More than 50 years ago I made some profound discoveries that I’ve used over the years to explore the world of mathematical physics and the relation between quantum mechanics and consciousness. This adventure began when I developed a proof for the famous mathematical conundrum known as Fermat’s Last Theorem in 1965.
My proof, known as FLT65, has never been refuted. Fermat’s Last Theorem, thought by most to be an abstract number theory theorem, has serious application to quantum physics and It is being currently being reviewed in connection with new research.
This is the first in a series of posts of sections of this paper currently being reviewed by number of mathematicians and scientists.
CLARIFICATION AND EXPLANATION OF FLT65
A 1965 PROOF OF FERMAT’S LAST THEOREM
By Edward R. Close, PhD
ABSTRACT:
Fermat’s last theorem (FLT) states that no three positive
integers X_{1}, Y_{1},
and Z_{1} can satisfy the equation x^{n}
+y^{n }= z^{n} for
any integer value of n greater than two. Fermat penned the following statement
in Latin in the margin of a book on Diophantine equations in 1637:
“Concerning whole numbers, while certain
squares can be separated into two squares, it is impossible to separate a cube into two cubes or a fourth power
into two fourth powers or, in general any power greater than the second into
two powers of like degree. I have discovered a truly marvelous demonstration,
which this margin is too narrow to contain.”

Pierre
de Fermat, circa 1637 ^{[1]}
Fermat’s “marvelous” proof was never found and the
theorem remained officially a conjecture
without proof until Andrew Wiles published a lengthy treatment in 1995 that was accepted by number
theorists as a valid proof ^{[3]}.
Prior to that, however, Edward R. Close completed a proof in 1965 (FLT65), submitted it to the first of many
reviewers in 1966, and published it in 1977 ^{[4]}. The Close proof though never refuted, presented difficulties for some reviewers because of
unconventional notation, and at least three reviewers have suggested that the
difference between applying the division algorithm to algebraic polynomial
factors and integer factors of the equation as used in the proof make it
questionable and either incomplete or incorrect. This concern is addressed in
this paper and shown to be unwarranted: There
has been a tendency for reviewers to ignore the uniqueness requirements for the
division algorithm stated in FLT65 while looking for mistakes which they assume
must be there, for reasons explained in this paper. When everything presented
in FLT65 is appropriately considered, the concern over the application of the
division algorithm is eliminated, removing the only serious objections to the
proof.
______________
In this paper, the author provides a brief description and
history of FLT65, followed by presentation and refutation of the objections and
supposed counterexamples offered by some reviewers in efforts to invalidate
FLT65. Several appendices are attached, including a copy of the original FLT65
proof. The
other appendices contain supporting arguments and information confirming that
FLT65 was and is, a valid proof, justifying the statement found on the last
page of FLT65:
“… we have reached a complete contradiction by
assuming X, Y and Z to be integers, and may state that
the equation X^{N} + Y^{N}
= Z^{N} has no solutions in positive integers when N is an integer > 2. And so the
proof of Fermat’s last theorem is complete.”

Edward
R. Close, December 18, 1965 ^{[2]}
'Never been refuted'? It seems Close never accepted our email exchange (sorry the super/subscripts don't copy), which continued for quite some time but made no progress:
ReplyDeleteOn 12 Jan 2014, at 01:19, Edward Close wrote:
please look at the Addendum.
I responded
Fine! I’ve done that, and would query this part:
F(Z) = ZN – X1N  Y1N = 0, and thus can be expanded to produce:
F(Z) = (ZX1)( ZN1 + ZN2X1 + ZN3 X12 +•••+ X1N1)  (ZX1)(Z –a )N = 0.
By inspection of this equation, we see that G(Z) is a polynomial factor of F(Z) ...
We can see the error here by using z for the variable and Z for the constant figuring in the Fermat eqn. and I will keep track of the difference by using z and Z as appropriate.
We need an expression for F(z). The definition is F(z) = zN – X1N  Y1N
The problem is that the first part of this factorises into
(zX1)( zN1 + zN2X1 + zN3 X12 +•••+ X1N1)
and for the second part, by combining
g(Z)f(Z) = YN (eqn. 3)
g(z) = z  X1
and f(Z) = (Z –a )N
we end up with (ZX1)(Z –a )N.
Thus the first term of that expanded version of F(z) above involves the variable z, while the second involves the constant Z. So the factorisation of F that you rely on doesn’t apply.
In response Close wrote:
The only advantage I have is: I know FLT is true. Ergo there must be a way around it. When I find it (because I believe I will), I will send it along. Thank you for being interested in seeing it.