The conjecture Ax + By = Cz made by Mr. Andrew Beal is concerned with the common
prime factor for positive integers and their exponents.
"If ,ax + by = cz ,
where A,B,C, x, y and z, are positive integers x, y and z are all greater
than 2, then A, B and C must have a common prime factor." [Mauldin, 1997] This represents the original wording of the Beal Conjecture.
The Beal Conjecture requires positive integers
in the terms [A, B, C] and exponents [x, y, z] of the equation (the latter whose
value must be greater than 2). The products of the terms must reflect the selfsame
multiplication of the terms in whole numbers or positive integers. Obviously,
no fractional expressions are to appear in any of the three terms or three exponents
of the equation. And, the most significant part of the conjecture affirms the
necessity that the terms share a common prime divisor. Or, to the contrary,
In our view, the concept of a counterexample
stipulated by the Beal Conjecture is simply impossible by definition. An equation
with co-prime terms cannot have positive integer terms and exponents, as stipulated,
given the very definition of primes and co-primes, and their multiples. If
a co-prime pair of terms (divisible only by the greater common denominator of
1) were to exist, then that would represent a counterexample in our view. Now, the fact that the conjecture may wish to see a counterexample with
all exponents as whole numbers (as well as the terms) is simply expecting something
that cannot derive from co-primes and their relationship in the cited equation.
We shall explain the reason for this as of the concept of selfsame addition of terms.
The argument made in this essay is directed
at the system of notation that we have inherited throughout history, based upon,
in this case, the limiting method of selfsame multiplication of terms. In that
sense, these observations go beyond the Beal Conjecture. The insight posed by
Mr. Andrew Beal in his conjecture serves as encouragement for looking at old
problems in a new light. We are simply attempting to peel back the first layer
of algebraic notation in order to emphasize the relations occurring behind the
symbolic language. And, in our mind, that is precisely what Mr. Beal has afforded
in launching such a critical conjecture.