h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

This a Fermat's Last Theorem question. Well, actually, it's more of a statement than a question. Wiles appears to have taken the starch out of most amateurs (a great relief to the professionals, no doubt). I still see crackpot proofs on the Internet though. Most of these "proofs" involve manipulating the binomial theorem, extrapolating from Pythagorean tripes, or conjecturing on what Fermat might have had in mind. (My own belief is that there is no elementary proof of FLT and that you have to use cyclotomic fields or some such sophisticated mathematics to even scratch the surface of it.) My own pet theory is that you can extrapolate from binary quadratic forms (for example,(a**p+b**p)/(a+b) or (a**p+b**p)/(a+b)/p when p=3)to at least figure out to tackle FLT along classical lines. After all, there exist a and b such that (a**p+b**p)/(a+b) or (a**p+b**p)/(a+b)/p are pth powers when p=3. (I'm surprised that someone hasn't come up with a "proof" along these lines. Maybe they have and I don't know about it.) I don't think this is a totally preposterous approach; after all elliptic curves are cubic in nature. So, my question is; is there anyone interested in this?

>>> So, my question is; is there anyone interested in this?

Sorry, I'm afraid I didn't follow you. Does the "this" in your your question refer to the history of the search for a proof of Fermat's Last Theorem? Or guesses about the proof Fermat decided he could not fit into the margin?

In either case, may I recommend:
Fermat's Last Theorem for Amateurs
by Paulo Ribenboim



HN

There exist a and b such that (a**p+b**p)/(a+b) or (a**p+b**p)/(a+b)/p is a pth power when p=3 and "propositions" can be derived from examining this data (I've collected all solutions for 3,000,000>a>b>0); I was referring to generalizing these empirically derived propositions to p>3. Sorry about the confusion; I've found that what seems clear to me doesn't always come across very well. You may remember from Barlow's formulas (discussed in Ribenboim's "Fermat's Last Theorem for Amateurs")that (a**p+b**p)/(a+b) and a+b must be pth powers when a**p+b**p=c**p, p does not divide c, or (a**p+b**p)/(a+b)/p and p(a+b) must be pth powers when a**p+b**p=c**p, p divides c. So, in a sense, investigating a and b such that (a**p+b**p)/(a+b) or (a**p+b**p)/(a+b)/p is more basic than Fermat's Problem. So I'm trying to do a little more than duplicating the proof of FLT (plus, I don't like the elliptic curve approach [mainly because I know nothing about it] and prefer the classical approach using cyclotomic fields). Of course, I'll never succeed in proving FLT or any variants thereof; it's way out of my league. Although I have a degree in mathematics (and graduated summa cum laude), I don't have the talent to play with the big boys. I'm just trying to impart an approach to FLT that I've discovered (none of the mathematical journals will touch my stuff since it's empirically derived). It's funny that you should mention this book (I have a copy); I'd skimmed through it before and hadn't found much of interest except Fermat's Congruence, Wieferich's criterion, and Vandiver's criterion (and of course the epilogue where elliptic curves is discussed). Now I see that I've duplicated some work on Fermat's Congruence (about the probability that there exist solutions of a**p+b**p=c**p(mod p**2), p does not divide abc). If you're still interested, you can see my work at "http://www.planetmath.org" (look under the "papers" for "dkc" [you don't have to be a member]) or you can look at my website at "http://home.graysoncable.com/dkcox". There's a simple approach to Wieferich's criterion that should be accessible to everyone (all that is required is a basic knowledge of congruences). At the time (1909) almost no one could understand Wieferich's proof. One of Ribenboim's other books (besides his book "Classical Theory of Algebraic Numbers") is "13 Lectures on Fermat's Last Theorem". This is more advanced than "Fermat's Last Theorem for Amateurs", but it has some good stuff in it. (Unlike some other amateurs, I decided to find out what had been done before I charged off and tried to re-invent the wheel. Duh!)