IS FERMAT'S LAST THEOREM
PROVEN?
...........................................................................James
Constant.......................................................................
...................................................................math@coolissues.com...............................................................
Wile's
Proof of Taniyama's Conjecture is Trumped by Existence Proofs of
Fermat's Last Theorem and Frey's Equation
Introduction
Fermat claimed in 1637
to have discovered a marvelous proof of his last theorem.
(1) .....
..........m>2,
x,y,z,m integers
There are two current methods of proving Fermat's Last Theorem
(FLT) (1). Both use proof by contradiction. They start by
assuming the negation of FLT
(2) .....
..........m>2,
x,y,z,m integers
and then showing that (2) is not true for the
stated conditions. Later, I will assume z may also be
non-integer.
Wile's Proof Uses Hyperbolic Geometry 1
In 1954 Taniyama conjectured that for
every elliptic equation in Euclidean geometry, there is a
corresponding modular structure in hyperbolic geometry. Elliptic
equations have the form
(3)..... 
In 1983 Frey suggested a connection
between FLT and the Taniyama conjecture (TC). He connected (2) to
a "certain set of elliptic equations", the "Fermat
subset", which required setting a=af
, b=0, c=cf in (3) where af
,cf are obtained from the x and y
elements in (2). The result was Frey's elliptic equation (FEE)
(4) .....
Frey also noted that (4) could not be
represented having modular structures in hyperbolic geometry. In
1986 Ribet proved Frey's connection. Accordingly, there are two
possibilities from this connection: first, a direct proof of (2)
implies an indirect disproof of TC and, second, a direct proof of
TC implies an indirect disproof of (2).
In 1993, in a 200 page paper, Wiles
accepted Ribet's proof of Frey and then directly proved TC for
FEE (4) implying an indirect disproof of (2) and thus proof of
FLT.
Wiley's proof turns on how FEE (4) is
interpreted. As demonstrated by Frey, (4) is an elliptic equation
which is not modular. These facts, based on Frey's rigorous
mathematics, spell disaster for TC. Since (2) was assumed, Frey
questioned the "existence" of (4). He claimed that
since (4) is not modular it is prohibited to exist by TC implying
that (2) does not exist thus proving FLT.
Constant's Proof Uses the
Binomial Expansion 2
In a 3 page paper, Constant starts by
assuming the negation of FLT (2) and asking whether z can
be an integer, by rearranging (2) and expressing it as a binomial
expansion,
(5) .....
...............................................................
Since m>2 (5) is an (absolutely
convergent or divergent) infinite series, i.e,. no finite sum or
integer z exists. Thus, equation (2) is wrong proving FLT
for z integer. Equation (5) also proves that equality (2)
holds for z non-integer.
Objections to Wile's Proof
Vos Savant questioned Wile's proof of FLT
on several grounds including, first, Bolyai, one of the three
founders of hyperbolic geometry, squared a circle in hyperbolic
geometry, a task known not possible in Eucledean geometry. Vos
Savant says it is logically inconsistent to reject Bolyai's
hyperbolic method of "squaring the circle" and to
accept Frey's and Wile's hyperbolic method of proving FLT and,
second, a fatal flaw in Frey and Wile's proof is that it also
proves the Pythagorean theorem to be false.3
An even more serious objection to Frey's
and Wile's logic is now made. TC claims that every
elliptic equation must be related to a hyperbolic modular form.
If FLT(1) is wrong, Frey has shown that this leads to an elliptic
equation which is not modular (4), an indirect
contradiction of TC. Both Frey and Wiles contend that, since (2)
is assumed and TC is proved by Wiles, FEE (4) cannot
"exist" and, therefore, FLT is true. This is curious
logic not rigorous mathematics. The illogic that (4) does
not "exist" means that ``There exists no
x such that conclusion y is true'', an
obviously false statement. Like any function in mathematics, (4)
can be obtained in a number of ways. For example, it can be
derived from (2) on conditions that z is non-integer or
arbitrarily by assigning special values of coefficients in the
general elliptic equation (3). It cannot be obtained or derived
from TC.
In mathematics, proofs
which state ``There exists an x such that conclusion y
is true'' are known as existence proofs. Many, but not all,
proofs of this type are constructive proofs so named
because one finds elements x that do satisfy the
conclusion y. For example y
in equation y2=x is a real number when x is real
positive and y is imaginary when x is real
negative.These are
values of x that are solutions to equation y2=x.
Of course not all proofs are this simple and to produce an
element(s) x that will work in more complex equations may
require ingenuity. Consider now FEE (4) and ask
"does there exist an x such that y is
true". An existence proof of (4) can be easily made by using
the rules of arithmetic in the real number system. We know that
if x is a real number and the right hand side of (4) is
positive then so too is y a real number. Frey's and
Wile's claim that (4) does not exist under TC is made without
constructive proof of its existence. A constructive
proof of (2) was made by Constant.
Since FEE (4) exists for some
values of x, TC is wrong. As noted, Frey and Wiles claim
that (4) does not exist because it is not modular and thus
prohibited by TC but does exist because it is connected to (2)
which they say fails with (4). Frey and Wile's illogic is
analogous to claiming that a coin (4) does not exist because one
face is heads (non-modular) but does exist because its other face
is tails (connected to (2)). But, as stated, the existence of (4)
depends from constructive proof of its existence not from its
modular attributes or the source (2) of obtaining its
coefficients.
To further illustrate and emphasize these
important objections, the coefficients af
,cf of FEE (4) can be obtained in a number
of ways:
1. In this method the coefficients af
,cf are derived from (2) on condition that z
is non-integer, as proven by Constant. In this case the equality
signs in (2) and (4) hold, FLT is disproven and TC fails for z
non-integer.
2. In this method the coefficients af
,cf cannot be derived from (2) on
condition that z is an integer, as proven by Constant. In
this case the equality sign in (2) does not hold, FLT is proven,
(4) cannot be obtained from (2), and TC is unaffected by FLT (1).
The method used by Frey, which assumes that coefficients af
,cf can be derived from (2) on condition
that z is an integer, as required by FLT, is hypothesis
disproved by Constant.
3. By setting coefficients af
,cf arbitrarily. There is no mathematical
reason FEE (4) must necessarily be connected to (2). It may
(equality holds for z non-integer) or may not (equality
does not hold for z integer). Connecting simply is one way
of determining coefficients. This is another way of saying that
(3) is valid for some values of coefficients a,b,c
including the arbitrary setting of a=af
, b=0, c=cf. In this case TC fails
because (4) is not modular, as shown by Frey.
Endnotes
1 Annals
of Mathematics 141 (3), May 1995 (difficult to read); Simon
Singh, Fermat's Last Theorem, Fourth Estate Ltd, 6 Salem
Road, London W2 4BU, England. 1998. pages 216-223.
2 James, Constant, Proof
of Fermat's Last Theorem, http://fermat.coolissues.com/fermat.htm
3 Marilyn Vos Savant, Is
it Solved? St Martin's Press, New York 1993. Pages 18,60-62.
Copyright 
2002 James Constant
By same author: http://www.coolissues.com/mathematics/sameauthor.htm