Fermat’s Enigma Resolved: An Algebraic Proof of His Last Theorem

Pierre de Fermat (1601-1665), a high-ranking jurist in the reign of Louis XIV (1643-1715), and considered to be one of the leading mathematicians in the first half of the 17th century, created one of the greatest enigmas in the history of mathematics:In the equation

an + bn = cn,                                                                         (1)        

he asserted that there are no integer solutions if n is greater than 2. In the Pythagorean form,

a2 + b2 = c2,                                                                        (2)

there are an infinite number of integer solutions. But for a3 + b3 = c3, or a4 + b4 = c4, … , Fermat claimed nointeger solutions. This became known as Fermat’s last theorem.

Unfortunately, practically all of Fermat’s work appeared in the margins of books he was studying, or in the margins of letters to mathematical colleagues. For an + bn = cn, n > 2, he added that he had discovered “a marvelous proof that would not fit in the margin.”

What appeared to be an unusually simple theorem baffled the most famous mathematicians for some 350 years. Finally, in 1997, Fermat’s last theorem was “officially” solved by Andrew Wiles of Princeton University with the help of his associates, Kenneth Ribet in particular. They devised a proof of extraordinary complexity employing hundreds of pages of the most esoteric mathematics of modern-day number theory. Because the knowledge of mathematics needed to develop this proof could not have been known in Fermat’s time, it leaves Fermat’s enigma unresolved. Naturally, any proof that Fermat had would of necessity be based on 17th-century mathematics.

But a relatively simple algebraic proof of Eq. (1) was developed by Roger Gilmont, and that is the reason for the present essay. Gilmont failed to achieve publication in mathematical journals, all of whose editors consider the work “unsuitable.” Gilmont is a thermodynamicist, and I attribute his failure to get published to the fact that he doesn’t write in a language and style to which mathematicians are accustomed. Also, only a big-shot mathematician can solve Fermat’s last theorem.

Gilmont has finally published his proof in a booklet: “Fermat’s Enigma Resolved: An Algebraic Proof of His Last Theorem,” Dorrance Publishing, 2005, 1-800-788-7654, www.dorrancepublishing.com.

Gilmont’s proof is based on two steps. First, use a different form instead of Eq. (1):

(an/2)2 + (bn/2)2 = (cn/2)2,                                                            (3)

which is obviously a form of the Pythagorean equation with

x = an/2, y = bn/2, z = cn/2.                                                            (4) 

The second step is based on the rules governing odd and even integers. If O is odd and E is even, for example,

O X O is O, E X E is E, O X E is E,     
and                                                                                                                                     (5)                                                                                                                                                                
O +  O is E, E +  E is E, O + E is O,

and so forth. Now let

z = y + m,                                                                         (6)

where m is an integer. Substituting into x2 + y2 = z2, we get

y = (m/2)[(x /m)2-1].                                                                   (7)

Next, examine all possible cases in which y is an integer and x is odd [and, from Eq. (7), m has to be odd]. That is followed by all possible cases in which y is an integer and x is even [and, from Eq. (7), m has to be even]. The complete proof also involves rational fractions. The work is lengthy but not tedious.

For there to be a solution to Fermat’s equation, then, x2 = an, y2 = bn, and z2 = cn must all be of perfect degree n. However, Gilmont shows that if x2 is of perfect degree n, then in all possible integer solutions of x2 + y2 = z2, y2 and z2 cannot be of perfect degree n. That is, if a is an integer, then b and c cannot be integers; therefore, there is no solution to Fermat’s equation.

This relatively simple algebraic proof employs mathematics known in Fermat’s time. Although we cannot, of course, prove it, there can be little doubt that Pierre de Fermat used Roger Gilmont’s method. The fact that Gilmont’s proof has not been accepted by mathematics journals is inexplicable.

 

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