Summation of Fermat's Last Theorem
(xn + yn = zn)

Charles William Johnson

Pierre de Fermat stated that it is impossible to resolve the equation xn + yn = zn whose terms (x,y,z) are whole numbers with the same exponent (n) of three or greater. Most attempts at explaining this conjecture are effected at the level of the possible terms and exponents. A recent attempt at a proof of the equation deals with exponents to [ 4000000]. The theorem can only be explained, however, at the level of the products of those terms and their exponents. The products of the terms xn and yn are ultimately added together in order to produce an equivalency with the product of the term zn. In consequence, the product of the term z shall always represent a value that is greater than those of x or y, since the product z is the sum of the products of x and y.

The exponents n3, n4, n5, and n6 establish four distinct serial patterns of last-digit terminations for all powers of all the natural numbers. These shall determine therefore the terms x, y, and z. Any natural number raised to any of the series of powers shall produce patterns of last-digit terminations, which are thus repeated throughout the series. The last-digit termination patterns for the natural numbers (1 - ¥) for the four series are:

Series I: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 (For powers: 1,5,9,13,17,21...)
Series II: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 (For powers: 2,6,10,14,18,22...)
Series III: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0 (For powers: 3,7,11,15,19,23...)
Series IV: 1, 6, 1, 6, 5, 6, 1, 6, 1, 0 (For powers: 4,8,12,16,20,24...)

According to the cited equation, no matter which terms or powers are employed, the last digit of the product of term z must correspond to the rules of simple addition, thereby reflecting the sum of the two last digits of terms x and y. For example, if the products of terms x and y terminate in a one (1) and a five (5) respectively, then the product of the term z must end in a six (6). For any two given terms (x,y) on the same series, the next product of equivalency of a whole-number term (z) on the same series (whose last-digit termination corresponds to their sum), shall always correspond to the product (z) of a term in a fractional expression. The next closest product z on the series, whose last-digit termination corresponds to the sums of the last-digit terminations of terms x and y, and, which also corresponds to a whole-number term z, shall always represent a value for z that is much greater and non-equivalent to the sum of the products of the x and y terms. Given the sequential pattern established by the last-digit terminations for all natural numbers for all powers, it is impossible to resolve the equation in whole numbers. (Further, even the resolution of the equation in fractional expressions of equivalency represents an approximation of the equivalency of values.)

The proof of Fermat's Last Theorem was achieved centuries ago when the powers n3, n4, n5, and n6 were computed. It is unnecessary to effect analyses of all of the possible exponents above n6, since the last-digit termination patterns established by these four powers are repeated for all subsequent powers to infinity. The rules of simple addition that apply to the products of these four powers also apply respectively to all other powers. In other words, whatever the last digits may be of the products of the terms x and y, these last-digit terminations must add up to a correspondingly correct last-digit for the product z. The next last-digit termination on the same series (and Fermat's Last Theorem establishes that the equation must work within the same series) shall always correspond to a product whose term (z) shall be a fractional number. The next whole-number term z, with a corresponding last-digit termination on the same series for any power ---that obeys the sum of the products of terms x and y--- shall always be of a value greater and non-equivalent to the sum of the products of x and y.

Although Fermat's Last Theorem is represented in alpha-symbolic terms, its point of reference is the behavior of numbers. Our summation reflects a direct explanation of the behavior of the terms, exponents and products in numerical terms, directly examining the numbers themselves and the patterns established thereof. There is little or no need for an indirect alpha-symbolic explanation of the equation, which often requires explanation itself.

Charles William Johnson email:

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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
Fermat's Last Theorem

Copyrighted © 1995-2016 by Charles William Johnson. All rights reserved. Reproduction prohibited. Printed in the United States of America. Published simultaneously in Mexico. This publication, or parts thereof, may not be reproduced in any form of photographic, electrostatic, mechanical, or any other method, for any use or purpose, including information storage or retrieval, without written permission from the author, except for the inclusion of brief quotations in a review.

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