A Submission to the Beal Conjecture Competition

Last-Digit Terminations and the Beal Conjecture:
An Explanation

By

Charles William Johnson

Before considering the Beal Conjecture below, we should make a few remarks about Fermat's Last Theorem or conjecture, since both of these conjectures are related. With regard to Fermat's Last Theorem, the last-digit terminations determine the possibility of solutions for that theorem. The stipulations of the equation cited by Fermat refer to computations on a vertical plane for each particular power (greater than two) of the natural numbers.

x n + y n = z n

The relation of equivalency concerns whole integers for the terms and exponents greater than two (2).

There are no solutions to this equation as recently purported by Andrew Wiles and other mathematicians in their study of elliptic curves and modular forms.

However, there appears to be an easier method for obtaining a clear proof of this equation and the absence of any relations of equivalency as we have illustrated in different essays and extracts in the Earth/matriX series (Cfr., Earth/matriX, Extract 38, 1999). One need only review the exponents of n3, n4, n5, and n6 for the first ten natural numbers in order to determine the fact that there are no solutions for this theorem. The reason for this, is that the first ten numbers in relation to the exponents of n3, n4, n5, and n6 produce a repeat pattern of last-digit terminations for all natural numbers (1 to infinity) and for all exponents (1 to infinity). Therefore, what holds true for the first ten numbers, holds true for all numbers given the existence of the four patterns of the last-digit terminations and the rules of simple addition.

In other essays and extracts, we have offered many examples of this proof. However, we shall list a few examples here, merely to offer an illustration of the patterns and their behavior according to the terms of Fermat's Last Theorem. For a more detailed analysis, one should refer to the various essays and extracts posted in the Earth/matriX series.

Last-Digit Terminations in Fermat's Last Theorem:
A Vertical Scroll of the Natural Numbers and Their Powers

From the previous table, one may observe how it becomes impossible, in adding numbers that end in a one (1) and a five (5), to obtain a third product that ends in a six on the progression of natural numbers and their exponents.

The Beal Conjecture represents a similar proposition, whose limiting conditions makes it impossible to find a solution or relation of equivalency for the terms of the equation. As stated, with regard to the Beal Conjecture,

if Ax +By = Cz , where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common factor,

one need consider the same four patterns of the last-digit terminations as explained in Extract 38. However, now one must seek relations of equivalency on an horizontal plane. In other words, if one selects any number of one power (say n3) and then adds that term to another number (say of n4), then one would expect the third term's last-digit termination to correspond to the rules of addition.

However, the third term's product with the next last-digit termination corresponding thereof would always be much larger or smaller for any other power than that required. Such a relation obtains from the fact of geometric progression of the products of the terms. Furthermore, this also occurs because the Beal Conjecture stipulates that the three terms and exponents of the equation are each distinct from the other. If the terms could be of the same value, consequently, then one obvious example would be 26 + 43 = 27 (or, 64 + 64 = 128). Much like Fermat's Last Theorem, then, the Beal Conjecture also places conditions upon the equation that creates it insolubility.

Let us observe an example of addition of the products of the terms by scrolling horizontally along the rows of numbers.

Last-Digit Terminations in the Beal Conjecture: An Horizontal Scroll
of the Natural Numbers and Their Corresponding Powers

To offer an example, let us take 25 (32) and consider it in relation to other numbers: 32 + 81 ¹ 343, which is the nearest number with a last-digit termination ending in a three (3) of another power that could result from the possible addition of 32 plus 81. The last-digit termination number three (3) does not appear on n4 nor on n6, with no possible solutions thereof. And, if we sought a product with a last-digit termination on n7, the next possible answer would be 77 (or, 823543); again far too large and obviously non-equivalent.

No matter which two numbers are chosen for any combination of distinct powers, the third last-digit termination of a corresponding product/number, from some other power, shall always be exceedingly large or small, or even non-existent. Consider that two of the four patterns of last-digit terminations of the exponents do not even possess certain numbers as last-digit terminations. For example, no numbers terminate in three (3) in n6 for the particular case cited.

In the example of 2401 + 59049, which requires a third term last-digit termination to be a zero (0), one can observe how in either horizontal direction of the scroll, the next possible last-digit termination of other powers is either too large or too small.

 2401 + 59049 ¹ 1000 2401 + 59049 ¹ 1000000

One could show an infinite number of examples such as this one, all with the same relations of non-equivalency. As one attempts to add together two products from any two columns, be these two products far or close together vertically on two different columns, the adjacent answers with corresponding last-digit terminations on a third column shall always be exceedingly small or large and shall produce a relation of non-equivalency. In other words, there are no solutions of equivalency according to the conditions stipulated by the Beal Conjecture.

The alpha-symbolic expression of the equations (x,y,z, A,B,C, n) are simply short-hand expressions for the terms at an initial level of computation. The level that is relevant to the actual computations is the translation from the algebraic expression in alpha-symbolic lettering to the actual numbers of the products of the terms. In other words, one could work entirely with the table of products shown (terms times themselves each exponential times).

In this manner, the Beal Conjecture invokes the same last-digit termination laws of simple addition as invoked by Fermat's Last Theorem. The explanation of these two conjectures, that of Beal and that of Fermat, may therefore be easily illustrated, proven and explained in the manner as we have shown herein.

Given the fact that the four patterns of last-digit terminations established for all numbers and all powers to infinity, then , the small numbers theorem does not apply. For, as in this case, what occurs at the level of the first ten or twenty small numbers occurs for all numbers (1 to infinity) and all powers to infinity (n1 to infinity).

The four patterns of the series of products for each of the exponents/powers are as follows:

Most mathematicians have treated Fermat's Last Theorem at the level of the terms of the equation. We have treated, both Fermat's Last Theorem and the Beal Conjecture, at the level of the products of the terms of the equation in relation to the patterns of last-digit terminations and the simple rules of addition (which ultimately the products of the terms must obey).

Based on the foregoing, we therefore conclude that there are no solutions to the Beal Conjecture according to the stated conditions of said conjecture. The proof herein is offered at the level of the products of the terms and exponents, in relation to the last-digit terminations of those products and the rules of addition.

Proof: The addition of any two products/numbers of any natural number and any exponent greater than two, from any two distinct columns of numbers/products, shall always produce a third product whose last-digit termination shall not correspond to an number/product on any third column/pattern of numbers. The closest last-digit termination that might correspond to the last-digit termination of the first two terms' products shall always represent a larger or smaller value or, be absent from the third column of numbers/products. Given that the products of all of the exponents (1 to infinity) in relation to all natural numbers (1 to infinity) produce four distinct last-digit termination patterns, the absence of any solution for the first ten natural numbers and the exponents n3, n4, n5, and n6 reflect the same absence of solutions for all other natural numbers and exponents up to infinity.

Charles William Johnson e-mail: johnson@earthmatrix.com

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Link: The Beal Conjecture: Submission Number Two

Earth/matriX
Science in Ancient Artwork and Science Today
Last-Digit Terminations and the Beal Conjecture: An Explanation
21 August 1999

Earth/matriX,
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