Fermat's last theorem

Pierre de Fermat

Fermat's Last Theorem is a very famous idea in mathematics. It says that:

If n is a whole number which is higher than 2 (like 3, 4, 5, 6.....), then the equation

$x^n + y^n = z^n$

has no solutions when x, y and z are natural numbers (positive whole numbers (integers) or 'counting numbers' such as 1, 2, 3....).

Pierre de Fermat wrote about it in 1637 inside his copy of a book called Arithmetica. He said "I have a proof of this theorem, but there is not enough space in this edge." (In Latin, it was: "Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.") However, no correct proof was found for 357 years. It is widely speculated amongst mathematicians everywhere that Fermat, in fact, did not have a good proof of this exciting and revolutionary theorem.

Relations to other mathematics

Fermat's Last Theorem is a more general form of the equation: $a^2 + b^2 = c^2$. (This comes from the Pythagorean theorem). A special case is when a, b, and c are whole numbers. Then they are named a "pythagorean triple". There are an infinite number of them (they go on forever). Fermat's Last Theorem talks about what happens when the 2 changes to a bigger whole number. It says that then there are no triples when a, b, and c aren't equal to 0.(a=0 and b=c is always a solution, but not an interesting one.)

Proof

British mathematician Andrew Wiles

The proof was made for some values of n (like n=3, n=4, n=5 and n=7). Fermat, Euler, Sophie Germain, and other people did this.

However, the full proof must show that the equation has no solution for all values of n (when n is a whole number bigger than 2). The proof was very difficult to find, and Fermat's Last Theorem needed lots of time to be solved.

An English mathematician named Andrew Wiles found a solution in 1995. Richard Taylor helped him find the solution. Wiles did a lot of secret work. He wanted to be the first.

After a few years of debate, people agreed that Andrew Wiles had solved the problem. Andrew Wiles used a lot of modern mathematics and even created new math when he made his solution. This mathematics was unknown when Fermat wrote his famous note, so Fermat could not have used it. This leads one to believe that Fermat did not in fact have a complete solution of the problem.

More reading

• Aczel, Amir (30 September 1996). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Four Walls Eight Windows. ISBN 978-1-568-58077-7. 
• Dickson LE (1919). History of the Theory of Numbers. Volume II. Diophantine Analysis. New York: Chelsea Publishing. pp. 545–550, 615–621, 731–776. 
• Edwards, HM (1997). Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory. Graduate Texts in Mathematics. 50. New York: Springer-Verlag. 
• Friberg, Joran (2007). Amazing Traces of a Babylonian Origin in Greek Mathematics. World Scientific Publishing Company. ISBN 978-9812704528. 
• Kleiner I (2000). "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem". Elem. Math. 55: 19–37. doi:10.1007/PL00000079. http://math.stanford.edu/~lekheng/flt/kleiner.pdf. 
• Mordell LJ (1921). Three Lectures on Fermat's Last Theorem. Cambridge: Cambridge University Press. 
• Panchishkin, Alekseĭ Alekseevich (2007). Introduction to Modern Number Theory (Encyclopedia of Mathematical Sciences. Springer Berlin Heidelberg New York. ISBN 978-3-540-20364-3. 
• Ribenboim P (2000). Fermat's Last Theorem for Amateurs. New York: Springer-Verlag. ISBN 978-0387985084. 
• Singh S (October 1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8. 
• Stark H (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8. 

Other websites

• Daney, Charles (1997-10-29). "The Mathematics of Fermat's Last Theorem". http://cgd.best.vwh.net/home/flt/fltmain.htm. Retrieved 2011-08-17. 
• Kleiner, Israel (2000). "From Fermat to Wiles: Fermat’s Last Theorem Becomes a Theorem" (PDF). Elemente der Mathematik. http://math.stanford.edu/~lekheng/flt/kleiner.pdf. Retrieved 2011-08-17. 
• Shay, David (2003). "Fermat's Last Theorem". http://shayfam.com/David/flt/index.htm. Retrieved 2011-08-17. 
• Weisstein, Eric W.. "Fermat's Last Theorem". MathWorld. http://mathworld.wolfram.com/FermatsLastTheorem.html. Retrieved 2011-08-17.