From: Richard Carr Subject: Re: A generalization of FLT. Date: Fri, 4 Aug 2000 01:03:36 -0400 Newsgroups: sci.math Summary: [missing] On 4 Aug 2000, Bill Taylor wrote: :Date: 4 Aug 2000 03:38:30 GMT :From: Bill Taylor :Newsgroups: sci.math :Subject: A generalization of FLT. : :I just saw this in a newly-published book, (forgotten the name and author, :of course - it's down in our library display), and thought I'd mention it. : : :Apparently there are NO natural number solutions to : : a^j + b^k = c^n, with j,k,n > 2, and NOT necessarily equal. : 2^n+2^n=2^{n+1}. : :Seemingly the same sort of probabilistic evidence applies to this as it did :to regular Fermat. Interestingly, a small number of cases are known where :one of j,k is allowed to be 2, but I gathered that even these are thought :to be finite in number. It is all closely connected to the ABC conjecture. : :That's all folks. : :------------------------------------------------------------------------------ : Bill Taylor W.Taylor@math.canterbury.ac.nz :------------------------------------------------------------------------------ : Sweeping generalizations are always wrong. :------------------------------------------------------------------------------ : ============================================================================== From: andybeal@my-deja.com Subject: Re: A generalization of FLT. Date: Sun, 20 Aug 2000 15:03:48 GMT Newsgroups: sci.math In article , Gerry Myerson wrote: > In article <3999885A.EBD6F55D@home.com>, Jon and Mary Frances Miller > wrote: > > => If Beal had shown a lot of interest before offering the prize, the > => name of the conjecture would (to me) not be offensive, merely wrong. Apparently some misconception exists about my discovery of the "beal conjecture" relationship. It is the discovery, not the posting of the prize that is significant. It is also unfair to suggest that I am self promoting in offering the prize money. I made the discovery and widely disseminated it years before a UNT professor suggested an interest in writing an article about it. The fact that I have not even published some related discoveries is also inconsistent with "self promotion". The relationship was clearly unknown prior to my work and discovery of it in 1993. This fact remains true despite Andrew Granviles assertion that he once asked a class of his to look at solutions for similiar equations, and despite Alf Vanderportens assertion in the first edition of his 1996 book that he was "proposing" the conjecture (when informed of his error, Vanderporten changed the assertion in subsequent editions). I received responses in 1994 from leading mathematicians that ackowledged the relationship as unknown. That Vanperporten would even suggest in 1996 that he was "proposing" the relationship is compelling that Vanderporten must agree that it was unknown prior to my 1993 discovery of it. What is hard to understand is who is posting these critical e-mails and why??? best regards --andrew beal Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Hull Loss Incident Subject: Re: A generalization of FLT. Date: Mon, 21 Aug 2000 17:31:16 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > It is also unfair to suggest > that I am self promoting in offering the prize > money. Fair or not it appears to be true. That impression is reinforced by your publicly insisting on priority, in the face of polite attempts by experts such as Granville to disabuse you. > I made the discovery and widely > disseminated it years before a UNT professor > suggested an interest in writing an article about > it. When did you make the discovery, and how did you disseminate it? By the way, the article by the UNT professor (not a number theorist) mentions that you fund scholarships at the UNT math department, and that professor also sits on and represents the committee adjudicating your prize fund. (UNT = univ. of North Texas. I'm sure if you had asked the number theorists at UT Austin they could have set you straight immediately.) > The fact that I have not even published some > related discoveries is also inconsistent > with "self promotion". First, it is extremely consistent. If the discoveries are of the same nature as the Beal Conjecture itself, they have very little self-promotion potential, and would be unpublishable in most non-obscure math journals. That you "disseminated it widely for years" without generating much interest, already indicates what the promotional value of the discoveries was on their own, sans prize. The prize on the other hand generated publicity and the fawning AMS article, and just may succeed in attaching your name to a bit of history. Second, since you did not publish, consider how many number theorists also worked on and compiled computer data about standard conjectures such as yours without publishing, long before you studied the problem. Euler's problem on sums of N'th powers is a test case for the heuristics that predict your conjecture, and