A SUMMARY OF AN EXTENSIVE ESSAY ON THE THEME OF:

The Beal Conjecture: Ax + By = Cz

A Proof and Counterexamples [Ax + By = zC]

Charles William Johnson

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, present counterexamples.

The Beal Conjecture has been clarified by Professor R. Daniel Mauldin with the following examples of the resolution of the equation: "Here are some examples of solutions to the equation Ax + By = Cz. Note that all values are positive integers, all exponents are greater than or equal to 3, and A, B, and C always share a common factor."

    Common Prime Factor
23 + 23 = 24 => 2
29 + 83 = 45 => 2
33 + 63 = 35 => 3
39 + 543 = 311 => 3
274 + 1623 = 97 => 3
76 + 77 = 983 => 7
335 + 665 = 336 => 11
345 + 514 = 854 => 17
194 + 383 = 573 => 19

The Beal Conjecture stipulates, then:

1- the positive integer terms may be assigned any numerical value in any given order of distinction or repetition; 2- the positive integer exponents of the terms, may be assigned any numerical value greater than 2 in any given order of distinction or repetition and; 3- the terms must have a common prime factor. In the absence of a proof, one must present counterexamples.

The Beal Conjecture is concerned with the appearance of the common prime factor in relation to the positive integer terms and exponents of the equation. The conjecture is not questioning the positive integers and like exponents, as in Fermat's Last Theorem, but rather the surprise caused by the presence of common prime divisors in the terms of the equations. But, nothing is really specified in the conjecture about the counterexamples that might have terms without a common prime factor. The conjecture appears to affirm, that a counterexample would have all the stipulations fulfilled, and yet have no common prime factor of the terms.

The conjecture Ax + By = Cz is concerned with the common prime factor for positive integers and their exponents [greater than 2].

Any and all resolutions of Ax + By shall yield either a fractional term or a fractional exponent or both in the third term.

The following resolutions, for example, are thereby denied by the stipulations of the Beal Conjecture:

First Term

Second Term

Third Term

173
+
184
=
47.898076553
173
+
184
=
18.207006844
173
+
184
=
10.190390915
etc.
173
+
184
=
27.559582533.5
173
+
184
=
12.469803604.6
173
+
184
=
7.6626313765.7
etc.

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 terms and multiples of the terms are all positive integers, although the selfsame multiplication procedure produces a fractional exponent for the third term of the equation. This relationship would be translated into today's symbolic notation as:

173 + 184
=
134.525323672
4913 + 104976
=
109889

The terms and their multiples are all positive integers [ 289 (17s); 5832 (18s); 8453 (13s) ], while the procedural method of selfsame multiplication causes the notation to produce a fractional exponent for the third term [ 134.525323672 ]. In other words, the symbolic procedural numbers, 134.525323672 , reflect 8453 [13s] just as 134.525323672; 47.898076553 , 47.898076553 , 18.207006844 10.190390915 , 6.9208436306, ...n, each equals 8453 thirteens. The problem with the conjecture's notation is the method of selfsame multiplication of the terms and their exponential expression. An additional problem arises from the discourse in that one begins to believe that relations of equivalency whose terms have no common prime factor do not exist or, only "fractional" expressions may exist.

There exist then two distinct methods for deriving multiples and subsequent relations of equivalency. The conjecture made by Mr. Andrew Beal employs the method of selfsame multiples through multiplication [ xn ]. the second method is that of selfsame multiples through addition. The end result of selfsame addition is the obtaining of relations of equivalency whose terms are positive integers, their multiples are positive integers, and they do or do not share a common prime factor. If one wishes to obtain counterexamples within the nature of the conjecture, then, one must abandon the method of algebraic notation based on selfsame multiplication used in the conjecture. One must employ selfsame addition in order to view the numerous examples where no common prime factor is shared by the terms of the equation.

Mr. Andrew Beal, in our view, is correct in his conjecture. If one employs the algebraic notation of the conjecture based on selfsame multiplication, then, the proof of the conjecture is as stated by Mr. Beal, and there are no counterexamples. By using selfsame addition, one may observe the innumerable counterexamples.

A reading of the exponent is quite difficult to imagine: thirteen to the 4.525323672nd power power!

Based upon the above, possibly one may create a distinct notation for expressing the method of selfsame addition in the counterexamples, the terms of which obey relations of equivalency, and have no common prime factor.

289 [17s] plus 5832 [18s] equal 8453 [13s]

One could express this example as xA + yB = zC

28917+ 583218 = 845313

in order to differentiate it from today's algebraic notation, thus reading:

289 seventeens plus 5832 eighteens equal 8453 thirteens.

