The Beal Conjecture
Submission Number Two

By

Charles William Johnson

The Beal Conjecture has been clarified and restated as follows:

"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."

23 + 23 = 24
29 + 83 = 45
33 + 63 = 35
76 + 77 = 983
39 + 543 = 311
335 + 665 = 336
194 + 383 = 574
345 + 514 = 854
274 + 1623 = 97

The proof of the Beal Conjecture is quite actually quite redundant. "The Beal Conjecture does not stipulate that the three terms and exponents of the equation are distinct".

Given the above conditions and examples, the ultimate computational step of the Beal Conjecture concerns the addition of the products of the terms A and B, which must equal the product of the term C. The fact that equation is expressed in alpha symbols does not alter the last step in the computation. The combined products of term A and B must be added together in order to equal the product of term C.

Ultimately, one is simply adding multiples of terms (A, B) together in order to see if they equal another multiple ( C) of those same two terms terms. This aspect of the Beal Conjecture is stated in terms of a "common factor", which means a common divisor/factor of those terms.

Let us look at one of the examples that has been offered by the competition committee.

345 + 514 = 854

The common factor, as it is referred to in the Beal Conjecture, in this example is 17, the minimum(whole number) common divisor of the three terms:

 34 / 17 = 2 51 / 17 = 3 85 / 17 = 5

Basically, the equation is showing multiples of the common divisor being added to produce a third multiple of that same common divisor.

345 + 514 = 854

45435424 + 6765201 = 52200625

Now, notice the common divisor:

45435424 / 17 = 2672672

6765201 / 17 = 397953

52200625 / 17 = 3070625

Subsequently,

2672672 + 397953 = 3070625

The aforegoing means that:

2672672 seventeens plus 397953 seventeens equal 3070625 seventeens

In percentiles:

.8704 + ..1296 = 1.0

In other words, the equation simply states that there are 2672672 common divisors of 17 in the term 345 ; there are 397953 common divisors of 17 in the term 514 ; and, there are 3070625 common divisors of 17 in the term 854.

The Beal Conjecture is redundant, in that the terms A + B, irrespective of the number of multiples of each, merely represents multiples of a common divisor/factor (fractional or whole numbers), which when added together shall equal a third multiple of that same common divisor.

Now, another fractal common divisor could be a fraction, since the nature of the common factor is not specified, which in this example could be that of 2.5 for the three integers.

45435424 / 2.5 = 18174169.6

6765201 / 2.5 = 2706080.4

52200625 / 2.5 = 20880250

Subsequently,

18174169.6 + 2706080.4 = 20880250

18174169.6 two-point-fives plus 2706080.4 two-point-fives equal 20880250 two-point-fives

Since nothing was stated in the Beal Conjecture about the nature of the common divisor, one could suspect this to be a valid perception, as well as any other fractal within this series (2.5, 1.25, .625, .3125, etc.). One need only add zeros and allow the decimal point to float to obtain whole numbers for all of the terms.

3405 + 5104 = 8504

The Beal Conjecture simply reflects the expression of the elementary equations of addition with the terms taken to the power of one (1):

 2 + 4 = 6 2 and 4 share a common factor (2), and, therefore the result 6 shares the same common divisor 3 + 4 = 7 3 and 4 share no common divisor, and, therefore the result 7 has the same characteristic of not sharing a common divisor

The Beal Conjecture takes this simple rule of addition of terms and relations of equivalency to the level of distinct multiples thereof. The same rules apply to the common divisor/factor at the power of one, as at the level of an infinite number of multiples (n to infinity) of the common divisor/factor, and infinite powers thereof of the exponents.

Mathematical formulae exist to produce some of the cited relations of equivalency of the Beal Conjecture. However, the resolution of the Beal Conjecture may be proven without the aid of mathematics as well:

A multiple of a common divisor/factor plus a multiple of a common divisor/factor equals a third multiple of that same common divisor/factor. Multiples of two numbers with no common divisor/factor when added together produce a number, which shares that same trait: the absence of a common divisor/factor. Obviously, if A and B are multiples of the same number (a common divisor/factor), then their combined addition shall produce a third multiple of that same number. If A and B are not multiples of the same number (a common divisor/factor), then their combined addition shall produce a third number which is neither a multiple of that in-existent number. It is similar to ordering a hamburger without cheese, or without anything else for that matter. If hamburgers A and B have no cheese on them, hamburger C shall have no cheese either.

In other words, like produces like (presence of common divisor/factor), and unlike produces unlike (lack of common divisor/factor) in this case. In other words, when certain multiples of terms A and B share a common factor/divisor, then term C shall share that same common divisor. If terms A and B do not share a common divisor, then C shall have that same trait, although it may share a common divisor/factor with either A or B, but not with both, since A and B do not share a common divisor.

One may understand this better with oranges and apples. If you add apples together, you get apples. The resulting numbers are multiples of apples. If you add oranges and apples together, then you get oranges and apples together. The resulting numbers are multiples of either oranges or apples, with the total representing a multiple of combined oranges and apples.

In spite of the alpha-symbolic expression of the terms and exponents in the Beal Conjecture, the equation represents a problem that is solved ultimately with simple addition. The multiples of the terms of the equation expressed in alphabetic letters for the integers of the terms and their exponents, make it appear as though something is being said other than the simple relation of addition. If one translates the alpha-symbolic terms of the equation into their products, the multiples of numbers, then the reasons for which the Beal Conjecture are correct become evident.

In conclusion, once the conditions of the Beal Conjecture have been clarified, the nature of the equation becomes self-evident.