was the subject of computer calculations in the 1960's and theoretical work in the 70-80's, and that's just the published stuff. It is extremely unlikely people did not try out computationally easier case of the generalized Fermat equation X^a + Y^b = Z^c. Did they publish about it? Maybe not, and it wouldn't necessarily have been notable if they had. > The relationship was clearly unknown prior to my > work and discovery of it in 1993. It was very well known by probabilistic arguments, that if you look at FLT-like equations f(X)+g(Y)=h(Z) where f,g,h are polynomials of degrees a,b,c, you expect many/few/no solutions according to whether 1/a + 1/b + 1/c is greater/equal/smaller than 1, and that this parallels the famous trichotomy of genus 0/1/higher for algebraic curves. This includes your conjecture and much more, and goes back to the 1920's or earlier. > This fact > remains true despite Andrew Granviles assertion > that he once asked a class of his to look at > solutions for similiar equations, i.e. it was standard enough to make a student exploration. That says a lot already. > and despite Alf > Vanderportens assertion in the first edition of > his 1996 book that he was "proposing" the > conjecture (when informed of his error, > Vanderporten changed the assertion in subsequent > editions). He may have been the first to state it in print, but it was folklore for years. I doubt he was under any illusions as to the novelty. > I received responses in 1994 from > leading mathematicians that ackowledged the > relationship as unknown. Can you quote any leading number theorist who unambiguously supports the claim that you made a discovery that was either significant or not already well-known? > That Vanperporten would > even suggest in 1996 that he was "proposing" the > relationship is compelling that Vanderporten must > agree that it was unknown prior to my 1993 > discovery of it. More likely, Van der Poorten meant to draw attention to a specific FLT-like problem as a new challenge, even though it was well-known to insiders as a plausible conjecture. Van der Poorten has posted to sci.math before, and I think Granville may have also visited at one point. It would be interesting to see whether they agree with your interpretation of their statements. ============================================================================== From: andybeal@my-deja.com Subject: Re: A generalization of FLT. Date: Tue, 22 Aug 2000 03:47:02 GMT Newsgroups: sci.math In article <39A19FA4.4C8AEEF2@y.z.com>, Hull Loss Incident wrote: > andybeal@my-deja.com wrote: 1.) First you suggest that the beal conjecture is so trivial as to be beneath you - then you suggest that I may attatch my name to a piece of mathematical history - which is it?? 2.) Granville made no "polite attempts" to "disabuse" me. Granville was abusive, ill informed, and totally unable to demonstrate any prior knowledge, despite claims to the contrary. 3.) It was not known previously by "probabilistic arguments". Again - you make these unsubstantiated incorrect statements - why and for what purpose??? 4.) Vanderportens text was clear and unambiguous - he was claiming credit in 1997 for my 1993 discovery. He appropriately changed the text in subsequent editions. This fact is totally inconsistent with your assertions. 5.) Harold Edwards, author of "Fermat's last theorem, a genetic introduction to algebraic number theory" wrote me in 1994 and stated that the discovery was "remarkable" and unknown. Who are you - why must you hide behind the "hull loss incident" name and criticize and mischaracterize me. I have told readers who I am - who are you?? --andrew beal [previous article quoted in entirety, here deleted --djr] Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Hull Loss Incident Subject: Re: A generalization of FLT. Date: Tue, 22 Aug 2000 03:00:16 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > 1.) > First you suggest that the beal conjecture is so trivial as to be > beneath you - I have not suggested the mathematical problem is trivial. It is difficult and there are substantial articles by leading young researchers such as Darmon, Granville, Merel, Poonen and others making inroads into the problem. I hope they receive a share of your prize money. I do suggest that arriving at the mathematical problem as a conjecture is trivial, given well-known principles in number theory. > then you suggest that I may attatch my name to a piece of > mathematical history - which is it?? I suggest that a possible result of the prize money and consequent publicity may be that people increasingly refer to the problem as "Beal conjecture" or similar, especially in the absence of a pre-existing consensus title for either the equation or the conjecture. I do not suggest you "may attach" your name in the sense that doing so would be a legitimate move. > 2.) > Granville made no "polite attempts" to "disabuse" me. Granville was > abusive, ill informed, and totally unable to