26. https://doi.org/10.5281/zenodo.3974352.

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The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z $ with $\mu, \xi $ and $ \nu$ odd primes at least $3$. A proof of this longstanding conjecture is given.
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2

James, Joseph. "A Proof of Beal's conjecture." Journal of Progressive Research in Mathematics 9, no. 3 (2016): 1411–12. https://doi.org/10.5281/zenodo.3976690.

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It is proved in this paper t that the equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z,$ with $\xi, \mu, \nu$ odd primes at least $3.$  This is equivalent to Fermat\rq{}s Last Theorem which is stated as follows: If $x.y, z$ are  positive integers, and $\pi$ is an odd prime satisfying $z^\pi=x^\pi+y^\pi,$ then  $x, y, z$ are not relatively prime.
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3

Gandhi, K. Raja Rama, and Reuven Tint. "Proof of Beal's Conjecture." Bulletin of Mathematical Sciences and Applications 5 (August 2013): 48–49. http://dx.doi.org/10.18052/www.scipress.com/bmsa.5.48.

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4

Nishad, T. M., and Mohamed M. Azzedine Dr. "Proofs of Beal's Conjecture, Fermat's Conjecture, Collatz Conjecture and Goldbach Conjecture." Indian Journal of Advanced Mathematics (IJAM) 3, no. 1 (2023): 1–7. https://doi.org/10.54105/ijam.A1137.043123.

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5

Guła, Leszek W. "The Proof of the Beal’s Conjecture." Bulletin of Society for Mathematical Services and Standards 12 (December 2014): 21–28. http://dx.doi.org/10.18052/www.scipress.com/bsmass.12.21.

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The truly marvellous proof of The Fermat's Last Theorem (FLT). Two false proofs of FLT for n = 4. The incomplete proof of FLT for odd prime numbers n ∈ Ρ. The proof of The Beal's Conjecture. The Beal's Theorem.
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6

Joseph, James E., and Bhamini P. Nayar. "Another Proof of Beal's Conjecture." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 2 (2018): 7878–79. http://dx.doi.org/10.24297/jam.v14i2.7587.

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7

Bosco, Adriko. "The Beal's Conjecture and Fermat's Last Theorem." International Journal of Science and Research (IJSR) 10, no. 11 (2021): 435–41. https://doi.org/10.21275/mr211031162015.

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8

Alexander, Bolotin. "The Propositional Lattice of Divisibility and Beal's Conjecture." British Journal of Mathematics & Computer Science 22, no. 2 (2017): 1–8. https://doi.org/10.9734/BJMCS/2017/33315.

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9

dos Santos Godoi, Valdir Monteiro. "Disproof the Four Counterexamples for Beal's Conjecture." Bulletin of Mathematical Sciences and Applications 13 (October 2015): 13. http://dx.doi.org/10.18052/www.scipress.com/bmsa.13.13.

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In journal, probably by mistake, was published a paper which intended give some counterexamples of the Beal’s conjecture. Unfortunately these counterexamples are wrong, and the Beal ́s conjecture hold true.
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10

James, Joseph. "UNIQUE FACTORIZATION FERMAT'S LAST THEOREM BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 10, no. 1 (2016): 1434–39. https://doi.org/10.5281/zenodo.3976651.

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In this paper the following statememt of Fermat\rq{}s Last Theorem is proved.  If  $x, y, z$ are positive integers$\pi$ is an odd prime and  $z^\pi=x^\pi+y^\pi, x, y, z$ are all even. Also, in this paper, is proved (Beal\rq{}s conjecture): The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.
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11

Joseph, James E. "ALGEBRAIC PROOFS FERMAT'S LAST THEOREM, BEAL'S CONJECTURE." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 9 (2016): 6576–77. http://dx.doi.org/10.24297/jam.v12i9.130.

