Is Fermat's Last Theorem Proven?

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=a_f , b=0, c=c_f in (3) where a_f ,c_f 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 nox 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 a_f ,c_f of FEE (4) can be obtained in a number of ways:

1. In this method the coefficients a_f ,c_f 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 a_f ,c_f 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 a_f ,c_f 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 a_f ,c_f 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=a_f , b=0, c=c_f. 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://www.coolissues.com/mathematics/Fermat/fermat.htm

³ Marilyn Vos Savant, Is it Solved? St Martin's Press, New York 1993. Pages 18,60-62.

By same author: http://www.coolissues.com/mathematics/sameauthor.htm