FERMAT'S LAST THEOREM PROVED 30 YEARS BEFORE SIR ANDREW WILES' PROOF

Over the years I have submitted my proof to more than 50 professional mathematicians. The mathematicians who have rejected my 1965 proof have done so primarily
because of the

*belief*that there cannot be a proof of FLT using simple mathematical concepts.The few who have actually tried to refute FLT65, have attempted to support this belief with the fact that the division algorithm*may or may not*apply to integer constants obtained by substituting specific integer values into integer polynomials and reducing them to single integers. But there is no proof that it is true for the integer polynomials of the Fermat equation, and so three of them have resorted to demonstrations that have no relevance to actual solutions of the Fermat equation, to try to make the point that FLT65*not be valid.***may**
Their idea that the division algorithm might not apply to the
integer polynomials of the Fermat equation factor f(Z) = Z

^{p-1}+ Z^{p-2}X + Z^{p-3}X^{2}+ ••• + X^{p-1}arises from the fact that, for given integer values of X and Z, f(Z) can be reduced to a single integer (a constant), and if that single integer is not prime, in general, one of its integer factors__may or may not__contain the integer equal to the integer value of Z_{1}– a.
FLT65 provides a way to determine whether or not any specific single
integer value of f(Z) (a polynomial factor of the Fermat equation) can contain
the specific single integer value of Z –a (a polynomial factor of Y in the
Fermat equation) as a factor, using the division algorithm and its three
corollaries. For an integer solution of the Fermat equation, f(Z

_{1}) must not only contain Z_{1}– a, it must be equal to (Z_{1}– a)^{p}.
The division algorithm and its corollaries, by definition, apply to
all polynomials with real number variables, so they apply to polynomials of
integer variables in the same way they apply to all polynomials of real numbers
because integers are real numbers which, along with non-integer rational and
irrational numbers, comprise the field of real numbers. Finally, an integer
solution of the Fermat equation, if there is one, is simply one of the infinite
number of solutions to one of the Fermat equations, and the three numbers of

*any*solution are a set of three numbers existing in the field of real numbers. FLT65 demonstrates the fact that for the Fermat integer polynomials f(Z) and Z – a, where both polynomials must be factors of Y^{p}, there are no integer values of a, X and Z for which Z – a divides f(Z), because the remainder will always be non-zero.
After more than 40 years, I still have hope that more mainstream mathematicians will join the small, but growing number of mathematicians who agree that there are no fatal flaws
in the logic of FLT65.

Edward R. Close, June 26, 2017

For those who are not familiar with Fermat's Last Theorem, I've pasted in a previous discussion and some relevant links below.

For those who are not familiar with Fermat's Last Theorem, I've pasted in a previous discussion and some relevant links below.

**The Basic FLT65 Proof**

The following steps summarize the logic and mathematics of FLT65.
For brevity, I will not present proofs of the steps here because they are so
easily proved that they can be proved by a bright high school algebra student.
If these steps aren’t obvious go to http://www.erclosetphysics.com/search?q=Fermat%27s+Last+Theorem+Part+1.

STEP #1: The first step in FLT65 was to provide a rigorous proof
of the division algorithm and its three corollaries. The reason I provided this
proof first, even though it was well known to mathematicians, was to show that
it applies to all polynomials across the field of real numbers,

*including integers*, and to highlight the fact that the*uniqueness*of the dividend and remainder allows the all-inclusive “*if and only if*” of Corollary III. These points were pointed out in FLT65.
STEP #2: If there is an integer solution for Fermat’s equation:

**x**, to prove or disprove it, we need only consider^{n}+ y^{n}= z^{n}**n**as prime numbers, p**>2, and****x, y,**and**z**as relatively prime positive integers. Proof of this is included in FLT65 allowing us to proceed to Step 3.
STEP #3: Fermat’s equation can be rewritten as

**z**, and since all prime numbers >2 are odd, factored as follows: z^{p}– x^{p}= y^{p}^{p}– x^{p }**=**(z-x)( z^{p-1}+ z^{p-2}x + z^{p-3}x^{2}+•••+ x^{p-1}) = y^{p}
Similarly, z

^{p}– y^{p }**=**(z-y)( z^{p-1}+ z^{p-2}y + z^{p-3}y^{2}+•••+ y^{p-1}) = x^{p}.
For the next step, and throughout this discussion, keep in mind
that we have assumed that there are integer solutions to Fermat’s equation, so
the approach is to determine whether this assumption leads to a contradiction.
If it leads to a contradiction, FLT is proved.

