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 , and Kummer. 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):
(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.