Fermat's Last Theorem: Proof for n=7: x + y + z ≠ 0

In today's blog, I will continue the proof for Fermat's Last Theorem n=7. Today's proof is used in step 3 of FLT n=7 proof.

The details for today are based on Paulo Ribenboim's Fermat's Last Theorem for Amateurs.

Lemma 1: x⁷ + y⁷ + z⁷ = 0, x,y,z integers, then x+y+z=0

(1) Assume that x+y+z ≠ 0

(2) We know that there exists Q,M (see here) such that:
p = x + y + z
q = xy + xz + yz
r = xyz
m = pq - r
Q = q/p²
M = m/p³
M² - M(1 - Q + Q²) + 1/7 = 0

(3) We can then conclude that there exists u (see here) such that:
u² = s⁴ + 6t²s² - t⁴/7
u is a rational number
s/t = 2Q-1
gcd(s,t) = 1
t is greater than 0

(4) We can also conclude that there exists an integer v (see here) such that:
t = 7^ev
7 does not divide v
gcd(s² + 3 * 7^2ev² + u, s² + 3 * 7^2ev² - u) is a power of 2 or it is 1.

(5) But v is neither odd (see here) nor even (see here) so we have a contradiction.

QED