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

DOI : https://doi.org/10.26634/jmat.7.2.14127

“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.

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