Sometime around 1630 while he was reading through Diophantus' Arithmetica, Pierre de Fermat, an amateur mathematician and one of the originators of number theory, jotted the following theorem in the margins of the book:

has no non-zero integer solutions for x, y, and z when n > 2.  Along with this theorem he wrote, "I have discovered a truly remarkable proof which this margin is too small to contain."  Fermat never did reveal his proof, however, and his theorem, which has become known as "Fermat's Last Theorem," might have disappeared from the annals of history if his eldest son, Clément-Samuel, had not discovered and published his notes after Pierre had died.

Historians now believe that Fermat did not actually have a proof of his theorem and that Fermat probably realized that his proof was faulty.  Fermat may have been able to prove his theorem for the cases n = 3 and n = 4, but, if he could, he never published his proofs.  Later mathematicians, however, would.

From 1630 until 1994, many mathematicians attempted, and failed, to prove Fermat's Last Theorem.  Among them were such famous mathematicians as Euler, Sophie Germain, Legendre, Dirichlet, Lamé, Gauss, and Kummer.  Many of them were able to prove Fermat's Last Theorem for specific cases of n, but none were able to offer a generalized proof that would establish Fermat's Last Theorem for all n.  In fact, using computers, mathematicians had proven Fermat's Last Theorem true for n up to 4,000,000 by 1993.

On October 6, 1994, Andrew Wiles, a British mathematician, submitted a proof of Fermat's Last Theorem that the majority of the mathematical society found acceptable.  His proof, however, relies not upon number theory, as many of the earlier proofs had, but upon the properties of elliptic curves, i.e., those curves with the form y² = x³ + ax + b for constants a and b.  His proof is based upon the following two theorems:

Theorem A
 If there is a solution (x, y, z, n) to the Fermat equation, then the elliptic curve defined by the equation

Theorem B
 All semistable elliptic curves with rational coefficients are modular.

Theorem A was first conjectured by Gerhard Frey in 1982 and was later proven in 1986 by Ken Ribet with help from Jean-Pierre Serre.  Theorem B is a special case of the Taniyama-Shimura-Weil Conjecture, which Taniyama, Weil, and Shimura proposed in 1955.  Wiles was able to construct a proof of the theorem using these two theorems and claimed to have established the proof of Fermat's Last Theorem on June 23, 1993.  Later evidence revealed that his proof was not complete, however, and, with the assistance of Richard Taylor, Wiles resumed his efforts to establish a proof.  Finally, a year-and-a-half later, Wiles was able to produce the final form of his proof that has since become accepted.

Author: Tony Brinsko

References:
 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

http://www.mbay.net/~cgd/flt/flt01.htm

Christiaan Huygens, was born two years earlier in 1629 in the Netherlands.  He parents were from an important Dutch family so he received an excellent education.  He was tutored at home until the age of sixteen. Descartes, a friend of the family influenced his mathematics.  He studied mathematics and law at the University of Leiden and then College of Orange at Breda.  He published his first mathematical paper, Cyclometriae, in 1651.  In it he disproved the method at which Gregory of Saint-Vincent who had claimed to square a circle.  Then he worked with the optics of telescopes, finding a new method of grinding and polishing lenses.  With improved lenses he observed the first moon and the shape of the rings of Saturn.  He made a theory for the phases and changes in shape of Saturns ring which was published in Systema Saturnium.  Because he needed accurate timekeeping for astronomy, he patented the first pendulum clock, which was based on his mathematical work on the cycloid.  He was convinced that by making clocks that would function on boats, regardless of climate or motion of the ship, the longitude of the ship could be found, thus he made several pendulum clocks for ships.  He also worked on constructing clocks with springs, but they were less accurate than his pendulum clocks.  In 1673 Huygens published Horologium Oscillatorium sive de motu pendulorum which contained the law of centrifugal force for uniform circular motion.  He also worked with optics, claiming that light was a wave and that it traveled at a finite speed (Which was proven by Romer during his lifetime).  In mechanics he addressed the compound pendulum, and the dynamics of bodies.  Huygens worked on curves including the Cissoid of Diocles, the Cycloid and the Epicycloid.  He died on July 8, 1695.

Author: Charles DeBoer

References:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude1.html#57