demonstrate any prior > knowledge, despite claims to the contrary. It would be pretty delicious to see his "abusive ill informed" comments posted here, I encourage you to do so. Since you're already vilifying him it would not be too much of a stretch at this point. Anyway, Granville is hardly ill-informed. I doubt he would go to the trouble of a detailed literature search to educate someone as determined as you are to claim credit until definitively proven wrong. At the moment, there is no mathematical equivalent of patent searching that could conveniently resolve disputes like this one. That might be a worthy project for you to fund... > 3.) > It was not known previously by "probabilistic arguments". It was definitely known by probabilistic arguments. In detail: Consider the equation f(X,Y,Z) = X^a + Y^b - Z^c = 0. For large N, the number of triples (X,Y,Z) with X^a, Y^b, and Z^c all between 1 and N, is roughly N^(1/a + 1/b + 1/c). On the other hand, for such triples f(X,Y,Z) is between -N and 2N, a total of 3N possibilities. Assuming the values of f(X,Y,Z) are "randomly" (uniformly) distributed among the possible values, we thus expect each value to occur about 1/3 * N^(1/a+1/b+1/c -1) times. So if 1/a + 1/b + 1/c > 1 we expect many solutions, and if 1/a + 1/b + 1/c < 1, we expect a finite number of solutions (and in that case can refine the argument to take into account a search for all solutions up to some given N, and guess/estimate the chances of finding more solutions). I'm pretty sure one can find documented instances of this heuristic reasoning going back 50-100 years. (Hardy, Littlewood, Artin, Erdos, Davenport, ...) > 4.) > Vanderportens text was clear and unambiguous - he was claiming credit > in 1997 for my 1993 discovery. He appropriately changed the text in > subsequent editions. This fact is totally inconsistent with your > assertions. If you quote the relevant paragraph from Van der Poorten (maybe once from each edition) we can determine what is and isn't consistent with my assertions. > 5.) > Harold Edwards, author of "Fermat's last theorem, a genetic > introduction to algebraic number theory" wrote me in 1994 and stated > that the discovery was "remarkable" and unknown. What did Edwards actually say, in full? It may indeed be "remarkable" that you arrived at the conjecture without much exposure to current number theory. In light of that theory, however, your conjecture is well known and not all that remarkable, though it is of course a difficult and interesting problem. The meaning would depend on what Edwards actually said, and whether he qualified it as his private opinion or a professional judgement on your conjecture in relation to prevailing mathematical knowledge. Edwards, by the way, is not a number theory researcher. His FLT book, like his other books and published articles, is primarily an exposition of mathematics developed before 1900. His books that I have seen are well-written and scholarly, but do not qualify him as an expert on FLT or related mathematics. Granville is an expert and a current active researcher in that area, and certainly is in a better position to tell you what is and isn't known. ============================================================================== From: Gerry Myerson Subject: Re: A generalization of FLT. Date: Wed, 23 Aug 2000 09:42:04 +1000 Newsgroups: sci.math In article <39A22500.92FBAEBE@y.z.com>, Hull Loss Incident wrote: Since Andrew Granville's contribution to the Western Number Theory problem list has come up in this discussion, I want to put it on record here. The December 1992 Western Number Theory meeting was held in Corvallis. The problem list was edited by Richard Guy and is dated 9 June 93. The relevant part of Problem 92:12 reads as follows. ***************************************** 92:12 (Andrew Granville) Find examples of x^p + y^q = z^r with 1/p + 1/q + 1/r < 1 other than 2^3 +1^7 = 3^2 and 7^3 + 13^2 = 2^9. [Blair Kelly III gave 2^5 + 7^2 = 3^4 and Reese Scott 17^3 + 2^7 = 71^2.] *************************************************** In Guy's write-up of the 1993 problems, dated 3 March 94, there is a comment about 92:12, wherein Granville agrees with the suggestion that it was intended that x, y and z be relatively prime, and gives 3^5 + 11^4 = 122^2 as another example. Peter Montgomery gave 5 larger examples found by Beukers & Zagier. Gerry Myerson (gerry@mpce.mq.edu.au) ============================================================================== From: Bob Silverman Subject: Re: A generalization of FLT. Date: Wed, 23 Aug 2000 16:29:28 GMT Newsgroups: sci.math In article <8o03vl$sts$1@nnrp1.deja.com>, andybeal@my-deja.com wrote: > I'm not interested in attatching any credibility to your arguments as a > function of who you are - I am trying to understand who you are and how > I obviously have offended you in such a manner that you would spread > these lies BZZT.... I have read these threads. I have seen no "lies" posted by "Hull Loss Incident". >in some kind of attempt to create contempt toward me. You are trying to claim credit for a discovery that is NOT rightfully yours. This conjecture has been part of mathematical culture for a LONG time. I doubt whether any one individual can lay claim to it. -- Bob Silverman "You can lead a horse's ass to knowledge, but you can't make him think" Sent via Deja.com http://www.deja.com/ Before you buy. ==============================================================================n From: andybeal@my-deja.com Subject: Re: A generalization of FLT. Date: Tue, 22 Aug 2000 06:24:42 GMT Newsgroups: sci.math In article <39A19FA4.4C8AEEF2@y.z.com>, Hull Loss Incident wrote: 1.) Granvile has never made "polite attempts" : Granvile was abusive, ill informed, and unable to demonstrate any prior knowledge of my conjecture. His efforts and inability to do so increased confidence that none existed. 2.) first you suggest that the conjecture is trivial and unpublishable, then you say that I may succeed in attatching my name to a bit of history. Which is it: trivial, or history making??? and why do care either way??? 3.) You incorrectly assert that the conjecture was known by probabalistic arguments. This is another untrue statement - what motivates you to say these things??? 4.) Vanderpoortens text in his 1997 book was clear and unambiguous - he was claiming credit for my 1993 discovery. He appropriately corrected subsequent editions. these facts are entirely inconsistent with your assertions. 5.) who are you "hull loss incident"?? what is your problem?? why must you hide behind this name, misrepresent the truth, and mischaracterize my efforts???? -- who are you and what did I ever do to offend you??? You criticize me for being self promotional -- even if true, who cares??? why does this bother you so??? I believe I discovered the "beal conjecture". However, I have been wrong before in my lifetime, so why don't you simply demonstrate prior knowledge of the concept and prove me wrong??? I regularly admit my errors, which I make regularly. why don't you cut the bullshit and simply cite a reference?????????????? Instead of doing something meaningful like the above, you simply run your mouth off criticizing me. 6.) To cite but one expert, Dr. Harold Edwards, NYU, and author of "Fermat's Last Theorem, a genetic introduction to number algebraic number theory" wrote me in 1994 saying my discovery was unknown and "quite remarkable". For those of us not as brilliant and informed as you, perhaps this doesn't seem quite so trivial. --andrew beal ------------------------------------------------------------------ [previously quoted article quoted _again_, here again deleted --djr] Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: A generalization of FLT. Date: 22 Aug 2000 07:50:32 GMT Newsgroups: sci.math In article <8nt6as$g4v$1@nnrp1.deja.com>, wrote: >I believe I discovered the "beal conjecture". What does it mean to "discover a conjecture"? To be the first to formulate it? It's often rather hard to ascertain who first enunciates an idea which is simple and speculative (and somewhat amorphous). Who, for example, "discovered" the conjecture that there may be aliens on other planets -- Carl Sagan? The "canals on Mars" guy? The writer of Revelation? Likewise, the "Beal Prize conjecture" can be said to have originated with Beal, with Granville, with Catalan, with Euler, or with Fermat, to name but a few candidates. I can't really see much point arguing who first formulated a conjecture, unless the conjecture has such a detailed statement as to suggest its own proof ("I conjecture the finite simple groups are precisely the following three general classes and twenty-six exceptional cases:..."), in which case something of real value may have been contributed. A succession of increasingly refined or more challenging problems is more common. It's rather like school children teasing each other back and forth with, "Yeah, sure, but can you do _this_?". Sometimes the teacher demands to know, "Who started this?" but it's not clear that there is an answer, nor that the answer is very important. dave ============================================================================== From: Bob Silverman Subject: Re: A generalization of FLT. Date: Wed, 23 Aug 2000 16:37:08 GMT Newsgroups: sci.math In article <8nt6as$g4v$1@nnrp1.deja.com>, andybeal@my-deja.com wrote: > In article <39A19FA4.4C8AEEF2@y.z.com>, > Hull Loss Incident wrote: > > 1.) > Granvile has never made "polite attempts" : Granvile was abusive, ill > informed, Actually, Granville knows so much more about this subject and its history than you do that a comparison between the two of you would be unfair. You are beginning to sound like a crank. > and unable to demonstrate any prior knowledge of my > conjecture. It would be hard to find *anything* in print. It is part of mathematical folklore/culture. > You incorrectly assert that the conjecture was known by