Beal Conjecture Problem Spurious Solutions

James Constant

math@coolissues.com         

        Beal's conjecture (BC) is disproved for the same reasons Fermat's last theorem is proved. See http://www.coolissues.com/mathematics/Beal/beal.htm  However, solutions are claimed to exist for  Beal's Conjecture (BC) stated as

z^m=x^a+y^b . . . . . . . . . . . . . . . . . . . . to be proved . . . . . . . . . . . . . . .(1)

        Assuming that the equality in equation (1) exists, BC does not change if a common term fm^mor factor f exists. In such case  equation (1) is replaced by
        (fmz)^m=(fax)^a+(fby)^b . . . . . . .  . . . . to be proved . . . . . . . . . . . . . . . (2) 

Thus, if one assumes that z,x,y have no common factors f, the only way common terms can be cancelled and equation (2) can be reduced to equation (1) is when terms fm^m=fa^a=fb^b

        I now distinguish from spurious proofs unrelated to BC. Some claim that "solutions" to BC are not very scarce. See at http://planetmath.org/encyclopedia/BealsConjecture.html However, they assume different what is to be proven. Each equation in the cited reference assumes an equality without proof of origin in BC. If common terms are canceled, each equation originates from an equality other than the presumed BC equality. Some examples of equations which do not originate from BC, are 3⁵ = 6³ + 3³ and 9⁷ = 27⁴ + 162³. If common terms are canceled, both equations reduce to

3²=1 + 2³ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . .(3)

which falls outside BC because while no common term fm^mor factor f exists  m=2 and a=1 which fall outside the m,a>2  required by BC. Equation (3) is not equation (1), the presumed BC equality.

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