The proof for Fermat's Last Theorem: n=5 turns out to be more complicated and more difficult than any of the previous proofs. This is perhaps not too surprising since otherwise, Euler or Gauss would have most likely found the proof.
The final proof was the work of two very talented mathematicians: Adrien-Marie Legendre and Johann Peter Gustav Lejeune Dirichlet. The solution rested on the work done on the Binomial Theorem and Continued Fractions.
Since the purpose of this blog is to present a complete set of proofs, I will need to take a step back from Fermat's Last Theorem and review the developments of these two ideas. The Binomial Theorem was the result of three major mathematicians: Pierre de Fermat, Blaise Pascal, and Sir Isaac Newton. Major work on continued fractions was done by Joseph Louis Lagrange.
I will not attempt to do a complete survey of either of these very broad fields. Instead, I will review the lives of the mathematicians mentioned above and go through the proofs of the major results which are used later in the proof for n=5.