ON THE FULL BEAL CONJECTURE

The purpose here is to confirm Beal’s Conjecture, specifically as a consequence of Fermat’s Last Theorem (FLT) and a defining transform, T:, used to generalize the problem, all in context with conditions necessary to uniquely describe the values of all supposed solutions.

Consider the sum of a and b such that a>0, b>0 and p>0 where a and b are solution pairs (prior to further specification).

a + b = p^q.

a and b can be “cross-defined,” (one in terms of the other) by the transform T: a to b^m, b to aⁿ, with m,n and q, all greater than 2, uniquely defining all Integer contributors.

Applying T,

T¹(a + b) = b^m + aⁿ = p^q,

which is the general representation of Beal’s Conjecture when equivalence on the right is negated.

T, once incepted, may be repeated as T^z such that z is an element of {2, 3, . . . }, where continued application neither extends or restricts the validity of its first application, subject to the same limitations.

Reapplying T:,

T²(a + b): = T(b^m + aⁿ) :

b^mn + a^mn = p^q ,

which is a broad rendering of contradiction to FLT, i.e. q = mn is possible.

For simplification relative to FLT, x and y are substituted to yield:

x^m + y^m= p^q,

best described as “the sums of like powers that add to an exponential.” The negation of FLT lies within the range of this generalization. Because q = m is potential, the expression is false (or rather an inequality) consequent to recent proof of FLT (see: WWW). Contradiction of the equivalence negates T²(a + b), a direct implication of T¹(a + b). Reasoning by nature of contrapositive; “ b^mn + c^mn unequal to p^q, a transformation under T:, implies b^m + aⁿ unequal to p^q , under the same transform,” which initially preceeded it. Thusly, the inequality of T is true over all its application. Conclusion: the Beal Conjecture is true.

The object has thus been satisfied except for when p^q is a power of 2, x=y=2^m, in which case q must be reduced to an equivalent 2² + 2² = 2³, which is an excepted case; and further, unless a common divisor is present, in which case factorization alters the general representation of the problem.

QED

Suggested reading: “In Defense of Mr. Fermat,” on WWW at geocities.com/kerryme47714.

By same author: “An ‘Expository’ Proof of the Goldbach Conjecture,” at geocities.com/carryme47714.

email: [email protected] .