Question

Fermat's Last Theorem 3 Proof help

Answer 1

Proof:

Answer 2

Answer the following:
1. From Part 2, prove that gcd(x0, y0) = 1.
2. In Part 3 the author writes “If z0 is even then x0 and y0 are odd.” What justifies this statement?
3. In Part 3, find expressions for x0 and y0 in terms of p and q.
4. From Part 3, use gcd(x0, y0) = 1 to prove that gcd(p, q) = 1.
5. From Part 3, prove that p not≡ q (mod 2).
6. From Part 3, prove that z30 = 2p(p2 + 3q2).
7. From Part 4, prove that when z0 is odd, there exist p and q so that (a) gcd(p, q) = 1 (b) p not≡ q (mod 2) (c) 2p(p2 + 3q2) is a cube (Hint: begin with “Since z0 is odd, exactly one of x0 or y0 must be even. Without loss of generality, suppose x0 is even.”)
8. In Part 7, what mathematical statements should appear in the first two blanks and what proposition should be cited in the third blank?
9. In Part 7, verify that 2p = (2c)(c − 3d)(c + 3d).
10. In Part 8, verify that x^3 + y^3= z^3.
11. In Part 8, justify that z1 < z0.