x + By = Cz , 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 Ax + By = Cz”. Some basic and fundamental properties of quadratic equation are also used in the proof.

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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 :

#### Abstract

“If Ax + By = Cz , 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 Ax + By = Cz”. 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.

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

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