Fermat's Last Theorem

In Number Theory, Fermat's Last Theorem states:

No three positive integers x, y, and z can satisfy the equation

xn + yn = zn where n is greater than two.

Excerpts from the book Fermat's Enigma by Simon Singh.


"During the autumn of 1984 a select group of number theorists gathered for a symposium in Oberwolfach, a small town in the heart of Germany's Black Forest. One of the speakers, Gerhard Frey, a mathematician from Saarbruchen, could not offer any new ideas as to how to attack the conjecture, but he did make the remarkable claim that if anyone could prove the Taniyama-Shimura conjecture then they would also immediately prove Fermat's Last Theorem.

"When Frey got up to speak he began by writing down Fermat's equation:

xn + yn = zn where n is greater than 2.

"Fermat's Last Theorem claims that there are no whole number solutions to this equation, but Frey explored what would happen if the Last Theorem were false, i.e., that there is at least one solution. Frey had no idea what his hypothetical, and heretical, solution might be and so he labeled the unknown numbers with the letters A, B, and C:

An + Bn = Cn.

Frey then proceeded to "rearrange" the equation. By a deft series of complicated maneuvers Frey fashioned Fermat's original equation, with the hypothetical solutions, into

y² = x3 + (An - Bn) x² - AnBn.

"Although this rearrangement seems very different from the original equation, it is a direct conseqeunce of the hypothetical solution. That is to say if there is a solution to Fermat's equation and Fermat's Last Theorem is false, then this rearranged equation must also exist. ..... Frey pointed out that his new equation was in fact an elliptic equation, albeit a rather convoluted and exotic one. Elliptic equations have the form

y² = x3 + ax2 + bx + c,

but if we let

a = An - Bn, b = 0, c = -An Bn,

then it is easier to appreciate the elliptical nature of Frey's equation.

"By turning Fermat's equation into an elliptical equation, Frey had linked Fermat's Last Theorem to the Taniyama-Shimura conjecture. ..... The Taniyama-Shimura conjecture claims that every elliptic equation must be related to a modular form. ..... Frey claimed that his elliptic equation is so weird that .... it could not be modular. Therefore the existence of Frey's weird elliptic equation defies the Taniyama-Shimura conjecture."

The situation at this point was:
(1) If the Taniyama-Shimura conjecture can be proved to be true, then every elliptic equation must be modular.
(2) If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist.
(3) If the Frey elliptic equation does not exist, then there can be no solutions to Fermat's equation.
(4) therefore Fermat's Last Theorem is true!

If you wish to find out what happened afterwards, read the book.


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