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In this paper, the following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; then x; y; z are all even. Also, in this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:
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12

James, E. Joseph, and M. P. Nayar Bhamini. "EQUIVALENCE OF FERMAT'S LAST THEOREM AND BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 14, no. 1 (2018): 2289–91. https://doi.org/10.5281/zenodo.3974322.

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It is proved in this paper that (1){ \bf Fermat's Last Theorem:} If $\pi$ is an odd prime, there are no relatively prime solutions $x, y, z$ to the equation $z^\pi=x^\pi+y^\pi,$ and (2) { \bf Beal's Conjecture :} The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z$ with $\mu, \xi, \nu$ odd primes at least $3$. It is proved that these two statements are equivalent.
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13

JAMES, E. JOSEPH. "Proofs of Fermat's Last Theorem and Beal's Conjecture." Journal of Progressive Research in Mathematics 10, no. 1 (2016): 1446–47. https://doi.org/10.5281/zenodo.3976670.

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If $\pi$ is an odd prime and $x, y, z,$ are relatively prime positive integers, then $z^\pi\not=x^\pi+y^\pi.$ In this note, an elegant simple proof of this theorem is given  that if $\pi$ is an odd prime and $x, y, z$ are positive integers satisfying $z^\pi=x^\pi+y^\pi,$ then  $x, y, z,$ are each divisible by $2:$ (Beal\rq{}s conjecture) The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.$ is also proved; that is $x, y, z $ are all even.
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14

Bolotin, Alexander. "The Propositional Lattice of Divisibility and Beal's Conjecture." British Journal of Mathematics & Computer Science 22, no. 2 (2017): 1–8. http://dx.doi.org/10.9734/bjmcs/2017/33315.

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15

VINAY, KUMAR. "PROOF OF BEAL'S CONJECTURE AND FERMAT LAST THEOREM USING CONTRA POSITIVE METHOD." i-manager’s Journal on Mathematics 7, no. 2 (2018): 1. http://dx.doi.org/10.26634/jmat.7.2.14127.

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16

Moussounda Mouanda, Joachim. "On Beal's Conjecture for Matrix Solutions and Multiplicative Commutative Groups of Rare Matrices." Turkish Journal of Analysis and Number Theory 12, no. 1 (2024): 1–7. http://dx.doi.org/10.12691/tjant-12-1-1.

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17

Gandhi, K. Raja Rama, Reuven Tint, and Michael Tint. "The Methods of Solving Equations Ax+By=Cz with Co-Prime A, B, C, where x≥2,y≥2,x≥2 Are Natural Numbers, Equal the Two only in one of the Three Possible Cases - The Proof of Catalan's Conjecture." Bulletin of Society for Mathematical Services and Standards 8 (December 2013): 17–25. http://dx.doi.org/10.18052/www.scipress.com/bsmass.8.17.

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One of the principal problems of the Beal's conjecture, as we see that, is methods for finding a pairwise coprime solution which is defined below. First found methods and identities, allowing receiving infinite number solutions of equations as Ax+By=Cz for co-prime integers arranged in a pair (A,B,C)=1 are natural (whole) numbers, where a fixed permutation (x,y,z)corresponds to each of the permutations (2,3,4), (2,4,3), (4,3,2) Here we obtain also our method and identities of all not recurrent and not co-prime solutions of the above type, part of which has already been published, in contrast t
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18

T M, Nishad, and Dr Mohamed M Azzedine. "Proofs of Beal’s Conjecture, Fermat’s Conjecture, Collatz Conjecture and Goldbach Conjecture." Indian Journal of Advanced Mathematics 3, no. 1 (2023): 1–7. http://dx.doi.org/10.54105/ijam.a1137.043123.

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19

Saravanan, S. B. E. "Beal’s Conjecture - Counter Examples." Bulletin of Mathematical Sciences and Applications 12 (May 2015): 39–40. http://dx.doi.org/10.18052/www.scipress.com/bmsa.12.39.

