I am not a mathematician. Probably this is an incredibly obvious question, but I have never read anything about it. Likely because I can’t ask the right question to find an explaination. So I turn to the Teeming Millions:
Fermat’s Last Theorem states that no three positive integers a, b, and c can satisfy the equation a[sup]n[/sup]+ b[sup]n[/sup]=c[sup]n[/sup] for any integer value of n greater than two.
a[sup]2[/sup] + b[sup]2[/sup]= c[sup]2[/sup] having solutions is known to all and sundry. Fermat’s theorem states there are no solutions for a[sup]3[/sup] + b[sup]3[/sup]= c[sup]3[/sup] etc etc.
My question is, why isn’t the progression seen as:
Which I guess would be the equivalent of a four-dimensional right triangle, whatever you’d call it.
I am not claiming to have discovered a previously unknown principle of math. Instead, I am wondering why I never read anything about it. It seems like explaining it this way would help explain FLT. I don’t even know the terminology for the question. What do you call a general equation of the form a[sup]2[/sup] + b[sup]2[/sup]= c[sup]2[/sup]? (I DID pay attention in math! Honest.)
I know I am running the risk of looking incredibly stupid with this question, but curiosity overcomes my shame at missing something important and/or obvious. Besides, none of you know me.
The simple non-expert answer to the OP is that while that other type of equation is interesting and exhaustively studied, it has nothing directly to do with the question asked by Fermat. They are different types of equations requiring different techniques of analysis.
This is a very minor point of Robert Forward’s novel Rocheworld (AKA Flight of the Dragonfly). Astronauts from earth talk to intelligent (but not industrial) aliens who are brilliant mathematicians and describe Fermat’s Last Theorem. The aliens say that a more sensible result is you increase the number of terms along with the exponent. But as I recall they do offer a proof of the Theorem after some thought. The novel was written before the current inelegant proof.
I think the reason you have never seen anything about this is because from a mathematical standpoint, it’s just not interesting.
I don’t mean to say that you are not interesting. It’s just that the more complex the formula, the more likely it has some solution. Fermi deliberately chose the three terms because it was simple enough to be interesting.
I don’t see how it isn’t interesting that adding one extra term is enough to change the game. I’m not saying that it would be important to the math, but I think it would be useful in explaining the problem.
Honestly, I don’t see how this has a factual answer.
I think the problem here is in understanding the fundamental nature of math. There is no one single question that is the question to ask. There are always a multitude of potential questions one can ask, some of them more interesting than others, and opinions on which are more interesting will vary from person to person. Fermat happened to have been interested in the question of a^n + b^n = c^n. Why? Does it matter? And a lot of other people have since become interested in the question. Again, it really doesn’t matter why people are interested: They are, and so they study it.
Maybe in that sense, it shouldn’t have been in GQ. I debated before starting the thread.
I was hoping there was at least a starting point, like maybe ‘what you want is basic Fendel equation theory’, and then I read about it and say “OK, now I understand.” It’s not that I am looking for an ‘answer’, so much as direction.
But as I noted, I think my question is too simple. I think I am looking for a ‘rule’ or a theory that isn’t there because there isn’t anything surprising about the math. It just seems to me like there should be.
I once heard John Conway describe what makes a sequence interesting. He said it is interesting if there is a simple rule to generate the next term and there are easy to state, but hard to prove theorems about the sequence. Fermat’s Last Theorem is interesting because it is easy to state and hard to prove, compounded by the fact that he left a note saying he’d found a simple and elegant proof (which he almost certainly had not found).
You partially answered your own question: We know that those equations do have solutions. This general type of question is called Diophantine analysis, and the fields of math dealing with it are algebraic number theory, analytic number theory, Galois theory, algebraic geometry, arithmetic geometry, etc. It’s a broad field in which the questions are usually very difficult to answer, but one of the most common is something along the lines of, “For polynomial equations of a certain ‘type’ over a number field k*, how many solutions are there?”At the very least, we’d like to know whether there are no solutions at all, only finitely many, or infinitely many. For specific cases, like elliptic curves, we can ask more detailed questions about the structure of the space of solutions. Along similar lines, you can consider the Waring problem: For each n, is every integer the sum of some bounded number of nth powers? (The answer is yes. It’s not that hard to prove, but getting explicit bounds on the required number of *n/I]th powers is tricky.)
To give you an idea of the flavor of some of the results, consider the following statement for a fixed r:
In other words, if you have enough variables (with the cutoff depending on the degree of the polynomial f) , then there has to be a solution somewhere. The details are complicated and technical, but you can that prove that certain fields have property (C[SUB]r[/SUB]) and certain others do not. Why pick out this particular property? Again, the details are very complicated, but there are two reasons: It’s useful in proving other results, and it’s a result that’s amenable to some of the machinery in (in this particular case) algebraic geometry.
To answer your specific question: Since no one could find any solutions to the polynomial in Fermat’s last theorem, it was natural to ask if there were any at all. What was so tantalizing about it is that there were so many partial results that could be proved. It’s not hard to show (at least with modern techniques), for example, that there are only finitely many solutions for a given n. It’s easy to show that we can reduce to the case where n is prime, and one of the earliest major results was to prove FLT for ‘regular’ primes. (It’s thought that about 60% of primes are regular, though I don’t think it’s been proved.) Most interestingly, it was proved in the 1980s that FLT followed from the Taniyama-Shimura-Weil conjecture, which is an incredibly important result about elliptic curves and of far greater importance than FLT itself. (That conjecture is actually what Wiles proved.) And, of course, it’s interesting the Fermat claimed to have a proof of his statement, although that’s certainly considered nonsense now.
It’s not a stupid question at all.
If you’re unfamiliar with the idea of a number field, take it to mean the rational numbers. (Number fields seem to be the right thing to consider in these sorts of problems, at least as a first step. The questions are trivial in the complex case, and we want to avoid complications that arise in finite characteristic. There’s usually a lot of Galois theory involved, and we want to avoid separability issues.)
** A homogeneous polynomial is one in which every term has the same (total) degree: the polynomial x[SUP]3[/SUP] + yz[SUP]2[/SUP] + z[SUP]3[/SUP] is homogeneous of degree 3, and the polynomial x[SUP]3[/SUP] + yz is not homogeneous.
Wait, is it obvious that for every (positive integer) n, one can find a sum of n many nth powers (of positive integers) which is itself an nth power? It’s not obvious to me.
Where on Wikipedia does it say that? All I’ve found is that Euler’s sum of powers conjecture is open for n >= 6, but this has to do with whether one can find a sum of less than n many nth powers which is itself an nth power, and thus does not directly answer my question.
At any rate, the progression in the OP is potentially interesting (just, a different interesting question than Fermat’s Last Theorem), and whether it continues or not is apparently non-obvious (even though the OP may be rashly presuming it must).
Indeed, we have that the sum of the 7th powers of {127, 258, 266, 413, 430, 439, 525} is the 7th power of 568 and the sum of the 8th powers of {90, 223, 478, 524, 748, 1088, 1190, 1324} is the 8th power of 1409, so those instances at least are not open.
(It does seem from this site that no solutions are known for n = 6, nor for n from 9 to 32 (and presumably not for any larger n either, but I am reticent to jump to conclusions).)