STEP #4: It is easy to show by simple algebraic division that the
only common factor that may be shared between the factors of the Fermat
equation is the integer p, and since

**x, y,**and**z**are relatively prime integers, if either**x**or**y**contains p as a factor, the other cannot. See the proofs of this in the original proof in the link above. So we can let**y**represent the one that does not contain p. It then follows that the two factors of the left hand side of the first equation of step 3 are relatively prime and thus are perfect p-powers of integers. Thus, by inspection of
(z-x)(z

^{p-1}+ z^{p-2}x + z^{p-3}x^{2}+•••+ x^{p-1}) = y^{p}, we see that we can write
(z-x)= B

^{p}, and (z^{p-1}+ z^{p-2}x + z^{p-3}x^{2}+•••+ x^{p-1}) = A^{p}, where A and B are positive integers.
STEP #5: If there is an integer solution, then
x and y are specific integers X

_{1}and Y_{1}, and the p-1 polynomial in z, f(z)=(z^{p-1}+ z^{p-2}X_{1}+ z^{p-3}X_{1}^{2}+•••+ X_{1}^{p-1}) = A^{p}, and B^{p}A^{p}= Y_{1}^{p}. That is, the two factors must be perfect p-powers, integers raised to the pth power.
STEP #6: In a positive integer solution, z
>Y

_{1}>A, and by closure of integers, there is a positive integer a, such that A= (z – a), and by corollary II of the division algorithm, when f(z)=(z^{p-1}+ z^{p-2}X_{1}+ z^{p-3}X_{1}^{2}+•••+ X_{1}^{p-1}) is divided by Z – a, the remainder is equal to f(a) = a^{p-1}+ a^{p-2}X_{1}+ a^{p-3}X_{1}^{2}+•••+ X_{1}^{p-1}.
STEP #7: Corollary III of the division
algorithm says that f(z) is divisible by z –a

**, f(a) = 0. But, since f(a) is the sum of p positive integers, it can***if, and only if*__never__equal zero. Thus by assuming there is an integer solution of**z**, we have produced a contradiction proving Fermat’s Last Theorem.^{p}– x^{p}= y^{p}**Discussion**

Note
that the case n = 4 is not addressed in this proof. It was overlooked in FLT65,
but this was not a problem because there were several known proofs for n = 4,
including one by Fermat himself.

So FLT65 is effectively a complete and valid proof of FLT; but
approximately 90% of the mathematicians to whom the proof was submitted over
the years did not respond at all. This is because, before Sir Andrew Wiles’
proof was accepted, professional mathematicians received hundreds of supposed
proofs of Fermat’s last theorem per year.

If you’ve ever taught mathematics and had to evaluate proofs
developed by students, you know it can often be very challenging and time
consuming, and attempts at proofs by amateur mathematicians are usually filled
with all kinds of errors. In addition, because in more than 300 years, so many
first-rate mathematicians had tried to prove or disprove FLT and failed, most
mathematicians consider reviewing such ‘proofs’ a waste of time.