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20

Thiagarajan, Raj C. "A Proof to Beal’s Conjecture." Bulletin of Mathematical Sciences and Applications 8 (May 2014): 66–69. http://dx.doi.org/10.18052/www.scipress.com/bmsa.8.66.

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In this paper, we provide computational results and a proof for Beal’s conjecture. We demonstrate that the common prime factor is intrinsic to this conjecture using the laws of powers. We show that the greatest common divisor is greater than 1 for the Beal’s conjecture.
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21

Gandhi, K. Raja Rama, and E. A. S. Sarma. "The Beal’s Conjecture (Disproved)." Bulletin of Society for Mathematical Services and Standards 6 (June 2013): 35–37. http://dx.doi.org/10.18052/www.scipress.com/bsmass.6.35.

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22

Bodie, John. "Proof of Beal’s Conjecture." Journal of Mathematical Problems, Equations and Statistics 5, no. 1 (2024): 161–62. http://dx.doi.org/10.22271/math.2024.v5.i1b.137.

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23

Samuel, Bonaya Buya. "Proof of Beals conjecture." African Journal of Mathematics and Computer Science Research 11, no. 8 (2018): 109–13. http://dx.doi.org/10.5897/ajmcsr2018.0764.

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24

Ghanouchi, Jamel. "A Proof of Beal’s Conjecture." Bulletin of Mathematical Sciences and Applications 5 (August 2013): 30–34. http://dx.doi.org/10.18052/www.scipress.com/bmsa.5.30.

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More than one century after its formulation by the Belgian mathematician Eugene Catalan, Preda Mihailescu has solved the open problem. But, is it all ? Mihailescu's solution utilizes computation on machines, we propose here not really a proof of Catalan theorem as it is entended classically, but a resolution of an equation like the resolution of the polynomial equations of third and fourth degrees. This solution is totally algebraic and does not utilize, of course, computers or any kind of calculation. We generalize our approach to Beal equation and discuss the solutions. (Keywords: Diophantin
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25

Marshall, Stephen M. "An All-Inclusive Proof of Beal’s Conjecture." Bulletin of Society for Mathematical Services and Standards 7 (September 2013): 17–22. http://dx.doi.org/10.18052/www.scipress.com/bsmass.7.17.

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This paper presents a complete and exhaustive proof of the Beal Conjecture. The approach to this proof uses the Fundamental Theorem of Arithmetic as the basis for the proof of the Beal Conjecture. The Fundamental Theorem of Arithmetic states that every number greater than 1 is either prime itself or is unique product of prime numbers. The prime factorization of every number greater than 1 is used throughout every section of the proof of the Beal Conjecture. Without the Fundamental Theorem of Arithmetic, this approach to proving the Beal Conjecture would not be possible.
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26

Guła, Leszek W. "The Proof of the Fermat-Beal Theorem." Bulletin of Mathematical Sciences and Applications 11 (February 2015): 4–5. http://dx.doi.org/10.18052/www.scipress.com/bmsa.11.4.

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27

Mohamed, Halima Jibril, Adela Zyfi, and Ghedlawit Futzum. "Breaking Boundaries: Discovering the Impossible Counterproof of Beal’s Conjecture." JOURNAL OF ADVANCES IN MATHEMATICS 17 (July 23, 2019): 12–18. http://dx.doi.org/10.24297/jam.v17i0.8279.

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This paper will attempt to logically differentiate between two types of fractions and discuss the idea of Zero as a neutral integer. This logic can then be followed to create a counterexample and a proof for Beal’s conjecture.
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28

Nyambuya, Golden Gadzirayi. "A Simple and General Proof of Beal’s Conjecture (I)." Advances in Pure Mathematics 04, no. 09 (2014): 518–21. http://dx.doi.org/10.4236/apm.2014.49059.

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29

Diez, Enfer. "Beal’s Conjecture on the Polynomials with Root of Powers." Universal Journal of Applied Mathematics 2, no. 4 (2014): 195–97. http://dx.doi.org/10.13189/ujam.2014.020405.

