Fermat's Last Theorem: Proof for n=5

For those interested in the history behind this proof, you may want to start here.

The proof presented is based on two books: Harold M. Edward's Fermat's Last Theorem: A Genetic Introduction and Paulo Ribenboim's Fermat's Last Theorem for Amateurs.

Theorem: Proof for FLT: n=5

x⁵ + y⁵ = z⁵ → xyz = 0 if x,y,z are integers.

(1) Assume there is a solution where xyz ≠ 0

(2) From previous results, we know that we can assume that gcd(x,y,z)= 1 (see here).

(3) We can also assume that x,y are odd and z is even since:

(a) We know that at least two of the values are odd since gcd(x,y,z)=1 tells that only 1 at most can be even.

(b) If two are odd, then the third is even since odd + odd = even and odd - odd = even.

(c) If z is even we are good so let's assume that x is even.

We know that there exists z', x' such that z' = -z and x' = -x so x' is even:

(-1)⁵(x')⁵ + y⁵ = (-1)⁵(z')⁵

First, we add (z')⁵ to both sides to give us:

(-1)⁵(x')⁵ + y⁵ + (z')⁵ = 0.

Then, we add (x')⁵ to both sides to give us:

y⁵ + (z')⁵ = (x')⁵

Even in this case, we have derived a form z⁵ = x⁵ + y⁵ where z is even.

(4) We can assume that 5 divides xyz from Sophie's proof.

(5) Now, it can be shown that if 5 divides z, then there is no integer solution (see here for proof) and likewise, it can be show if 5 doesn't divide z, then there is no integer solution (see here for proof).

(6) So we have a contradiction and can reject our initial assumption.

QED