Fermat's Last Theorem

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

 If  is a whole number larger than 2, then the equation  has no solutions when x, y and z are natural numbers.

Or,
 It is impossible to express in whole numbers two cubes, which added equal a third cube. Furthermore, it is impossible with anything higher than squares.

This means that there are no examples where ,  and  are natural numbers, i.e. whole numbers larger than zero, and where  is a whole number bigger than 2. 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 margin". However, no correct proof was found for 357 years. It was finally proven in 1995. Most mathematicians do not think that Fermat, in fact, ever had a margin proof of this theorem.

In its original the problem is as follows:

Cubum autem in duos cubos, aut quadratoquadratum in duosquadratoquadratos & generaliter nullam in infinitum ultra quadratum
potestatem in duos eiusdem nominis fas est dividere cuius reidemonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. Overview 
Fermat's Last Theorem is a more general form of the Pythagorean theorem, which is an equation that says:

 

When ,  and  are whole numbers this is called a "Pythagorean triple". For example, , and since  we can say  is a Pythagorean triple. Fermat's Last Theorem rewrites this as

 

and claims that, if you make the  a larger whole number than 2, then ,  and  cannot all be natural numbers. For example,  and , and so  is an example that confirms this.

On the equation's quadratic
The x and y are two unknown sums, summing imaginary third sum z. Despite there being 4 terms: n, x, y & z, the n is a function summing the total of unknown sums. Zero is missing from this equation by the rule of "1 plus 1 is 2 and no more", written 1+1=2+0.

To give clarification, the n is known to be a sum.

 Proof 

The proof was made for some'' values of , such as , ,  and , which was managed by many mathematicians including Fermat, Euler, Sophie Germain. However, since there are an infinite number of Pythagorean triples, as numbers count upwards forever, this made Fermat's Last Theorem hard to prove or disprove; the full proof must show that the equation has no solution for all values of  (when  is a whole number bigger than 2) but it is not possible to simply check every combination of numbers if they continue forever.

An English mathematician named Andrew Wiles found a solution in 1995, 358 years after Fermat wrote about it. Richard Taylor helped him find the solution . The proof took eight years of research. He proved the theorem by first proving the modularity theorem, which was then called the TaniyamaShimura conjecture. Using Ribet's Theorem, he was able to give a proof for Fermat's Last Theorem.  He received the Wolfskehl Prize from Göttingen Academy in June 1997: it amounted to about $50,000 U.S. dollars.

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 maths when he made his solution. This mathematics was unknown when Fermat wrote his famous note, so de Fermat could not have used it. This leads one to believe that de Fermat did not in fact have a complete solution of the problem.

Criticism of proof
Vos Savant wrote in 1995, that Wiles' proof should be rejected for its use of non-Euclidean geometry. She said, "the chain of proof is based in hyperbolic (Lobachevskian) geometry", and because this geometry allows things like squaring the circle, a "famous impossibility" despite being possible in hyperbolic geometry, then "if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat's last theorem."

Proof without elliptic
Where n is known to sum two ordinal values, it cannot exceed the counted value 2 if the larger is taken as 1 unit.

Generalization
Beal's Generalization Conjecture, or the Beal Conjecture, posed by investor Andrew Beal, asks why there are always common factors (like cells in batteries), in equations like this, of the general form
aˣ+bʸ=cᶻ.

References

More reading

Other websites 

Theorems in number theory