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 = 7ev
7 does not divide v
gcd(s2 + 3 * 72ev2 + u, s2 + 3 * 72ev2 - 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.