For at the level of simple addition, and in reality, there is a relation of equivalency among positive integer terms and exponents, with no common prime factor. The Beal Conjecture seeks a positive integer in the number of times the procedural step of selfsame multiplication is effected (the exponential number). In our view, a primary significance is to understanding that the relations of equivalency exist among the positive integers of the prime/prime and prime/composite, and composite/composite terms (for the first two terms) with no prime common factor among the three terms of the equation.

According to these findings, then, Mr. Andrew Beal is correct in his conjecture based upon of selfsame multiplication, because counterexamples exist only in the terms identified by selfsame addition.

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.

Given the fact that the Beal Conjecture is based upon the algebraic notation of selfsame multiplication, the counterexamples exist as of selfsame addition, and follow the notation below:

Ax + By = zC

To follow a previous example, this counterexample would represent

173 + 184 = 8453(13)

A distinct notation for:

173 + 184 = 134.525323672

4913 + 104976 = 109889

289 [17s] plus 5832 [18s] equal 8453 [13s]

In conclusion, the Beal Conjecture is seeking a counterexample with no common prime factor in a specified sense of algebraic notation based upon the method of selfsame multiples through multiplication Ax + By = Cz. Such relations of equivalency with no common prime factor exist, but at the level of selfsame multiples through addition Ax + By = z(C). The whole number (positive integers) of terms and their selfsame multiples through addition (not multiplication), share no common prime factor among the terms. To seek counterexamples at the level of exponents greater than two based upon selfsame multiplication represents a contradiction of terms, and ultimately, a contradiction of the relation of equivalency. In the final analysis, the relation of equivalency in positive terms, with or without a common prime factor, determines the nature of the conjecture.

Terms with a common prime divisor produce whole number terms with whole number multiples (exponents). Terms without a common prime divisor produce whole number terms and whole number multiples but fractional proportions among the multiples. Terms without a common prime divisor with whole number terms and whole number exponents with whole number proportions among the multiples do not exist.

According to our findings, then, Mr. Andrew Beal is correct in his conjecture based upon of selfsame multiplication, because counterexamples exist only in the terms identified by selfsame addition.

This may be better understood when we show surprise at finding whole numbers in some of the computations. History is strange like that. When mathematics was in its infancy, prejudice had it that everything in Nature was thought to be whole numbers. When fractions were first discovered, they were suppressed in the name of whole numbers and order. Today, we marvel at mathematical expressions of whole numbers; we are terribly used to the precision of fractional expressions. In fact, as we have attempted to illustrate in this essay, we have even designed our algebraic notation and its computational methods to reflect the fractions more than the whole numbers.

Therefore, today's view on math might cause us to think that another counterexample such as 6859 [19s] plus 1444 [38s] equal 20577 [9s] is quite primitive and simplistic. Consider how this particular relation of equivalency comes into conflict with the algebraic notation based on selfsame multiplication and finds its symbolic expression as 194 + 383 = 94.5202157889 in today's algebraic language. Possibly, we may have developed a scientific algebraic notation that hides such elementary truths as the counterexamples illustrated in this essay. Think about it: we are ready to throw the relations of equivalency out the window because of the notation of the chosen method of computation. For if we do not want to see solutions such as 194 + 383 = 94.5202157889, then, that means we shall not see such relations of equivalency as those likening to 6859 [19s] plus 1444 [38s] equal 20577 [9s].

When, in fact, the counterexamples follow the reasoning:

Ax + By = z(C)

194 + 383 = 20577 [9s].

6859 [19s] plus 1444 [38s] equal 20577 [9s],

The counterexample resolution reveals terms in positive integers with no common prime factor and, multiples greater than 2 in positive integers.

In our mind, it is more important to teach that relations of equivalency, such as 6859 [19s] plus 1444 [38s] equal 20577 [9s] exist, rather than emphasize concepts in an algebraic notation such as 194 + 383 = 94.5202157889 , which may even suggest the impossibility of such equivalencies. The Beal Conjecture stipulates the terms and their exponents through today's algebraic notation of selfsame multiples through multiplication. Had the equation been written in terms of the method of selfsame multiples through addition, then one would observe the infinite number of counterexamples that exist where the terms have no common divisor.

If the difference between selfsame addition and selfsame multiplication were commonly perceived, then one would not propose the idea of counterexamples through selfsame multiplication, knowing the infinite number of counterexamples in selfsame addition. Furthermore, today's algebraic notation concentrates upon selfsame multiples through multiplication, which is an extremely limited number of possible relations of equivalency, a fraction of the possibilities attainable through selfsame addition. For a more exhaustive treatment of any mathematical subject, we should be teaching our students selfsame addition, which embraces the total universe of possibilities in these analyses.

Jefferson, Louisiana
22 August 2002


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The Beal Conjecture: A Proof and Counterexamples
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