Earth/matriX
Science in Ancient Artwork

Fermat's Last Theorem

By
Charles William Johnson


36

:

64

:

100

48

:

52

:

100


Extract Nș2
Science in Ancient Artwork Series


Fermat's Last Theorem

Pierre the Fermat's Last Theorem is a conjecture about the impossibility of the equation

(1)

when the terms (x ,y ,z ,n) are whole numbers with n greater than 2. In order to understand how the numbers behave, the equation must be considered where the terms may be fractions and/or whole numbers and n may be any number. Aside from x + y = z, the initial equation begins then with

x2 + y2 = z2 (2)

which represents the Pythagorean Theorem, reflecting the relation of equivalency among the sides of right triangles. Equivalency may be expressed in fractions/whole numbers combined (right triangles), or in whole numbers alone (perfect right triangles).

Whenever fractions are involved, the relation of equivalency always represents an approximation. Nonetheless, mathematicians and geometrists accept that numerical expression as being equivalent. Perfect right triangles represent exact equivalencies expressed in whole numbers. The 3-4-5 perfect right triangle, and its series of multiples (6-8-10...), alongwith its series of variations (5-12-13...) are determinant for understanding Fermat's Last Theorem. Irrespective of the power involved, the addition of the terms x and y must share complementary percentiles of the lue z (100%).

32 + 42 = 52 62 + 82 = 102
9 + 16 = 25 36 + 64 = 100
25 = 25 100 = 100

        furthermore,


All multiples of the 3-4-5 perfect right triangle reflect this 36:64:100 percentile relationship.

An alternative expression of the Pythagorean Theorem for the variation of the 3-4-5 perfect right triangle involves

x2 = y + z (3)
52 = 12 + 13
25 = 25

        furthermore,


Applying the original equation (2) of the Pythagorean Theorem to this particular variation of the perfect right triangle would produce a distinct percentile relationship (.14792899408:.85207100591) from the 48:52:100 relation.

The 3-4-5 perfect right triangle series (including multiples and variations) is the only expression of the equation where all of the terms within the equation itself are whole numbers: the roots (3-4-5); the products of those roots (9-12-25) and; the percentile numbers (36:64:100). This same combination of numbers in whole number terms would have to occur at powers greater than 2 in order to obtain relation of equivalency that would involve whole numbers only. The appearance of a fractional number in any one of the numbers would impede such an appearance.

At higher powers, this particular relation of equivalency produced by the 3-4-5 perfect right triangle disappears: for example,

63 + 83 103 6 + 8 10
216 + 512 1000 1296 + 4096 1000
728 1000 5392 1000

The relation of equivalency in 62 + 82 = 102 breaks down due to geometric progression. However, one may than consider applying the Pythagorean Theorem in another manner; consider the following arrangement:

63 + 83 = z 6 + 8 = z
216 + 512 = z 1296 + 4096 = z
728 = z 5392 = z

        further,

        then,

6 3 + 8 3 = 8.9958828913 6 + 8 = 8.56914459

Within the Pythagorean Theorem, these higher power relations represent acceptable examples of equivalency even though they are expressed in fractions.

By adding zeros to the above numbers, the decimal place is removed and whole numbers appear on the calculator:

60000000003 + 80000000003 = 89958828903
6000000000 + 8000000000 = 8569144590

however, that is not the case since, for example, the calculator rounded off to 8569144590 which is actually 85691445899... as a root. Nonetheless, for computational needs, this relations of equivalency is perfectly acceptable, just as any fractional expression of the Pythagorean Theorem is considered to be equivalent. Fractional expressions constitute acceptable relations of equivalency for any power.

In general, then, the equation x + y = z represent relations of equivalency in fractions alone, fractions with whole numbers, or whole numbers alone. However, the relations of equivalency expressed in whole numbers alone occur only at the power of 2 for measurements of multiples and variations of the 3-4-5 perfect right triangle. All other combinations of fractions or fractions and whole numbers together occur at any power.

The particular conditions imposed by Fermat, that this equation represent only whole numbers for powers greater than 2 deny the possibility of achieving a relation of equivalency given the procedure of the addition of two of the terms. The hidden numbers of the percentiles must also be whole numbers in relationship to the whole numbers of the terms (x, y, x, n) and their products. Consider the percentile numbers of the previous example which appeared to represent whole numbers on the calculator:

216/728 = .2967032967 ... 1296/5392 = .24035608308 ...
512/728 = .70329670329 ... 4096/5392 = .75964391691 ...
+
+
.99999999999 ... .99999999999 ...

No two whole numbers, above the square, after undergoing the multiplication process known as powers of itself, will produce two numbers whose products will add up to 100% the value of a third whole number's product to that same power. Such a relation of equivalency only occurs for the series of perfect right triangles represented by the measurements 3-4-5 and their multiples (6-8-10 ...) and variations (5-12-13 ...). One may even visualize how the 3-4-5 numbers of the perfect right triangle, and their multiples and variations, represent a cut-off point on a table of roots and powers.

WHOLE NUMBERS: ROOTS AND POWERS
x x x x x x
root/product root/product root/product root/product root/product  
1 1 1 1 1 1 1 1 1 1  
2 4 2 8 2 16 2 32 2 64  
3 9 3 27 3 81 3 243 3 729  
4 16 4 64 4 256 4 1024 4 4096  
5 25 5 125 5 625 5 3125 5 15625  
6 36 6 216 6 1296 6 7776 6 46656  
7 49 7 343 7 2401 7 16807 7 117649  
8 64 8 512 8 4096 8 32768 8 262144  
9 81 9 729 9 6561 9 59049 9 531441  
10 100 10 1000 10 10000 10 100000 10 1000000  
11 121 11 1331 11 14641 11 161051 11 1771561  
12 144 12 1728 12 20736 12 248832 12 2985984  
13 169 13 2197 13 28561 13 371293 13 4826809  
14 196 14 2744 14 38416 14 537824 14 7529536  
15 225 15 3375 15 50625 15 759375 15 11390625  
16 256 16 4096 16 65536 16 1048576 16 16777216  
17 289 17 4913 17 83521 17 1419857 17 24137569  
18 324 18 5832 18 104976 18 1889568 18 34012224  
19 361 19 6959 19 130321 19 2476099 19 47045881  
20 400 20 8000 20 160000 20 3200000 20 64000000  
... ... ... ... ...  

Due to the geometric progression that sets in, no other combination of roots and powers greater than 2, reproduce the distinguishable 100% relation of the addition of terms and equivalency together, for all of the terms involved: roots:products:percentiles in whole numbers. And, because numbers behave in this manner for perfect right triangles, different ancient cultures may have chosen the 3-4-5 right triangle for their artwork. The 3-4-5 perfect right triangle, alongwith its multiples and variations, not only distinguishes itself in equations to the power of two, but allows for understanding its absence in all powers greater than 2. This may also explain one of the reasons why ancient mathematicians and geometrists preferred whole numbers.

*****

For more detailed analyses of the ideas treated in this extract:

Fermat's Last Theorem: A Brief Summation
Fermat’s Last Theorem: Powers and Last-Digit Patterns

Summation of Fermat's Last Theorem


 

©1998-2016 Copyrighted by Charles William Johnson. All rights reserved.

Your comments and suggestions are greatly appreciated:
e-mail: johnson@earthmatrix.com


Earth/matriX
Science in Ancient Artwork
Fermat's Last Theorem
Extract No.2


©1998-2016 Copyrighted by Charles William Johnson. All Rights Reserved

Earth/matriX,
P.O. Box 231126 New Orleans, LA 70183-1126, USA;
Reproduction prohibited without written consent of the author.


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