Saturday, June 18, 2005

Proof for n=4 using Gaussian Integers

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.

(1) First, I will show that if there is a solution in rational integers, then the following is true:

There exists α, β, γ, ε, λ such that:
ε * λ4n * α4 + β4 = γ2 where n ≥ 2. [See here for proof.]

(2) Second, I will show that if (1) is true, then there is another set of values: ε1, α1, β1, γ1 such that:

ε1 * λ4(n-1) * α14 + β14 = γ12. [See here for proof.]

(3) But then, repeating step (2) again and again, we eventually get to values: αi, βi, εi, γi where

εi * λ4i4 + βi4 = γi2

(4) Which contradicts step (1) where n ≥ 2. So we reject that there is a solution to FLT for n=4.



Scouse Rob said...

Perhaps n is used too many times for different things.
Could be confusing.

Beautiful Proof.



Larry Freeman said...

Hi Rob,

Thanks very much for your comment.

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.



Scouse Rob said...

"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.


Scouse Rob said...

Perhaps λ=1-i should be introduced before the proof starts.

Currently following the path through the proof leads you to step (5) in the first linked lemma:

(5) We know that λ divides either α or β

Which made me think:
"What on Earth is λ?"

Then you have to dig though the links:
i-1 and Fermat's Last Theorem n=4

Then again to:
Gaussian Integers: properties of 1-i

This just seems to flow wrong.(Although in chronological published order it flows perfectly.)

I would suggest introducing the reader to λ through a link to
Gaussian Integers: properties of 1-i before the proof of the Thoerem.