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Proof of Beal's Conjecture and Fermat Last Theorem using Contra Positive Method

Vinay Kumar*

Professor, Vivekananda School of IT, VIPS, GGSIPU, New Delhi, India.

Periodicity:April - June'2018
DOI : https://doi.org/10.26634/jmat.7.2.14127

Abstract

“If A^x + B^y = C^z , for integers A, B, C ≥2 and integers x, y, z greater than 2 , then A, B, C must have a common prime factor”. The statement is known as Beal's conjecture (Rubin & Silverberg, 1994). Without loss of generality, integers B and C can be expressed in terms of A. Assuming B = A + m and C = A + n, the present study proves the conjecture for all the four cases: i) m = 0, n = 0; ii) m = 0, n≠ 0; iii) m≠0, n = 0; and iv) m≠0, n ≠0. A, B, and C can be ordered (sequenced) in six different ways. A theorem that is proved for one sequence, the same theorem can easily be proved for other five sequences. Contrapositive approach together with integer division algorithm is used to prove the conjecture. Contrapositive statement of Beal's x conjecture is “if A, B, and C have no common prime factor then no integers A, B, C and integers x, y, z > 2 such that A^x + B^y = C^z”. Some basic and fundamental properties of quadratic equation are also used in the proof.

Keywords

Beal’s conjecture, Fermat’s last theorem, Fermat’s general proof, division algorithm, exponent, prime factor, contra positive proof.

How to Cite this Article?

Kumar. V. (2018). Proof of Beal's Conjecture and Fermat Last Theorem using Contra Positive Method. i-manager’s Journal on Mathematics, 7(2), 1-7. https://doi.org/10.26634/jmat.7.2.14127

References

[1]. Aczel, A. D. (1996). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. N. Y.:Dell Publishing.

[2]. Boston, N. (1995). A Taylor-made plug for Wiles' proof. The College Mathematics Journal, 26(2), 100-105.

[3]. Cornell, G., Silverman, J., and Stevens, G., (eds.) (1997). Modular Forms and Fermat's Last Theorem. Springer- Verlag.

[4]. Granville, A., & Monagan, M. B. (1988). The first case of Fermat's last theorem is true for all prime exponents up to 714,591,416,091,389. Transactions of the American Mathematical Society, 306(1), 329-359.

[5]. Ireland, K., & Rosen, M. (1990). A Classical Introduction to Modern Number Theory (Vol. 84). Springer Science & Business Media.

nd [6]. Jacobson, N. (1989). Basic Algebra II, 2 Ed. New York: W. H. Freeman.

[7]. Kumar, V. (2010). Restricted backtracked algorithm for Hamiltonian circuit in undirected graph. BIJIT-BVICAM's International Journal of Information Technology, 2(2), 23-32.

[8]. Kumar, V. (2012). Mathematics Is Science: A topic revisited in context of FCS of India. International Journal of Modern Education and Computer Science, 4(6), 17-26.

[9]. Kumar, V. (2014). Proof of Fermat last theorem based on odd even classification of integers. Int. J. Open Problems Compt. Math., 7(4), 23-34.

[10]. Kumar, V., (2015). Discrete Mathematics. New Delhi: BPB Publication.

[11]. Mauldin, R. D. (1997). A generalization of Fermat's Last Theorem: The Beal conjecture and prize problem. Notices of the American Mathematical Society, 44(11), 1436-1437.

[12]. Radford, L., & Peirce, C. S. (2006, November). Algebraic thinking and the generalization of patterns: A semiotic th perspective. In Proceedings of the 28 Conference of the International Group for the Psychology of Mathematics

Education, North American Chapter (Vol. 1, pp. 2-21).

[13]. Ribenboim, P. (1999). Fermat's Last Theorem for Amateurs. New York: Springer-Verlag.

[14]. Rubin, K., & Silverberg, A. (1994). A report on Wiles' Cambridge lectures. Bulletin of the American Mathematical Society, 31(1), 15-38.

[15]. Silverman, J. H. (1986). Heights and elliptic curves. In Arithmetic Geometry (pp. 253-265). Springer, New York, NY.

rd [16]. Stewart, I., & Tall, D. (2002). Algebraic Number Theory and Fermat's Last Theorem, 3 Ed. AK Peters/CRC Press.

[17]. Washington, L. C. (1997). Introduction to Cyclotomic Fields (Vol. 83). Springer Science & Business Media.

[18]. Wiles, A. (1995). Modular elliptic curves and Fermat's last theorem. Annals of Mathematics, 141(3), 443-551.

[19]. Zelator, K. (2011). Integer Roots of Quadratic and Cubic Polynomials with Integer Coefficients. arXiv preprint arXiv:1110.6110.

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