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Fermat's Last Theorem(FLT) New Simple Proof

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Introduction:

"The French mathematician Pierre De Fermat while reading Bachet's Diophantus recorded the results of his meditations in brief marginal notes in his copy. The margin was not suited for the writing out of proofs. Thus commenting on the eighth problem of the second book of Diophantus' Arithmetic, which asks for the solution in rational numbers (fractions or whole numbers) of the equation x ² + y² = a² , Fermat comments as follows:

' On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or generally, any power above the second into two powers of the same degree: I have discovered a truly marvellous demonstration [ of this general theorem] which this margin is too narrow to contain' (Fermat, oeuvres, III, p.241). This is his famous Last Theorem, which he discovered about the year 1637.' " (The above is quoted from E.T Bell's "Men of Mathematics", Simon and Schuster: New York 1937.) ________________________________________________________

We will establish our proof Of Fermat's Last Theorem (FLT) on the fundamental properties of numbers and facts in mathematics which will be understandable to a High School graduate who knows the binomial theorem. FIRST we will set out the following Lemma which will be used extensively in various stages of the Proof.

Lemma:
"If a composite integer K can be resolved into many pairs of factors such as (x₁, y₁), (x₂ , y₂ ), (x₃, y₃ ) etc, then these pairs of factors in turn when multiplied together must produce K and only K, and cannot produce any other integer."

Proof of the Lemma:
This is axiomatic based on the properties of composite integers. For instance:
18 can be factorised into (1,18), (2,9) and (3,6) which imply when the process is reversed each in turn must produce 18 and no other integer: e.g. 1 x 18 = 2 x 9 = 3 x 6 = 18.

Note: This Lemma is not derived from nor dependent on the truth or falsity of FLT. This is an independent property of all composite numbers in Arithmetic.

Now we will proceed as follows:
Consider the Equation:
aⁿ = bⁿ + c ⁿ where (a, b, c) are integers > 0,
                          and n an integer > 1..... (A)

From Observations on (A):
a > b, so let a = b + x.... ( i);
a > c, so let a= c + y.......(ii);
Also a < (b + c), so let a = (b + c) - z.... (iii);
( Note; If a >= (b + c), then the Left Hand Side (LHS) of (A) > the Right Hand Side (RHS) invalidating (A))
Adding (i) & (ii) 2a = (b + c) + (x + y).....(iv)
Subtracting (iii) from (iv) a = (x + y + z)....(v)
Solving (v) & (i)          b = (y + z).....(vi)
Solving (v) & (ii)         c = (x + z).....(vii)
From (v), (vi) and (vii) therefore, (A) implies :

(x + y + z) ⁿ = ( y + z) ⁿ + ( x + z)ⁿ .. where (x, y, z) are integers.. (B)

Now substituting values of n as n = 2, n=3, n = 4, n = 5 respectively in (B) and simplifying ( no need for alarm!, we want to obtain a pattern only), we get:
z²/xy = 2......................( C)
z³/xy = 3(x + y + 2z)..........(D)
z⁴/ xy = 2 ( x + y)² - 2 x² y² + 3z (x + y + z).........(E)
z⁵/xy=5{(x 3+ y 3) + 2xy (x + y) + 4z (x ² + y² + z ²)+ 6z (xy + yz + zx)}.....(F)

Now we will pause to inspect what we have obtained so far.
In ( C), RHS is an arithmetic integer with a definite numerical value, and we can see our Lemma in action. The first values which satisfy (C) are: z=2, x=2 and y=1. Next we can try out multiples of z and adjust values of x and y to obtain the same value for the LHS which keeps equation (C) a valid equation. Thus we can find out the whole infinite set of Pythagorean Triplets (PT) for (A) via (B) by finding values of (x, y, z) which satisfy (C). Perforce z must be an even number and in addition if (x, y) are chosen as co-primes, then we get all the Primitive Pythagorean Triplets (PPT). (Not a bad discovery in about 2500 years after Pythagoras as how to produce his triplets!)

