Harte, on 10 December 2012 - 08:56 PM, said:

Correct me if I'm wrong, but I believe that the proof was that **there is no solution** to the above. IOW, there exist no three numbers (x, y, z) for which the above is true.

The proof has since been done, but in a very unsatisfying way ( using computers.)

Harte

Harte, Im not mathmatican of any kind. So, this might have had many errors. Others mathmatician and historians are free to join.

He said that there is no solution for any n higher than two.

In 1657, Fermat attempted to solve Diophantine equation 61x2 + 1 = y2 (solved by Brahmagupta over 1000 years earlier). The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

Fermats in free time study mathemathics. He wrote to mathmatician in Europe .He read Diophantus work and problems

of finding whole number solution and Pythagoras equasion.

He said that we squares thing but nothing else. With whole numbers.

Its not possible to wrote one cube as sum of two cubes.

xn+yn=zn for all n>2

Euler made prooved with gap. But it was correct.

He was correct so in fact he proove it.

In 19 century Marie-Sophie Germain act that she was a man.She study at night mathemathics from father library.

Parents were against that idea.She correspondent with famous mathematicians such as Lagrange, Legendre, and Gauss.She wrote, again under the pseudonym of M. LeBlanc.Despite the friendship of Germain and Gauss, they never met.One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject.

Fermat deal with exponent 4. Euler with cubes. And Germain with all to 100.

**That was a huge leap.** Imagine.

To proove Fermat theorem for particular number you take that number and do serious of

operations to it. And you checked some things and some of these things is true then you know that Fermats theorem

is true for that exponent.She asked Gauss if her approach to the theorem was worth pursuing so called Sophie Germain's theorem. Gauss never answered.

Gauss in beside Euler one of top five mathmaticians of all time as Im sure you already know but Gauss wasnt so interested in Fermats enigma.Its more historical interest. Its not fundemental.

**It was proven by the British mathematician Andrew Wiles in 1994.**
Between him and Sophie...well here is from wiki:

*The ultimately successful strategy for proving Fermat's Last Theorem was by proving the modularity theorem or*
*Taniyama–Shimura–Weil conjecture. The strategy was first described by Gerhard Frey in 1984.[106] Frey noted that if Fermat's equation had a solution (a, b, c) for exponent p > 2, the corresponding elliptic curve[note 2]*
*y2 = x (x - ap)(x + bp)*
*would have such unusual properties that the curve would likely violate the modularity theorem.*
*"epsilon conjecture", was proven by Ken Ribet in 1986. Second, it was necessary to prove a special case of the modularity theorem. This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.*
*....*
**Edited by the L, 10 December 2012 - 09:20 PM.**