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30

Joseph, James E. "algebraic proofs of Fermats last theorem and Beals conjecture." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 9 (2016): 6586–88. http://dx.doi.org/10.24297/jam.v12i9.132.

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In this paper, the following statememt of Fermats Last Theorem is proved. If x, y, z are positive integers is an odd prime and z = x y , x, y, z     are all even. Also, in this paper, is proved (Beals conjecture) The equation   z = x  y has no solution in relatively prime positive integers x, y, z, with  ,, primes at least .
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31

Melo, Severino T., and Marcela I. Merklen. "On a conjectured noncommutative Beals-Cordes-type characterization." Proceedings of the American Mathematical Society 130, no. 7 (2001): 1997–2000. http://dx.doi.org/10.1090/s0002-9939-01-06270-0.

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32

de Alwis, A. C. Wimal Lalith. "Solutions to Beal’s Conjecture, Fermat’s Last Theorem and Riemann Hypothesis." Advances in Pure Mathematics 06, no. 10 (2016): 638–46. http://dx.doi.org/10.4236/apm.2016.610053.

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33

BA BS AA MLIS, James T. Struck. "Beal conjecture disproved as no common prime factors in 3 Counterexample." Global Research Review in Business and Economics 10, no. 4 (2024): 01. http://dx.doi.org/10.56805/grrbe.24.10.4.41.

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34

Gandhi, K. Raja Rama, and Reuven Tint. "The Proof of the Insolubility in Natural Numbers for n>2, the Fermat's Last Theorem and Beal's Conjecture for Co-Prime Integers Arranged in a Pair A, B, D in the Equations An+Bn=Dn and An+By=Dz (Elementary Aspect)." Bulletin of Mathematical Sciences and Applications 5 (August 2013): 44–47. http://dx.doi.org/10.18052/www.scipress.com/bmsa.5.44.

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We give the corresponding identities for different solutions of the equations: aAx+bBx=cDx [1] and aAx+bBy=cDz [2]: As for coprime integers a, b, c, A, B, D and arbitrary positive integers x, y, z further, for not coprime integers, if A0x0+B0x0=D0xo [3] and A0x0+B0yo=D0z0 [4], where x0, y0, z0, A0, B0, D0 - are any solutions in positive integers.
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35

Miller, Kenneth G. "Microlocal regularity on step two nilpotent Lie groups." Proceedings of the Edinburgh Mathematical Society 31, no. 1 (1988): 25–39. http://dx.doi.org/10.1017/s0013091500006544.

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A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with
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36

Mohammed, Ghanim. "A SHORT ELEMENTARY PROOF OF THE BEAL CONJECTURE WITH DEDUCTION OF THE FERMAT LAST THEOREM." GLOBAL JOURNAL OF ADVANCED ENGINEERING TECHNOLOGIES AND SCIENCES 8, no. 1 (2021): 1–16. https://doi.org/10.5281/zenodo.4568087.

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The present short paper, which is an amelioration of my previous article “confirmation of the Beal-Brun-Tijdeman-Zagier conjecture” published by the GJETS in 20/11/2019 [15], confirms the Beal’s conjecture, remained open since 1914 and saying that:       The proof uses elementary tools of mathematics, such as the L’Hôpital rule, the Bolzano-Weierstrass theorem, the intermediate value theorem and the growth properties of certain elementary functions. The proof uses also the Catalan-Mihailescu theorem [18] [19] and some methods developed in my paper o
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37

Wilson, Jonathan. "Machine Learning Guided Proof of Beal's Conjecture." SSRN Electronic Journal, 2024. http://dx.doi.org/10.2139/ssrn.4770977.

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38

Gholinejad, Jalal. "A Proof of Beal's Conjecture Using Wave Model." SSRN Electronic Journal, 2020. http://dx.doi.org/10.2139/ssrn.3693267.

