In today's blog, I will show how it is possible to use Gaussian Integers to establish a proof for Fermat's Last Theorem n = 4. Previously, I showed two proofs for this result. I showed a standard proof based on Pythagorean Triples (see here). Also, I showed how it could be derived from Fermat's proof that a right triangle cannot have a square area (see here).
The proof for the case n = 4 using Gaussian Integer is both very strange and very interesting. The proof that I present is taken from Paul Ribenboem's book Fermat's Last Theorem for Amateurs.
Please note that I use Greek letters to represent Gaussian Integers and Latin letters to represent rational integers. This is really to emphasize that reasoning with Gaussian Integers requires different assumptions. For example, the Well Ordering Principle no longer applies and 2 is no longer a prime.
Theorem: x4 + y4 = z2 has no solution in Gaussian Integers where xyz ≠ 0.
(4) Which contradicts step (1) where n ≥ 2. So we reject that there is a solution to FLT for n=4.
Perhaps n is used too many times for different things.
I am not clear on how "n is used for too many different things."
In the proof on this page, n is assumed to have the same value in all cases where it is used.
If you provide additional details, I am glad to review the proof and update it as needed.
"Proof for n=4 using Gaussian Integers"
"ε * λ^4n * α^4 + β^4 = γ^2 where n ≥ 2"
It is obvious that they have different meanings so I'm probably being too pedantic here.
Perhaps λ=1-i should be introduced before the proof starts.
"What on Earth is λ?"
Gaussian Integers: properties of 1-i before the proof of the Thoerem.