Examples are:

Ex. 1. z=2, x=2, y=1, giving the first PPT (5,4,3).
Ex. 2. z = 4, gives (x y)=8, so that we get (8,1) or (4, 2) as values of (x, y), yielding the PPT (13,12, 5) and an ordinary PT (10, 8, 6) which is a multiple of 2 of the first PPT (5, 4, 3).
Ex. 3. z =6, gives x y = 18, and possible factors (18, 1), (9,2), and (6, 3) for (x , y), yielding the PPT ( 25, 24, 7) ( note: this is distinct from the first PPT ( 5,4,3) multiplied by 5); and (17, 15, 8); and an ordinary PT (15, 12, 9) which is a multiple of 3 of the first PPT (5,4,3) found before.
Thus in (C) and (B) we have a hitherto unknown method* for generating all PPT and PT for equation (A), when n= 2 and we can carry on ad infinitum. *(This method was published in the Mathematical Gazette of March 1999 issue (pages 131/132) which is the periodical of The Mathematical Association U.K.)

The examples given are not pointless exercises. They show how we can find infinite number of values for the general integers (x, y, z), (and hence for (a, b, c) in (A), if we are given an expression involving a ratio of general integers (x, y, z) on the LHS and an arithmetic integer of fixed numerical value without generality for the RHS as in (C).

SO FAR SO GOOD.

We will now explore the possibility of integer solutions for (x, y, z) in (D) i.e. for the case n = 3.
Here we see the RHS does not have an arithmetic integer of fixed numerical value as in (C), so finding values ( if at all possible) is not going to be as straightforward. However as (x, y, z) are assumed integers, the RHS of (D) can be simplified to a single integer of assumed numerical value, say, R. Thus solution of (D) involves in effect solution of two simultaneous equations:

       z³ /x y = R .........(D₁)
       and, 3 ( x + y + 2z) = R .........(D₂ ).

Equation (D₁) is exactly of the same form as (C), and if the product (xy) is a composite number S with many possible pairs of factors then the Lemma becomes applicable and the equation (D₁) can be seen to remain constant for any given value of z and S, irrespective of how factors of S themselves can be varied.
Equation (D₂) on the other hand contains expressions of the form ( x + y + z).
So that variation of factors of S (i.e the various pairs of factors of the form (x,y) which when multiplied produce S) will produce a different R for each such set of factors of S with the same value of z.

Thus equation (D₁) is incompatible with to equation (D₂) and hence cannot be equal to it as seen in equation (D).

So equation (D) is self contradictory between its LHS and RHS and hence is not possible for a composite value of product (xy) = S.

Note on fallacy: Readers should note that equations (D), (E), (F), etc are all homogeneous equations (in x, y, z) derived from the premise or parent equation (A) which is homogeneous in (a,b,c).In these instances homogeneity means if one set of solution exits then infinite sets of solutions exists to the equations, as the same multiple or divisor of the integers in the equation will form another valid equation.After this important realisation of homogeneity the readers must be aware of the following type of fallacious criticism: Criticism: "Consider example(a): (xy) = (x + y); this has a set of solutions x=2, y=2. Example(b): (xy)z = (x+y+z); this has a set of solutions x=2, y=3, z=1. From these two examples it follows that all similar equations have at least one solution. Hence equation (D) in text must have one solution as well, so the contradiction shown by the author is invalid."
Author's Reply : The underlined sentence in the criticism contains the well known fallacy of "establishing a general proposition by analogy from particular examples. No logical argument or induction is used to establish the general proposition. Use of numerical examples for the underlined sentence is not permissible in logic. However numbers may be used to show that a particular set of solutions exists for equation(D) as has been found for the examples. This is the legitimate use of numerical examples to demolish an argument rather than establish a counter arguing general proposition. The criticism is fallacious and totally invalid in logic and so readers should not try to establish general propositions by particular means to set up counterarguments to muddy the water to no purpose. Also the readers must note that the examples quoted by the critic are all non homogeneous equations and hence only have one solution and the examples are therefore not applicable to equation (D), (E), (F)etc.

