Pierre de Fermat was a number theorist of the fifteenth and sixteenth centuries.  It is surprising to find that Fermat was a lawyer and an amateur mathematician.  His interests varied from analytic geometry to finding the extrema of functions.  His major works, however, were in the field of prime numbers.  Fermat's most interesting theorem referred to as "Fermat's Last Theorem" was written in 1637.  It sets out to prove that for the expression, xⁿ+yⁿ = zⁿ (when x, y, and z are non-zero integers), no numbers other than 2 are a possible value for n. Although Fermat claimed to have found a proof of the theorem, a proof has not been found.  Since it is the last remaining statement in the list of Fermat's works that needed to be proven, it became known as Fermat's Last Theorem.
 Although mathematicians have tried to prove Fermat's Last Theorem, they have all failed.  A recent case involves Andrew Weils, a British mathematician. In June of 1993, after a three day lecture on Fermat's theorem, Wiles announced his proof.  However, in December problems in Weils' proof were noticed.  Weils began to collaborate with a Richard Taylor.  By April 1995, Taylor lectured on Fermat's Last Theorem and gave the impression that no real doubts remained. Their proof involves solving for two theorems, A and B.  It is believed that together, they prove Fermat's Last Theorem.  Theorem A states the following: "If there is a solution (x,y,z,n) to the Fermat equation, then the elliptic curve defined by the equation Y² = X(X-xⁿ)(X = yⁿ) is semistable but not modular."  Theorem B states that "all semistable elliptic curves with rational coefficients are modular."  Although both these theorems are very difficult themselves, and both have been proven in the last ten years, a very small

Author: Henry Sheen

References:
Koblitz,Neil.  Number Theory Related to Fermat's Last Theorem.  Boston: Birckhauser, 1982.

Singh, Simon.  Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.  New York: Walker, 1997.