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 Positive Integer having both an infinite unique transcendental Logarithm as well as an infinite unique integer power sequence from (int)^N.

In his era, general power sequences above hundred 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 proof could never have been based on these concepts as they were unknown until hundreds of years later.

This paper explains the reasons why 'Z' can never exist mathematically; each obscuring the others.

The brightness of the "'Nth' power index perspective" is what has made the answer so difficult to find.

Can a number be both MODular and Dimensionally universal?

It was this ambiguity of 'N' being both the dimensional answer yet bound by the strict natures of the MODular and Integer definitions and assumptions.

This paper introduces two new terms "Replica" and "Transunique".

"Replica" represents 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 as this paper illustrates.

Replicas are formally defined for all Positive 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; while its Logs would be modular 'N'.

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 ".

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.

A Replica LOG of proper length can always be computed accurately from most significant downward.

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?". Or can two MODular's intersect?

Assumption 1: All Sums are local and never unique as they can be formed in many ambiguous ways; they are also artificial.

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 ordering process and have no single way to be formed.

All that can be ascertained is the number in question, is the result of just one function,

defined as a "precise repeatable sum or answer".

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.

Assumption 4: If Z exists at 'N', 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.

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.

A: "(X^N) + (Y^N)" must be Zero, at Modulo 'Z'.

A Given from the definition of modulo.

B: LOGs of Replica base 'Z' must exist and be Transunique.

A Given from the existence of integer 'Z'.

C: 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.

D: From assumption 4, If 'Zn' is the sum of more than two Replicas, all the pieces must be Transunique Replicas.

The definition of Replica.

E: (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

A: LOGs of base 'Z' do not exist.

B: The (Zn = X^N + Y^N ) sum is not a valid Replica value.

C: From assumption 5, the LOG of (Zn = X^N + Y^N ) is not Transunique due to 'Z' being a non-existant Replica base value.

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-evaluated 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 "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.

Ultimately, the non-replica nature of a local sum, makes its LOG's Transuniqueness fail due to its finite nature; whereas a Replica had an infinite nature.

The ambiguity of "MOD 'N'", while assuming a MOD 'Z' might exist; along with a 'Z' that is also a "LOG base integer"; combine together to obscure the resolution of the problem.

As the Theorem1.E example illustrates, 'Z' can not exist without compromising THE uniqueness of the Real and Complex Elementary Planes with infinite 'N' space intersections .

All due to Replica LOGs being Transunique and therefore never relatable to any other Replica LOGs of power 'N'.

it was obvious to me that this was his ultimate Puzzle challenge, to any who doubted his many unpublished Proofs.

Given Fermat's era and the informality of postal delays, proper responses to dozens of proof threads was often difficult.

Since this historical quest came about from a margin note claiming a proof,

it is very ironic that the whole puzzle effort evolved from having not enough room in the margin for the critical "Au Contraire".

Its lack would send all of math history on a wild chase; one that Fermat's spirit would enjoy,

knowing how futile all efforts of proof will be, due to under estimating the uniqueness of TWO's many

Dimensional powers and the hidden MODular ambiguities of the problem!

This WEB page address: "http://mister-computer.net/primesums/FLT-proof.htm"

Email of Author: 'RDo.meara@mister-computer.net'

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