n=3일 때의 페르마의 마지막정리

http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-proof-for-n3.html

Fermat's Last Theorem: Proof for n=3

Leonhard Euler came up with two proofs for Fermat's Last Theorem: n = 3.

One proof involved a very innovative method using irrational numbers. Unfortunately, Euler made a mistake in his proof. Despite this, his method revealed a very promising approach to Fermat's Last Theorem which was later taken up by Gauss, Dirichlet , andKummer. I discuss the details of this method and Euler's mistake in another blog.

The other proof is less generalizable but still brilliant. This is the proof that I will present in today's blog.

The details of this proof are based largely on the work by H. M. Edwards in his book:Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

Theorem: Euler's Proof for FLT: n = 3

x³ + y³ = z³ 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. [See here for the proof]

(3) First, we observe that there must exist p,q such that (see here for proof):

(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. (see here for proof).

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

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

(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.

QED