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ABSTRACT: A short Proof of Pierre de Fermat's Last Theorem based on 'N' = 2 to infinity , of all possible ( X^N + Y^N = Z^N ? )
This Proof of the FLT is based on every Integer having both an infinite unique transcendental Logarithm as well as an infinite unique positive integer power sequence.
Introduction
This paper is about the Route Pierre de Fermat might have taken to discover that 'Z' can not exist!
In his era, general power sequences above two were unexplored and Napier's logarithm table was newly published.
The classic FLT equation states that " X^N + Y^N never equals any Z^N" as N goes from three to infinity; with X, Y, Z and N all being positive integers.
Several years ago, A. Wiles proved it using a long detailed series of equations based on the Taniyama-Shimura conjecture that every elliptic equation must be modular.
Fermat's alleged proof could never have been based on these concepts as they were unknown until hundreds of years later.
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This paper introduces two new terms "Replica" and "Transunique":
"Replica" to represent the infinite power mathematical sequence of each Integer; Z^N is the Nth Replica of integer 'Z'.
and only a Replica had a LOG that was "Transunique" to signify the unique combination of its infinite Transcendental LOG with the Replica base integer factors.
Replicas are formally defined for all Integers:
As integer 'J' goes from 2 to infinity, each 'J^N' is a Replica value of the sequence as 'N' goes from 2 to infinity.
The Replica sequence of the base integer 'J' is "J^2, J^3, J^4, J^5, ... , J^N" as N goes to infinity.
The power 'N' is an "Index or position" of the Replica sequence that has a value of 'J' to the Nth power.
For example, the Replica 15 sequence first four values are 225, 3375, 50625 and 15^5 = 759375.
Replica values can only match at different positions of 'N';
eg. Value 4096 is the 6th Replica of base integer 4 (4^6), the 4th Replica of base 8 (8^4) or the 12th Replica of base 2 (2^12).
4096 is also the start of its own Replica sequence; 4096^2, 4096^3, 4096^4, etc.
In summary, Replica 5, Replica 3 and Replica 15 are independent of each other;
Replica 5 = "25, 125, 625, 3125 ... " and Replica 3 = "9, 27, 81, 243 .... " do not relate to Replica 15 in any mathematical manner."3^5 + 5^5 = 3368 ; not 15^5 ".
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Each value of a Replica sequence shares the "Prime" like feature of being divisible only by one, by itself or by its base integer or any integer factors of its base.
eg. all values of the Replica 15 sequence are only divisible by 15, 5, 3 , their powers and their products; eg. 15^4, 5^3, 3*3*5^3 or 15*3^5, etc.
Therefore, no Replica value can be partitioned or factored except by its base or its base factors.
All Replica sequence values are Non-Primes by definition; but have Primes as their Base Factors!
Each Replica integer value of the sequence has a Transunique LOG that can be computed exactly, indefinitely.
As the Replica value goes to infinity, its LOG tracks it precisely.
Power sequences like Replicas were the historical basis of the Logarithm concept. LOGs are Transcendental by their nature.
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A Replica LOG of proper length can always be computed accurately.
Compute the LOG of (X^N) so that its length allows a precise exact match to 'N'; ie. "LOG (Xn = X * X * X...* X * X * X) / LOG ( X ) = N" .
Compute the LOG of (Y^N) so that its length allows a precise exact match to 'N'; ie. "LOG (Yn = Y * Y * Y...* Y * Y * Y) / LOG ( Y ) = N" .
The computed variables (Zn = Xn + Yn) are thus accurately formed to a finite precision.
The FLT equation can be restated as:
"Can the unique Replica value at X^N plus the unique Replica value at Y^N ever equal any unique Replica value Z^N, over the infinite range of power 'N',
except at N equals Two?".
Assumptions
Only positive Integers are used in this paper.
Assumption 1: Sums are never unique as they can be formed in many ambiguous ways; they are also artifical.
Let C = (A + B) or C = (E + D - F); Given 'C' , it is impossible to determine which components comprise its sum.
Sums are the result of an arbitrarily external orderings process and have no single way to be formed.
Assumption 2: If the sum of X^N + Y^N is a Prime Number, then it can not be any Replica sequence value.
Since (Prime_V = X^N + Y^N) can have only a Proper factor, while every Replica value was a product,
Prime_V can never be the power of any integer.!
Assumption 3: If ( X^N + Y^N = Zn ) exists, then Mod Z of (X^N + Y^N ) must be zero; [ Zn%Z = 0 ]
Since Z^N must have 'Z' for a factor, Mod Z must be zero.
Assumption 4: If Z exists, then there would be an infinite number of multiples of the form "k * (X^N) + k * (Y^N) = k * (Z^N)".
