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Beal Fermat and Pythagora's Triplets

James Constant

math@coolissues.com

Proof that Beal And Fermat Triplets Cannot Exist1

Consider Beal's proposed equality

            zm=xa+yb         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

            (zm/2)2=(xa/2)2+(yb/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 22/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.


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