It now remains for us to show that a prime value of the product (xy) is not possible in the context of equation (A).

For this we need to go back to equation (B) and show some of its properties assuming (x,y) to be prime to each other to eliminate unnecessary factors.

First transposing elements in (B) we have:

[{(x+z) +y}ⁿ - (x+z)ⁿ] = (y+z)ⁿ.......... (B₁)
Expanding and simplifying LHS of (B₁) we get:

n (x+z)^n-1 y + n (n-1)/2!{(x+z)^n-2 y² }+ ...+ n (x+ z) y^n-1 + yⁿ and on the RHS on expansion we get y or a power of y in every term except the last term which is zⁿ.
As y divides every term on LHS, and RHS except zⁿ , y must also divide zⁿ. Thus all prime factors of y must be common in z......(B1.1)

Now the question arises: Does y divide zⁿ exactly without leaving any prime factors of y in zⁿ as surplus, or does it leave a surplus?
First suppose y leaves t, a factor of y, as surplus in zⁿ after dividing it. Then t will divide in turn every term in RHS, and every term of LHS except the first term i.e.
n (x+z)^n-1 unless t= n. (Assume n to be a prime for simplicity of this argument). This is because if x is prime to y, (as by assumption, see immediately before (B1)), and all primes of y are common with z, (see B 1.1), y must be prime to (x+z ) or any power of (x+z).

Thus y must divide zⁿ exactly except for t=n as a factor of y which implies :
y must either be 1, or n^n-1 , or wⁿ, or n^n-1wⁿ where w is any other number.

i.e y is either 1 or composite.......conclusion 1,(B)
By similar arguments transposing (y+z)ⁿ of (B) to the L.HS it can be established that:
x is either 1 or composite.........conclusion 2, (B)
Now we can go back to our equation (D) for n=3 and in the light of above two conclusions see that product (xy) cannot be a prime ( unless x=1 and y=1 which renders FLT impossible any way because then b nd c becomes equal.)
Note also: We are finding whether (xy) is prime or not for us to judge whether the Lemma ( which has no doubt) is applicable to it or not, and not for testing the product (xy) against truth or falsity of FLT as FLT itself is the subject of doubt.
Thus considering first (xy) as composite and then showing that (xy) cannot be a prime (except 1) as derived from (B), we have established by the Lemma that for n=3, equation (D) is false, and hence equation (B) and equation (A) are false.
Examining the equations (E) and (F) or any further values of an n or prime n > 5 ( not done in this text), we see the patterns of equation for LHS and RHS is the same as for the case n=3 and all the arguments as for n=3 hold and apply and hence FLT is proved for all values of n>2. Q.E.D (The proof ends here, but the contradiction mentioned immediately below is also of remarkable Note as an alternative conclusive Proof).

CONTRADICTION BASED ON HOMOGENEITY

It will be seen the premise euation(A) is homogeneous in a,b,c.
It will also be seen that the derived equation(C)is homogeneous in x,y, z.
It will be seen that if numerical values were found and substituted for R, neither equation D₁ not euationD₂ is homogeneous in x,y,z.
But these equations are mathematically derived from the premise which is a homogeneous equation.
This contradiction on homogeeity with the premise shows the numerical evaluation of R is never possible.
Exactly the same argumentis applicable for equations (E) and (F) etc. which conclusively proves FLT. (QED)

.

Added below extract of Chapter 4- Discussion of Goldbach Conjecture

CHAPTER 4
Discussion of Goldbach Conjecture Much effort of keen minds is probably now being wasted to prove ( or disprove) this conjecture after the removal of Fermat's Last Theorem from the Arena of desirable mathematical Proofs by Andrew Wiles around 1996. Also I have been told that fabulous financial awards have been declared for this exercise, so much so that it is not unlikely that this conjecture would reach the same notoriety of FLT soon, and reputable mathematicians would decline to look at any work on it.