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39

Yuri, A. Ivliev. "Solution of Beal's Conjecture in the Paradigm of Quantum Mathematics." March 15, 2018. https://doi.org/10.5281/zenodo.4626546.

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The article gives a tight review of the author’s work dedicated to a new trend in modern mathematics called quantum computing mathematics. Quantum computing methods are interpreted extendedly as relating to quantum mathematical information (quantum mathematics). Mathematical information quanta are whole numbers (integers) characterizing principal integrity of relevant mathematical objects. This approach was applied to solving Beal’s Conjecture (Generalized Fermat’s Last Theorem) that allowed not only to prove the famous problem but unmask excessive formalism of traditional qu
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40

MR, Chandramohanan. "A Simple Elementary Proof of Fermat's Last Theorem." Journal of Mathematical & Computer Applications, August 30, 2024, 1–7. http://dx.doi.org/10.47363/jmca/2024(3)187.

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In this article we attempt to describe a short proof of Fermat's Last Theorem. The Fermat's diophantine equation xn = yn + zn gives rise to an equation of (n-1)th degree, which can be proved to have no positive rational roots. This proves the case for any odd prime n. For n = 4 we use the principle of reductioad-absurdum along with a polynomial equation of degree 3. Further Beal's conjecture is examined and proved to be true.
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41

Gholinejad, Jalal. "A Proof of Beal’s Conjecture Using Pythagoras Perspective." SSRN Electronic Journal, 2023. http://dx.doi.org/10.2139/ssrn.4644101.

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42

E. Joseph, James, and Bhamini M.P. Nayar. "FERMAT’S LAST THEOREM IS EQUIVALENT TOBEAL’S CONJECTURE." International Journal of Mathematics and Statistics 4, no. 2 (2023). http://dx.doi.org/10.53555/eijms.v4i2.22.

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It is proved in this paper that (1) Fermat’s Last Theorem: If π is an odd prime, there are no relatively prime positive integers x,y,z satisfying the equation zπ = xπ+yπ, and (2) Beal’s Conjecture :The equation zξ = xµ +yν has no solution in relatively prime positive integers x,y,z with µ,ξ and ν odd primes at least 3. It is also proved that these two statements, (1) and (2), are equivalent.
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43

MR, Chandramohanan. "About Fermat’s Last Theorem." Journal of Mathematical & Computer Applications, December 31, 2024, 1–5. http://dx.doi.org/10.47363/jmca/2024(3)195.

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In this paper, the possibility of finding a simple proof of Fermat’s Last Theorem is discussed by using the principles of elementary algebra instead of using the Modularity theorem. For an odd prime n the Fermat’s diophantine equation xn = yn + zn gives rise to an equation of (n − 1)th degree which can be proved to be irreducible over the field Q of rational real numbers by using Eisenstein’s criterion. This proves the theorem for any odd prime n. For n = 4 we use a method of reductio-ad-absurdum to prove the theorem. Finally, we deduce Beal’s conjecture.
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44

Harimohan, Singh Aithani. "THE MATHEMATICAL SOLUTION FOR THE BEAL CONJECTURE." September 30, 2018. https://doi.org/10.5281/zenodo.6942995.

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The Beal conjecture is an unsolved number problem. It was formulated in 1993 by Andrew Beal ,A Banker and an amature mathematician during investigating generalization of Fermats Last Theorem [1] [2] since 1997. He offered a monentary prize for an impressive proof of this conjecture . At present no suitable proof has been produced . In this article I also provided the clear and a systematic proof of this conjecture .  
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45

Sousa, Francisco Rafael Macena de. "Demonstração da Conjectura De Beal. Revista Científica Multidisciplinar Núcleo do Conhecimento." Revista Científica Multidisciplinar Núcleo do Conhecimento 05, no. 11 (2019). http://dx.doi.org/10.32749/nucleodoconhecimento.com.br/matematica/conjectura-de-beal.

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