ANOTHER ATTEMPT TO DISPROVE FLT65 FAILS

I thought those of you who have followed my posts on Fermat's Last Theorem might be interested to know that another university math professor has tried to disprove my 1965 proof of Fermat's Last Theorem. I'm posting it and my reply below; but I am witholding the names because I don't want to embarass anyone.

I received this email message from a friend in ISPE who doesn't believe my 1965 proof is valid:

Dear Ed,                                                                                  June 2, 2017

I tried to send this note to you on May 31. I probably used the wrong e-mail address. I’m trying again with this address.

Because of a common interest in music for big jazz bands, I recently made friends with a retired professor who taught mathematics in a university for years. Even more recently, I showed him a copy of FLT65. He agrees with my claim that FLT65 is flawed, but this note is not just to tell you that. I want to give you his explanation.

He has shown me what purports to be a more rigorous way to describe what he and I see as the flaw. My “arm-waving” description of it never convinced you, but perhaps his approach will be more persuasive. In any case, I think that you should see it, so I’ve pasted it onto this note below.

                        Sincerely, _______

I think I may have a useful thought for you on FLT65.  Your criticism of the proof is, I believe, exactly correct.  Here is a possible way of putting it that might convince Mr. Close.  In his argument he sets  a = Z - A   and considers the divisor polynomial  g(Z) = Z - a, which he says is a polynomial of degree  1  in Z.  But  g(Z) = Z - a = Z - (Z - A) = A,  which is not of degree  1  but of degree 0.  When the divisor A  is a (nonzero) constant, the polynomial division algorithm over the reals just says there exists a (unique) polynomial  q(Z)  such that  f(Z) = Aq(Z) + 0, where, of course, q(Z) = f(Z)/A.  so there is no  f(a).

My Reply, sent  7 AM, June 3, 2017:

Dear _____,

It's nice to hear from you; I hope you are doing well. We're leaving tomorrow for a conference in Chapel Hill NC, and we're busy packing today, so I'll have to be brief. I've seen this argument a couple of times before. One time was from a university prof considered by many to be one of the best in number theory. It only took a few lines to reveal his error. FYI, he acknowledged his error, and could never produce a disproof of FLT65, but he still felt that it must be flawed! I’m sure you will agree that mathematics should not be about feelings or beliefs.

Think about it for a minute. If you argue that z - a is not a 1st degree polynomial, you destroy the division algorithm for all polynomials. Yes, for specific values of Z and a, Z - a is a constant. But that is true for any and every polynomial. And the variables of any polynomial factor of any equation will take on specific values determined by the solutions of that equation.   

Your friend also says "the polynomial division algorithm over the reals just says there exists a (unique) polynomial  q(Z)  such that  f(Z) = Aq(Z) + 0, where, of course, q(Z) = f(Z)/A.  so there is no  f(a)".

This is a fine example of circular reasoning! When you set Z - a equal to a constant, which you can do for any polynomial, this statement is true, but, if it disproves FLT65, then it means that every polynomial is divisible by Z - a because every polynomial has solutions for which z - a will be constant. The point of the division algorithm is that f(Z), defined for all real numbers, is only divisible by Z - a when the remainder disappears, i.e. f(a) = 0. And the point of FLT65 is that because of the unique form of the Fermat equation’s factor, f(a) can never equal zero, and integers are reals, as stated in FLT65, a fact that no one disputes. 

All the Best,

Sincerely,

Ed Close

NOTE (Not a part of the exchange copied in above.): The argument put forth by most who don't want to believe that FLT65 is actually a valid proof, is that the division algorithm works for polynomials of a continuous variable, but may not work for integers. This is handled in the 1965 proof by noting that the division algorithm holds over the field of real numbers, and integers are a set included in the field of real numbers. In the case of X to the nth power plus Y to the n power equals Z to the n power, when n = any prime number greater than 2, (the equation of Fermat's Last Theorem), no one has proved otherwise.