probabalistic > arguments. This is another untrue statement - what motivates you to say > these things??? Because it is true. The same probabilistic arguments which apply to FLT can be used for the case of unequal exponents. It is ** SO ** obvious to one with even average skills in the art, that it would never be worth publishing. > 5.) who are you "hull loss incident"?? what is your problem?? why must > you hide behind this name, misrepresent the truth, and mischaracterize > my efforts???? I agree with your concern over his anonymity. However, he does NOT misrepresent the true. > > I believe I discovered the "beal conjecture". You can believe in the tooth fairy, for all I care. What you believe is irrelevant. This conjecture is so obvious and has been around for so long that your belief comes across as megalomania. However, I have been > wrong before in my lifetime, so why don't you simply demonstrate prior > knowledge of the concept and prove me wrong??? In fact, I did. I cited (on another thread) discussions which took place in 1985. -- Bob Silverman "You can lead a horse's ass to knowledge, but you can't make him think" Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Hull Loss Incident Subject: Re: A generalization of FLT. Date: Wed, 23 Aug 2000 23:38:07 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > Vanderpoortens text in his 1997 book was clear and unambiguous - he was > claiming credit for my 1993 discovery. He appropriately corrected > subsequent editions. I checked the book, and the above appears to be a misrepresentation. Van der Poorten's book, _Notes on Fermat's Last Theorem_, was published in February 1996. There is one edition only. In the book, Van der Poorten doesn't mention Beal or his conjecture at all, let alone try to take credit for anything. He does discuss the generalized Fermat equation X^a + Y^b = Z^c in the chapter on pages 143-150, citing the paper of Darmon and Granville (and also the work of Beukers and Zagier). Van der Poorten's discussion is, quite naturally, based on the cases when 1/a + 1/b + 1/c is larger or smaller than 1. This corresponds to Granville's formulation of the problem at the 1992 number theory conference, though Van der Poorten doesn't mention this, nor does he try and assign credit for the problem to anybody. He explains why one should restrict to relatively prime solutions. > these facts are entirely inconsistent with your assertions. The book supports the assertion I and others have made, that the problem was well-known enough that nobody in the profession would try to assume or assign credit for it. > 6.) To cite but one expert, There are no experts on your side. I challenge you again to quote (in full, not single words out of context) even one number theorist or equivalent expert who unambiguously supports your claim to a new and significant discovery. ============================================================================== From: Hull Loss Incident Subject: Re: A generalization of FLT. Date: Mon, 21 Aug 2000 17:35:07 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > Apparently some misconception exists about my > discovery of the "beal conjecture" relationship. It was known for decades in various forms. Anyway, since you are publically claiming priority for what you call a "significant discovery", you should state clearly what that discovery is, specifically. (In your posting you use the vague term "relationship".) Are you claiming just the conjecture that certain equations of the form X^a + Y^b = Z^c have no solutions without common factor? Or some other formula(e) or "relationship" relevant to the analysis of such equations?? > It is the discovery, not the posting of the prize > that is significant. Again, what discovery is significant? The statement sometimes known as the Beal Conjecture, or something in addition to that? > The relationship was clearly unknown prior to my > work and discovery of it in 1993. > I received responses in 1994 from > leading mathematicians that ackowledged the > relationship as unknown. > [...] Vanderporten must > agree that it was unknown prior to my 1993 > discovery of it. ============================================================================== From: Hull Loss Incident Subject: Re: Against the term "Beal Conjecture". Date: Mon, 21 Aug 2000 18:00:35 -0400 Newsgroups: sci.math [I assume the person posting is the real-life Andrew Beal who offers the prize, judging from his comments in the other thread on "generalization of FLT".] andybeal@my-deja.com wrote: > [...replica of two other postings...] Deja News seems to have mangled your posting, only the quoted text of the previous message made it through. Anyway, here is a copy of the MathSciNet review of the article in AMS Notices on the "Beal conjecture", containing some comments on the naming: 98j:11020 11D41 Mauldin, R. Daniel(1-NTXS) A generalization of Fermat's last theorem: the Beal conjecture and prize problem. Notices Amer. Math. Soc. 44 (1997), no. 11, 1436--1437. This note announces the award of a substantial monetary prize, fixed at $$50,000$ since the article under review was written, to any person who provides a solution to the "Beal conjecture", stated as the following: Let $A,B,C,x,y,z$ be positive integers with $x,y,z>2$. If (1) $A\sp x+B\sp y=C\sp z$, then $A,B,C$ have a nontrivial common factor. The story of this conjecture is an interesting one, and told at slightly greater length in the author's follow-up letter [Notices Amer. Math. Soc. 45 (1998), no. 3, 359]. See also the article by K. J. Devlin [Math. Horizons 1998, Feb., 8--10; per revr.]