From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: A generalization of FLT. Date: 23 Aug 2000 23:32:49 GMT Newsgroups: sci.math Summary: [missing] In article <8ntbc8$bd4$1@gannett.math.niu.edu>, Dave Rusin wrote: >In article <8nt6as$g4v$1@nnrp1.deja.com>, wrote: >>I believe I discovered the "beal conjecture". > >What does it mean to "discover a conjecture"? I know it's bad form to follow up my own post, but I have been shaking my head in disbelief at this whole thread. After some suggestions I received in email I'm entertaining the idea that part of the difficulty is that mathematicians are, or are perceived to be, unwilling to acknowledge the contributions of amateurs in such things as the making of conjectures. The brief exchange above, for example, suggests we think differently about conjectures. Allow me to present what I think is representative of the professionals' point of view. I don't know that there's a "right" view, but those of us who do this for a living have had to develop a pretty serviceable outlook. I'll use an illustration. "FLT and the Rusin Conjecture" I have long been fascinated by Fermat's Last Theorem and so I have read many partial proofs of it. After reading one by Weierfrich, I am led to make the following conjecture: "If p is a prime then there cannot be four consecutive numbers x for which x^p - x is a multiple of p^2 ." (In fact, the proof I was looking at would consider primes p for which the word "four" could be replaced by something like "160" ! But that need not concern us.) This conjecture is important enough to me that I offer a day's pay (75 cents) to the first person who can solve it. I call it the "Rusin Conjecture". Let us note a few things about this conjecture: (1) It can be stated briefly and in elementary terms; furthermore, (2) Its structure is similar to questions which arose in the famous Fermat Conjecture. Also, a theorem by Fermat states x^p-x is a multiple of p. Taken together, (1) and (2) make this conjecture "natural". (3) It has now been officially "published" as of August 2000. I am fairly sure it has never appeared in print before. Certainly it wasn't in the two books I consulted, and no one else has found exactly this conjecture in print. Of course, if and when they do so, I will gladly stop calling it the Rusin Conjecture. (4) I checked it out empirically (I had to keep the numbers manageable but I took it as far as all p < 1000 and all x < 1000000). So far there are no exceptions. (5) By the time you read this, the following will surely be true: at least one mathematician will refer to this as "remarkable". With these facts before us, I think it is clear that I can lay claim to this discovery. OK, for those who are slow to catch on, all of the above is meant to illustrate the bizarre situation I see unfolding in this newsgroup in exchanges between andybeal@deja.com (assumed to be Andrew Beal) and several others (including me). The whole thing strikes me a bit as a tempest in a teapot, but I hope I can with this example demonstrate the difference in perspectives between practicing mathematicians and Beal (and others?) when we look at conjectures. (I hope the points above capture some of the main ideas behind Beal's position. I'm not intending to be caustic here. I really do appreciate amateurs' enthusiasm for math.) First, let me clarify that the following has in fact been proved, long before Wiles et al finished off the proof of FLT. (About 1911 I think.) Theorem: if p>2 is prime and there are integers x,y,z not divisible by p making x^p+y^p=z^p, then a^p = a mod p^2 for a=2, 3, 4, ... [it's a finite list; it ends around 80-something] So the "Rusin conjecture" arises naturally: it implies there are no prime exponents to which this FLT proof does not apply. Of course, everyone reading this theorem will ask whether there are such primes p or not; finding no small ones people will begin to suspect there are none. AND (and this is important) after applying some informal reasoning known to work "in general", one can convince oneself that it would be a remarkable event to find such a p. (More remarkable or less, depending on how many a's are involved.) In view of these remarks, no self-respecting mathematician would make much of a big deal about the, by now obvious, conjecture that there are no such primes p. Maybe, or maybe not, they would mention a paragraph like the previous one if they discussed the issue in a talk or a publication. Let me also comment on one little matter: a couple of lines back, I managed to sneak the word "remarkable" into my analysis. Experienced number theorists will recognize that this refers to an informal probabilistic argument as I will work out below. Inexperienced readers could be forgiven for thinking that the conjectures being discussed are somehow "remarkable". Frankly I cannot picture what would be a "remarkable conjecture", but to a person not professionally engaged in mathematics, I suppose it might sound plausible that such a thing exists. (After all, there are remarkable theorems, remarkable proofs.) One more thing before I address the "Rusin Conjecture". You'll notice that the _Theorem_ is stated with precision: the proof has to prove some specific thing. On the other hand, the _conjectural_ discussion after it is more informal: maybe there are some p's satisfying the conditions, but only finitely many of them. Or maybe there are no p's satisfying all the congruences, while there are some p's satisfying a few of them. That kind of imprecision is perfectly justified in the exploratory parts of mathematical progress. But it does mean that it's a little hard sometimes to pin down just when a conjecture has first appeared. Maybe someone had a conjecture which turned out to be true, then someone else conjectured a slight generalization; or maybe a first conjecture was false so some conditions were added in the hopes of suggesting something true; or maybe it was eventually discovered that the terms of the conjecture didn't even make sense in the generality first proposed. These things happen. Conjectures evolve. It's important to make them on the way to proving something else. But the conjecture, per se, doesn't have a particularly discrete mathematical existence. Now let's look at my precisely-stated conjecture. Mathematicians will recognize that it states there are no