I’m sure that this was the reason the first mathematician to whom
I sent it rejected it. The reason he gave, however, was that, if FLT65 were
true, it would also apply to the case n = 2. [When n = 2, we have

**z**, which^{2}– x^{2}= y^{2}*does*have integer solutions known as the Pythagorean triples, e.g. 3,4,5]. Of course by giving this reason for rejecting FLT65, he revealed the fact that he hadn’t read it, because it is clear to anyone with basic math skills reading the first page that the method of proof of FLT65 doesn’t apply to the case n = 2.
Of the 10% who did respond, most gave the opinion that there had
to be a mistake somewhere, but failed to point one out, or provide any
mathematical argument supporting their opinion. Of the remaining recipients of
FLT65, only a few provided any sort of mathematical demonstration supporting
their opinions. Those responses are presented in the article accessed by the
links provided above. Those arguments were all easily refuted. However, one of
those demonstrations, actually offered with different numerical values by three
reviewers, is worth mentioning here because it is a classic example of
inadvertent misdirection, and it also shows how tricky a proof of FLT can be.

The argument they put forth was that the division algorithm and
corollaries certainly apply to algebraic polynomials, but they may not
necessarily apply to the integers obtained when, for specific integer values of
z, a and X

_{1}, the algebraic polynomials z-a, f(z) and f(a) are reduced to single integer values. This is an interesting conjecture, but none of the reviewers attempted to prove or disprove it, instead they offered what they thought were counterexamples to FLT65 for n = 3. They selected integer values of z, a, and X_{1}that, when substituted into f(z) and f(a), produced an integer value for f(z) that contained the integer z-a as a factor, even though f(a) did not equal zero, appearing to violate corollary III of the division algorithm.
It is worth contemplating this argument a little more deeply for a
moment, because by doing so, we expose the fact that such a demonstration is
not actually a counterexample, but is in fact, an inadvertent misdirection,
shifting attention away from the fact that z must be part of an integer
solution to the Fermat equation. It is not hard to find positive integer values
for z, a, and X

_{1}such that f(z) is divisible by z-a, and of course f(a)= a^{p-1}+ a^{p-2}X_{1}+ a^{p-3}X_{1}^{2}+•••+ X_{1}^{p-1}will still be non-zero because all the terms are positive integers. But because the values of z and a selected have no relation to the Fermat equation, these demonstrations have no bearing on the logic of FLT65.
FLT65 started with the assumption that there is an integer
solution for the Fermat equation. This means that for a numerical example to be
relevant, z must be part of an integer solution of Fermat’s equation. The issue
is not whether you can find integer values for z, a, and X

_{1}that will make f(z) divisible by z-a; the relevant point here is that, if there is an integer solution, the value of z must satisfy Fermat’s equation. Then, because f(z) and z-a are both polynomials in z, the algorithm and corollaries apply, and the remainder__must__equal zero for f(z) to equal a perfect p-power, A^{p}, if the assumption of an integer solution is true. But, of course*for Fermat’s equation*, f(a) cannot equal zero. -- End of story!
I think these reviewers were so intent on trying to find a way to
disprove FLT65, which they were convinced from the beginning could not be
valid, that they were blinded to the fact that, if their ‘counterexamples’ were
valid, they would actually have provided integer solutions for

**z**, directly disproving FLT, and thereby also disproving Andrew Wiles’ proof.^{p}– x^{p}= y^{p}
So after fifty years, FLT65 still has not been refuted. Those who
tried have failed, but only a few besides myself have accepted it as valid, and
two of them have since passed away. Many of the mathematicians who have
reviewed it believe it cannot be valid, and two even claimed to have refuted
it, but their arguments were easily disproved. See the details in the links
provided above.

After fifty years, I would like to have closure; so anyone out
there who believes the proof is faulty or incomplete is challenged to provide
irrefutable mathematical proof that I can understand supporting that belief. If
you can prove to me that FLT65 is wrong, I will acknowledge you proof and send
you a check for $100.

Unlike Sir Andrew Wiles’ proof of FLT, which is hundreds of pages
long, drawing on a very sophisticated knowledge and understanding of elliptic
functions and modular algebra, FLT65 is a relatively simple proof relying only
on basic mathematical principles. I believe that Pierre de Fermat will rest
easier when FLT65 is recognized as valid, because it proves that he could have
proved his famous theorem with mathematics available in 1637. If FLT65
is is at last recognized as correct, I, and poor Fermat will have closure.

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