The proof for Fermat's Last Theorem n=5 came in 1825 thanks to Johann Dirichlet and Adrien-Marie Legendre. It would take 14 years before the proof for n=7 was offered by Gabriel Lamé. It is complicated and uses methods which do not seem generalizeable. This would be the last major proof for Fermat's Last Theorem before Ernst Kummer's breakthrough.
Today's blog is based on the proof offered in Paulo Ribenboim's book Fermat's Last Theorem for Amateurs.
Theorem: if x,y,z are integers, then x7 + y7 = z7 → xyz = 0.
(1) Let ζ = (-1 + √-3)/2
(2) Assume that xyz ≠ 0
(3) xyz ≠ 0 → x + y + z = 0 [ See here for proof ]
(4) x + y + z = 0 → x,y,z are proportional to ζ or ζ2 [ See here for proof ]
(5) But if x,y,z are proportional to ζ or ζ2, then they cannot be integers since:
(a) Let y = xζ where ζ is irrational and x is an integer.
(b) Assume that y is an integer.
(c) Then ζ = y/x where both y,x are integers.
(d) But then ζ by definition is rational which is a contradiction since we know that it is irrational. Therefore, we must reject (b).
(6) So we have a contradiction and we reject #2.