. Andrew Beal is a successful Texas businessman, with enthusiasm for number theory. He has had a particular interest in Fermat and his methods following the announcement in 1993 of Andrew Wiles' work on Fermat's last theorem, and formulated this conjecture after several years of computer-based study. So often the amateur number-theorist turns out to be a well-intentioned crank; what is remarkable here is how close the stated problem is to current research activity by leaders in the field. In fact, the problem is essentially many decades old, and apparently V. Brun [Arch. Math. Nat. 34 (1914), no. 2, 1--14; JFM 45.1219.13] asked many similar questions. The formulation in the 1980s by Masser, Oesterle and Szpiro of the $abc$-conjecture has had great influence on the discipline of number theory, and in fact a corollary of the $abc$-conjecture is that there are no solutions to the Beal Prize problem when the exponents are sufficiently large. The prize problem itself was implicitly posed by Andrew Granville in the Unsolved Problems section of the West Coast Number Theory Meeting, Asilomar, 1993 ("Find examples of $x\sp p + y\sp q = z\sp r$ with $1/p + 1/q + 1/r < 1$ other than $2\sp 3 + 1\sp 7 = 3\sp 2$ and $7\sp 3 + 13\sp 2 = 2\sp 9$"), and was stated and discussed in A. van der Poorten's book [Notes on Fermat's last theorem, Wiley, New York, 1996; MR 98c:11026]. The resolution by Wiles of Fermat's last theorem disposed of a special case of the prize problem; and H. Darmon and Granville [Bull. London Math. Soc. 27 (1995), no. 6, 513--543; MR 96e:11042] proved the deep result that if $1/x+1/y+1/z < 1$ then there can only be finitely many triples of coprime integers $A,B,C$ satisfying $A\sp x+B\sp y=C\sp z$ (ten solutions are known). Recently, Darmon and L. Merel [J. Reine Angew. Math. 490 (1997), 81--100; MR 98h:11076] showed that there can exist no coprime solutions to (1) with the exponents $(x,x,3), x \geq 3$. There has been some feeling expressed by number-theorists that the conjecture should best be referred to as the "Beal Prize problem", though there is no doubt that Beal independently arrived at and formulated the conjecture without knowledge of the current literature. With a nod to T. S. Eliot, the matter of naming conjectures can be as difficult as the naming of cats. Reviewed by Andrew Bremner ============================================================================== From: Hull Loss Incident Subject: Re: Against the term "Beal Conjecture". Date: Tue, 22 Aug 2000 01:35:06 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > Yes it is unfortunate that Andrew Granvile misrepresented the history > of the problem to Dan Mauldin who authored the following article that > you have referenced. Mauldin correspondingly included the erroneous > references to similiar historical problems. That's quite a claim. What did Granville "misrepresent" to Mauldin, exactly? According to the review I quoted from Mathscinet (98j:11020), Granville himself publicized (and published) by 1993 a problem that includes yours as a special case. Even that was not original, i.e. he was apparently drawing attention to an already well-known circle of ideas and saying that this was a concrete problem that deserved work. Granville by the way is well-versed in number theory and (I presume from the contents of some of his articles) much of the relevant history. I believe his advisor was Ribenboim who wrote numerous articles and at least two books on FLT. Mauldin is not a number theorist (as far as I know) so while he was right to take Granville's assertions on faith, he may not be in a position to assess your work accurately. > Unfortunately those of us > not so lucky to have the insight and knowledge that "hull loss > incident" has must simply rely on the leading number theory texts There's quite a lot of knowledge in mathematics that is "out in the community" as oral tradition or folklore. The fact that it may be hard to find a textbook containing a statement of your particular conjecture certainly does not mean it was unknown. I remember as a student, hearing mathematicians mention in an offhand "everyone knows it" sort of way that FLT was expected to be true on simple probabilistic grounds for exponent 4 and higher, and sketching exactly the argument that also leads to your conjecture, Euler's problem on N'th