Fermat's
Last Theorem (2/2)

Last month,
we discussed the Fermat's Theorem and the prime number stuff and it was
fun. Slowly, the discussion died out and was replaced with the voting issue.
Sad, isn't it?

Few days after
the [SCIENCE FOR EVERYONE] FERMAT'S LAST THEOREM article was sent out last
month, anh Phuong sent the news that the Fermat's last theorem problem
was solved by Andrew Wiles and a colleague (I can not remember his name).
What I heard at that time was that one of Wiles' colleague said that Wiles'
new proof had no hole in it. I haven't heard any more news about that since.
I'm wondering if Wiles' new proof has been published, or if the whole mathematical
community has accepted the new proof, or if anyone has found new holes
in Wiles' new proof. Do any of you hear any thing about that?

In that [SCIENCE
FOR EVERYONE] FERMAT'S LAST THEOREM article, I had three bonus questions
and hoped that someone would be interested in the rewards (pho+? ta`u bay)
and would answer them. I did not receive any answer to any of those three
questions. It seems that many of us are more interested in politics than
science. How sad! Is there a way to change that?

Anyway, here
are the three questions and the answers.

Q1/ Find a
counterexample to Fermat's Last Theorem.

A1/ If Wiles'
new proof is proved to be solid then the problem is solved. However, if
it is not then shall we resume the discussion and try to solve it numerically?

Q2/ a^n+b^n+c^n=c^n.
It is trivial for n=1. For n=2 then 1^2+2^2+2^2=3^2. For n=3 then 3^3 +
4^3 + 5^3 = 6^3. Now try to find n>3 that would satisfy the equation.

A2/ Yes, there
are some n>3 that satisfy the equation. You probably can find them (using
computer) in a second if you put your mind to it. Example: 95800^4 + 217519^4
+ 414560^4 = 422481^4 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4

Q3/ There
are a^2+b^2=c^2 and a^3+b^3+c^3=d^3. Is there a^4+b^4+c^4+d^4=e^4?

A3/ Yes. Ex:
30^4 + 120^4 + 272^4 + 315^4 = 353^4

And here is
one involving with 5th power 27^5 + 84^5 + 110^5 + 133^5 = 144^5

Now, here
is a new question for those who are interested in this matter.

Is there any
kind of relationship between Fermat's last theorem (a^n+b^n=c^n) and the
other equations (a^n+b^n+c^n=d^n or a^n+b^n+c^n+d^n=e^n or...)? If there
is, how do we find them and can we use them in proving or disproving Fermat's
last theorem? Happy number crunching, everyone!!!