So I feel a few words need to be said about this conjecture which is normally stated in text books* as quoted below:-

"Every even integer greater than 4 can be written as a sum of two odd prime numbers".
(* Elementary Number Theory by David M. Burton, Allyn and Bacon, Inc.)
The words "can be written as" are crucial in understanding this statement i.e these words also imply in semantic connotation of the context that "it cannot be written as". For instance 8 = 4+4, or =6+2 where neither 4 nor 6 is a prime number.

Thus this statement contains a self refuting implicit contradiction and as such cannot be worthy of any logical investigation, because attempts to prove it by the method of contradiction is futile.

So the conjecture as is usually worded is not precise enough for logical investigation, however more precise wording are not yet known.

So how did it arise? History of Mathematics informs us that the original Goldbach Conjecture ( in a letter to Euler 1742) was that "every even integer is the sum of two numbers that are either primes or 1". This takes care of all even numbers including 2 and 4.

However the reader must note the word "is" in place of "can be written as". Obviously the contention "is" can be shown to have contradictions eg 6 = 4+2 where 4 is not a prime. So the claim "is" is whittled down to "can be written as" i.e until a trap is set so that no contradiction is meaningful as the expression contains an inbuilt semantic contradiction. The conjecture is obviously a tip which may be useful for practical purposes, (e.g. a computer program verification exercise) however its use cannot be extended to rigorous logical argument. In the vocabulary of every language there are many such equivocal expressions useful for particular everyday purposes ( eg in politics, poetry, drama, polemics etc) but such equivocal speech is not permissible in logic.
After many years on this fruitless task the Russian mathematician Vinogradov* stated:
(*Ibid.)
"Almost all even integers are the sum of two primes". Again the phrase "almost all" denotes the imprecision of the target to be probed with logic. And the word "are" is manifestly false as shown above for 6, 8 etc. (i.e 4+2, 6 +2 etc.)

In the same book there is also a quote from the famous mathematician Landau, "The Goldbach conjecture is false for at most 0% of all even integers; this at most 0% does not exclude, of course, the possibility that there are infinitely many exceptions."

Landau with mathematical jargon had jokingly derided the expression "at most 0%" and has stated that this could be "infinitely many" affirming the imprecise mathematical nature of the Goldbach Conjecture. Landau as an expert mathematician had missed the falsity of the common everyday nonmathematical( but stricly logical) word "are". ( As a matter of fact there are many such imprecise conjectures strewn in the domain of the theory of numbers , and seasoned mathematicians take particular delight in quoting them to novices to generate their interest in mathematics.)

Vinogradov and Landau are fair examples of warning not to waste valuable time in attempting logical proofs of these imprecise conjectures. Knowing about them as part of the history of the Theory of Numbers is one thing, but attempting a logical proof based on argument is another.

Perhaps a better use of time would be to give logical precision to the conjectures first so that statements, phrases etc are not equivocal, i.e. do not contain inbuilt implicit contradictions as well, as so often happens in common parlance. For example "may be good" without any more ado implies "may be bad" ,or "neither good nor bad" etc. at the same time. The phrase "may be" implies at the same time "may be not"!

Mathematicians using language for logical arguments should be particularly careful about the expression "any" and the expression "some". For instance " any value >n" is the same as "any one value >n" in turn implying "all values >n", if the concept "any" or "any one" remain unspecified. On the other hand "some value >n" could easily imply "any one value >n" but not "all values >n". So the word "some" is equivocal implying "some" and " and not some others" or "any but not all" i.e both concepts ( positive and negative) at the same time. Such confusion between "some" and "any" on the part of mathematicians deprived me credit for my first proof of Fermat's Last Theorem published in 1971 which proof was by an even simpler argument than in this book. (See Appendix of the book, not on the website).

So those attempting to prove or disprove the Goldbach Conjecture beware of the equivocal nature of the Conjecture before taking any other step towards such a proof!

This conjecture appears to be best suited for computer programs to demonstrate its validity for larger and larger numbers as required rather than use of logical argument to prove it for all even numbers.
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