Beal Fermat and Pythagora's Triplets

James Constant

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Proof that Beal And Fermat Triplets Cannot Exist¹

Consider Beal's proposed equality

            z^m=x^a+y^b         x,y,z,m,a,b positive integers         m,a,b>2                                 (1)
whose negation is Fermat's Last Theorem when m=a=b>2, and when m=a=b=2 is Pythagora's Theorem. Write equation (1) as
            (z^m/2)²=(x^a/2)²+(y^b/2)²                                                                                   (2)
Euclid's formulas for triplets can be written as
            z=(r²+s²)^2/m             x=(2rs)^2/a             y=(r²-s²)^2/b                                        (3)
whichwhen r,s are integers are impossible triplet integers z,x,y when m,a,b>2. Accordingly, since 2^2/a is an irrational number for all a>2,  Beal and Fermat triplets do not exist. However, when m=a=b=2, Pythagora's Theorem admits triplets z,x,y integers.



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¹ Some say that "solutions" to Beal's Conjecture are not very scarce. See at http://planetmath.org/encyclopedia/BealsConjecture.html. This claim is refuted in http://www.coolissues.com/mathematics/BealProblem/bealp.htm

        Copyright © 2008 James Constant

        By same author:http://www.coolissues.com/mathematics/sameauthor.htm