only to expert (god mode )?

Question

pls derive Fermat's Last Theorem. I want to know

Answer 1

Fermat's Last Theorem states that
x^n + y^n = z^n

has no non-zero integer solutions for x, y and z when n > 2.

His proposition was about an equation which is closely related to Pythagoras' equation. Pythagoras' equation gives you:

x^2 + y^2 = z^2

You can ask, what are the whole number solutions to this equation, and you can see that:
3^2 + 4^2 = 5^2
and
5^2 + 12^2 = 13^2

And if you go on looking then you find more and more such solutions. Fermat then considered the cubed version of this equation:
x^3 + y^3 = z^3

He raised the question: can you find solutions to the cubed equation? He claimed that there were none. In fact, he claimed that for the general family of equations:
x^n + y^n = z^n where n is bigger than 2

it is impossible to find a solution. That's Fermat's Last Theorem.


Proof :

n = 3

x^3 + y^3 = z^3 has integer solutions -> xyz = 0

(1) Let's assume that we have solutions x,y,z to the above equation.

(2) We can assume that x,y,z are coprime.

(3) First, we observe that there must exist p,q such that

(a) gcd(p,q)=1
(b) p,q have opposite parities (one is odd; one is even)
(c) p,q are positive.
(d) 2p*(p2 + 3q2) is a cube.

(4) Second, we know that gcd(2p,p2+3q2) is either 1 or 3.

(5) If gcd(2p,p2+3q2)=1, then there must be a smaller solution to Fermat's Last Theorem n=3.

(6) Likewise, if gcd(2p,p2+3q2)=3, then there must be a smaller solution to Fermat's Last Theorem n=3.

(7) But then there is necessarily a smaller solution and we could use the same argument on this smaller solution to show the existence of an even smaller solution. We have thus shown a condition of infinite descent.


I can not have much explaination about in general n. Anyways, that was a real nice question.

I read this somewhere, its not my creation.

All the best and have nice time

Answer 2

In problem II.8 of his Arithmetica, Diophantus asks how to split a given square number into two other squares (in modern notation, given a rational number k, find u and v, both rational, such that k2 = u2 + v2), and shows how to solve the problem for k = 4. Around 1640, Fermat wrote the following comment (in Latin) in the margin of this problem in his copy of the Arithmetica (version published in 1621 and translated from Greek into Latin by Claude Gaspard Bachet de Méziriac) :

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. (It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)

In modern notation, this comment corresponds to the theorem mentioned above. Fermat's copy of the Arithmetica has not been found so far; however, around 1670, his son produced a new edition of the book augmented with comments made by his father, including the comment above which would be known later as Fermat's Last Theorem.

In the case n = 2, it was already known by the ancient Chinese, Indians, Greeks and Babylonians that the Diophantine equation a2 + b2 = c2 (linked with the Pythagorean theorem) has integer solutions, such as (3,4,5) (32 + 42 = 52) or (5,12,13). These solutions are known as Pythagorean triples, and there exist infinitely many of them, even excluding trivial solutions for which a, b and c have a common divisor. Fermat's Last Theorem is a generalisation of this result to higher powers n, and states that no such solution exists when the exponent 2 is replaced by a larger integer.

Answer 3

Pls refer to the following links. By the way, the proof requires at least graduate level geometry and algebra to understand.

Wiles lengthy version
http://math.stanford.edu/~lekheng/flt/wiles.pdf

Shorter version
http://www.ams.org/notices/199507/faltings.pdf

Answer 4

Pierre de Fermat's

Click on the URL below for additional information concerning Pierre de Femat's

en.wikipedia.org/wiki/Fermat's_Last_Theorem

mathworld.wolfram.com/FermatsLastTheorem.html