Furthermore, X^N and Y^N themselves could then be the sum of other Replicas.
eg. If X^N = (x1^N + x2^N) or Y^N = (y1^N + y2^N), thus making Zn equal to the sum of two or more Replicas.
Assumption 5: All Replica Logs are Transunique at power 'N' and can have no factors other than its base integer and its base factors; also, all Replica Logs are modular 'N'.
Case 1. Let Prime1 be a Replica base; the LOG of Prime1 times 'N' is Transunique .
Case 2. Let Replica base F be a non-Prime (F = Prime1 * Prime2 ); the LOG of 'F' times 'N' is Transunique because of the Transunique LOGs of Prime1 and Prime2.
Case 3. Let Replica base G be a non-Prime (G = k * Prime1 * Prime2 ); the LOG of ' G' is Transunique based on the LOGs of its local replicas .
'N' ran from two to infinity, for the Transunique Log range.
Theory
IF - ----- Theorem1. If "X^N + Y^N = Z^N" is TRUE , therefore 'Z' does exist,
THEN all of the following five conditions must be TRUE.
1. "(X^N) + (Y^N)" must be Zero, at Modulo 'Z'.
A Given from the definition of modulo.
2. LOGs of Replica base 'Z' must exist and be Transunique.
A Given from the existence of integer 'Z'.
3. The LOG of each "k * (X^N) + k * (Y^N)" must be Transunique and continue on to infinity as every Replica sequence does.
The definition of Replica.
4. From assumption 4, If 'Zn' is the sum of more than two Replicas, all the pieces must be Transunique Replicas.
The definition of Replica.
5. (X^2 + Y^2 = Z^2) is no longer unique; two dimensional planes intersect 'N' dimensional spaces!
When Z^N exists, its Z^2 also exists; actually there are infinite pairs from (J*Z^2 = k*Z^N).
eg. The Pythagorean triple ( J*X^2 + J*Y^2 = J*Z^2) would first intersect with 'Z^N' when J = (Z^(N-2)).
( (Z^(N-2)) * X^2 + (Z^(N-2)) * Y^2 = (Z^(N-2)) * Z^2 = Z^N )!
An infinite number of intersections follow, via (J = k*Z^(N-2) ).
ELSE - ---- Theorem2. If 'Z' does not Exist,
THEN the following three items are TRUE
1. LOGs of base 'Z' did not exist.
2. The (Zn = X^N + Y^N ) sum is not a valid Replica value.
3. From assumption 5, the LOG of (Zn = X^N + Y^N ) is not Transunique due to 'Z' being a non-existant Replica base value.
Discussion and Conjecture
Of the five conditions needed for Theorem1, one through four are true, while five is False for several reasons.
First; Assumptions 3 and 4 allow infinite intersections between "2 and 'N' dimensional spaces.
Second: The Complex plane would also have (J*Z^2 = k*Z^N) based intersections which could lead to many unknowns.
And lastly : Sine, Cosine, etc as well as Pythagorean rules would also need to be re-evalueated at the very least.
eg. The Sine of the angle between 'X' and 'Z' is (Y / Z ) for the N=2 case; is there even an angle involved in the N not 2 cases?
Theorem 2 correctly asserts that 'Z' does not exist.
And the sum ( X^N + Y^N) is not a Replica value; therefore its LOG is not Transunique.
Conclusions
Fermat's Last Theorem is TRUE because Theorem1.5 is shown False by example, allowing Theorem2 to assert that 'Z' can not exist.
Ultimatley, the non-replica nature of a sum, makes its LOG's transuniqueness fail due to its finite nature; whereas a Replica had an infinite nature.
All due to Replica LOGs being Transunique and therefore never relatable to any other Replica LOGs of power 'N'.
As the Theorem1.5 example illustrates, 'Z' can not exist without compromising THE uniqueness of the Real and Complex Elementary Planes with 'N' space intersections .
LINKS 
author: RD O'Meara Oak Park, IL. 60302
This WEB page address: "http://mister-computer.net/primesums/FLT-proof.htm"
Email of Author: 'RDo.meara@mister-computer.net'
Primes3D: A Construction Proof of Prime Numbers having a cubic Nature.
JID's SLOPE: The Universal Slope of Volume, Mathematically and Physically.
AKA, the Rydberg constant of 1.0973~!
5SPACE: "Stable Particle Masses mapped by (N/2)^5, N=1 to 22"
Monopole Aether Theory: A Classical model of the Aether based on paired monopoles.
My Real interests are in computer science.
I have spent my career as a cybernetic engineer/designer
who is concerned about the future of our digital data.
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especially in the areas of forgery and security, for data and their time stamps.
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Distributed Kernel SECURITY Architecture or 'DSA' ; based on
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22August2011