powers, and many others. Relying on number theory textbooks as a reference is fine. It's an obvious mistake, though, to take what is NOT mentioned in the books as a serious indication that something is unknown to the number theory community. Textbooks lag behind current research, they can't include more than a tiny fraction of what is known, and they are often written by people who have time to write textbooks precisely because they're not at the current edge of research. As far as your specific problem is concerned, there is probably a dearth of literature on it simply because nobody could prove much before Wiles' breakthrough, and (because it was fairly well known) just stating the conjecture would not be publishable. I do expect that there was computational work published, on your equations or similar ones, and that if you look at the publications they will make statements from which your conjecture can be easily inferred. The place to look would be a keyword search on Mathscinet (AMS database of reviews of math papers). > and corresponence with mathematicians such as Harold Edwards > from NYU and Earl Taft from Rutgers. Neither of these are people I would expect to be familiar with current number theory or its literature. Taft is an algebraist who as far as I know works on quantum groups, an area far from most number theory. Edwards' publications are studies and expositions of mathematics developed more than 100 years ago. If you want to learn the classical approaches to Galois theory or algebraic number theory his books are wonderful, but with all due respect he is not an authority on FLT-related mathematics at the level of someone like Granville, whom you now accuse of spreading misinformation. I don't know where you are in Texas, but UT Austin has several number theorists who are quite knowledgeable about these questions. At least one of them posts to sci.math occasionally, and of course you can usually get good answers by asking questions in these newsgroups. > While there are certaintly many similiar > diophantine forms, none of the texts and no-one I corresponded with > indicated any prior knowledge of the beal conjecture or the concepts > involved. the closest reference was to the ABC conjecture which > hypothesizes a finite number of solutions I'm not sure what you consider the concepts involved. As I mentioned, it has been known for a long time that if you have some equation between sums of powers like X^a + Y^b - Z^c + 3*T^d - 57*W^e = 0, what determines whether you expect many solutions or few/none is if the sum 1/a + 1/b + 1/c + 1/d + 1/e is larger/smaller than 1. You can also take into account calculations up to some bound and make statements of the form, "if I check that there are no solutions with arguments under 1000000, the expected number of larger solutions is 0.003 thus we expect no such solutions". As an accessible expository reference with the same kind of probabilistic reasoning in number theory, see for example P.T. Bateman, J Selfridge, S. Wagstaff The new Mersenne conjecture American Mathematical Monthly 96 (1989) no. 2 pp.125-128 > In any event, the term beal conjecture results from my "purported" > discovery, not from my offer of a prize. Conjectures in number theory are a dime a dozen. If not for the prize and associated publicity, nobody would have associated your name with the problem. Maybe over time people who actually produced scholarship and results in the direction of this conjecture, such as Darmon and Granville, might have had their names attached to it as their papers became standard references. But it would not have occured to anyone to give you special credit for just posing the problem. Unless the problem is some previously unconsidered type of statement, you generally don't get credit without making substantial intellectual contributions. > Since the conjecture is so > trivial to you, why don't you ignore it?? And if so trivial, why is > there so much interest in the problem?? Perhaps to those of us out here > that are not as gifted as you, the problem is not quite so trivial?? I see no evidence of "so much interest", and I did not trivialize the mathematical problem. Formulating the problem is what's trivial, given knowledge that has been standard in number theory for many years. > Why don't you criticize my claim of discovery (and demonstrate > otherwise) I did demonstrate otherwise. I can post the details of the probabilistic argument (about 1/a+1/b+1/c) if you like, as can several other people who posted to this discussion and innumerable people who may be reading this. I am also confident that I could find references predating your conjecture where such ideas are spelled out in print. > instead of creating contempt for me by lying and saying that > i simply posted a prize. What I said is that you made no intellectual contribution to the problem, which is apparently true. Intellectual contribution means things that advance the community's knowledge, such