four consecutive p-th powers in the ring Z/(p^2). There can be three in a row of course: -1, 0, and +1 for example. Apart from those three, the other p-3 pth powers appear to be more or less randomly distributed among the possible residues {2, 3, ..., (p^2 - 2)}. Well, it's not "random" at all; I have to emphasize the following reasoning is informal only. In our collective research experience, however, we have found this reasoning to work well to distinguish likely from unlikely conjectures. So you're selecting, at random, about p elements from about p^2 choices. When do you get three or four in a row? Each element has about a 1/p chance of being selected, so if x is selected, x+1 and x-1 are selected with probability about 1/p (each), and so the chance that both are selected is about 1/p^2. Repeat this for each of the p selected ones (that each, each of the p candidates to be the middle x) and you see you have about a 1/p chance of selecting three in a row. So it looks like, for an individual large prime p, it's kind of unlikely that you'll find three consecutive solutions to x^p = x mod p^2 (apart from the {-1, 0, 1} solution which is a given). Indeed, the first value of p for which this does happen is p=59. But wait -- there are infinitely many candidate primes p to look at! If you look at all the primes from e.g. 100 to 200, you have to argue that you have a ~1/101 "chance" that there are three consecutive x's for p=101, AND a ~1/103 "chance" of winning with p=103 AND ... Each lottery ticket is unlikely to win, but you've just bought a couple dozen! As a rough estimate, then, you might think you expected number of primes producing a winner is about 1/101 + 1/103 + ... + 1/199. Well, that comes to about 0.14, which is to say it's still sort of unlikely that we'll "win" with primes in this range. Continuing further, though, we see the number of "winning" primes less than N is very roughly the sum of all 1/p for primes less than N. Now, there's a way to estimate such sums: you sum up some numbers 1/n where the n's are integers selected randomly but distributed as densely as the primes. Well, about 1/log(n) of the numbers near n are prime (proved in 1899) so we sum (1/n)*(1/log(n)) over all integers instead of summing 1/p over all primes. NOW we have a sum we can handle with easy rigour: from calculus we learn this sum is about log(log(N)). So the number of primes p < N with three (or more) consecutive nontrivial solutions to x^p=x mod p^2 ought to be about log(log(N)). IN PARTICULAR: Taking ever larger N's, we ought to find infinitely many such primes p! I have to stress that this is _not_ a proof, but a _plausibility argument_. Returning to the "Rusin conjecture", we see that the existence of _four_ solutions in a row will require one of two things: either we extend the trivial solutions, requiring {-1,0,1,2} all to be solutions to x^p=x mod p^2, or we find _four_ x's in a row which are nontrivial pth powers mod p^2. The latter event we can estimate as above as happening with probability about 1/p^2. But if we sum over all primes p, we now get a sum which CONVERGES. And if we start this argument only with large primes p, it will converge to something very small. In short: if we don't find such a pattern for a small prime p, it is _very unlikely_ that there is any at all. Well, a straightforward check rules out all the primes less than 1000, so we're pretty confident the conjecture is true. It's this kind of analysis which gives some support to the conjecture, turning it into something "genuine" instead of merely a wild guess. (Again, this is not a proof, but a heuristic.) So this kind of heuristic supports the Rusin conjecture. Unfortunately, it is deliberately incomplete: I have "forgotten" the other possibility, that 2^p = 2 mod p^2. Whenever this holds, we'll get four solutions in a row to x^p = x mod p^2. You see where I said (twice now) that I checked all p<1000? That's true, I did. But I deliberately stopped there because I know that 2^1093 = 2 mod 1093^2. Aargh! My conjecture is false! (At least that saves me the 75 cents :-) ) I arranged it this way to show why mathematicians discount statements like (4) above, similar to one made here by Beal. The fact that a statement has been checked to "a good range" is sort of useless if there is no understanding of what happens to the rest of the infinitely many remaining cases. Comforting, perhaps, but not really valuable. As a matter of fact, no one knows for sure whether or not there are infinitely many primes p with 2^p = 2 mod p^2. (Only p=1093 and p=3511 are known and p's up to the zillions have been checked.) IF we use again the same probabilistic arguments as before, we can justify the conjecture that, in fact, there ARE infinitely many p's satisfying the condition. (So the Rusin Conjecture fails for infinitely many primes. Boy was I "wrong".) If you are curious why we can't even find 3 when there are "expected" to be infinitely many, let me just point out that when God looks at the complete list of these p's, we expect that the _number of digits_ in these p's grows exponentially as we read the p's in order! It's not like you're going to stumble on the next one any time soon. Anyway, statements like (4) can certainly be misleading. So what's my point? I guess I'm trying to explain why it can be that mathematicians are so nonplussed when confronted by someone who has expressed a genuine open question in mathematics. It's not that having open questions is not important. It's just that the process of coming up with questions which guide us to significant and useful parts of mathematics is really subtle. Amateurs probably resent the perceived cold shoulder they get from working mathematicians; it's not intended, but if it is there, it's in part a result of the converse perception by mathematicians that amateurs underestimate the subtlety of their work. The arguments above, for example, are considered mind-bogglingly trivial. (I think I know how cancer researchers feel when amateurs want to tell them about their Grapefruit Diet, for example.) You want to make conjectures? Fine. Name them after yourself? Fine. Expect mathematicians to appreciate your conjecture as a contribution to their work? Well, you've got to know something about what their work _is_ to get that. dave (Busy avoiding all kinds of work :-( )