The problem is prove $x^3 + y^3 = z^3$ has no positive integer solutions. I understand this was proven in the 1700s by Euler, but I cannot find any books or in-depth references on the proof.
Here are some proofs I have tried to follow:
In your professional opinion, what would be the first step to understanding the proofs linked above? Do you know of any books or other references that go into more detail on this problem?
Also this is one attempt I made:
Starting with assuming $x^3+y^3=z^3$ has a solution such that $xyz\neq0$ and x,y,z are co-prime.
$\therefore z^3-x^3-y^3 + (x+y-z)^3 = (x+y-z)^3$
I believe this simplifies to:
But aren't these factors also co-prime and therefore not cubic?