as publication of relevant research or expository articles, teaching activity, or similar. I asked and received no answer as to what you have done in that direction. Privately circulating a manuscript and then seeking recognition for "significant discoveries" is egomania, not contribution. > why are you misreprenting my role here and attempting to create > contempt toward me?? I expressed contempt for the idea of buying mathematical "namespace" for the paltry sum of $75000. If you want to do something more positive for mathematics (even just the part related to your conjecture) with your money, I can certainly make suggestions. > I am confident that I discovered the beal conjecture. As stated in the Mathscinet review, there is no dispute that you arrived at the conjecture on your own without knowledge of the number theory literature. I do dispute your claims of priority and the claim that the conjecture was a "significant discovery" for mathematics. ============================================================================== From: andybeal@my-deja.com Subject: Re: Against the term "Beal Conjecture". Date: Wed, 23 Aug 2000 09:26:22 GMT Newsgroups: sci.math As I said in another post, the fact is that I appear to be the first one to have reasoned that co-prime bases are impossible as asserted in the beal conjecture. I additionally actually tested the assertion to confirm some probability of the concept. Granville publicly stated in 1998 that he doubted the beal conjecture was true - hardly evidence of common prior knowledge Vanderpoorten stated in the first edition of his 1997 book that he was "proposing" that co-prime bases were impossible (ie: the beal conjecture). --hardly consistent with prior common knowledge Mathematicians wrote me in 1994 after I disseminated my assertions and research agreeing that the concept was unknown and calling it "quite remarkable". I certaintly didn't simply "pose the problem", I reasoned a suspected conclusion and impirically tested it within reasonable boundries and disseminated my results. In any event, my agreement to post a prize when someone expressed interest in writing about the conjecture is hardly my only contribution to the problem.---andrew beal As I've said before, simply demonstrate that anyone ever suggested previously that co-prime bases were impossible and I'll walk away from the conjecture - that seems pretty simple doesn't it ???? --andrew beal In article , christian.bau@isltd.insignia.com (Christian Bau) wrote: > In article <39A21109.682E5844@y.z.com>, Hull Loss Incident wrote: > > > Relying on number theory textbooks as a reference is fine. > > It's an obvious mistake, though, to take what is NOT mentioned > > in the books as a serious indication that something is unknown to the > > number theory community. Textbooks lag behind current research, > > they can't include more than a tiny fraction of what is known, and they > > are often written by people who have time to write textbooks precisely > > because they're not at the current edge of research. > > In number theory, there are just so many things that are quite plausible > conjectures... I think every student of mathematics should be able to come > up easily with half a dozen conjectures that his professor cannot prove to > be wrong, or find in the literature within one day. Of course, the reason > is that most of these conjectures will not be considered worth writing > down. > Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Bob Silverman Subject: Re: Against the term "Beal Conjecture". Date: Wed, 23 Aug 2000 16:04:17 GMT Newsgroups: sci.math In article <8o05bg$ue0$1@nnrp1.deja.com>, andybeal@my-deja.com wrote: > As I said in another post, the fact is that I appear to be the first > one to have reasoned that co-prime bases are impossible as asserted in > the beal conjecture. This is nonsense. It is anything BUT a fact. I can recall discussions at the 1985 conference on Computational No. Theory in Arcata CA. Tate gave a lecture surrounding Frey's brand new discovery that Taniyama-Shimura was associated with FLT. There was quite a bit of *informal* discussion as to whether Frey's result also applied to the case of *unequal* exponents. This conjecture has been around a LONG time. It is anything but new. However, simply posing a conjecture is not worth a formal paper, unless there were techniques to suggest why the conjecture is true. You claim of being the first to think of this conjecture is ridiculous. It is an obvious extension of FLT to anyone who works in number theory. -- Bob Silverman "You can lead a horse's ass to knowledge, but you can't make him think" Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Hull Loss Incident Subject: Re: Against the term "Beal Conjecture". Date: Thu, 24 Aug 2000 03:00:48 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > As I said in another post, the fact is that I appear to be the first > one to have reasoned that co-prime bases are impossible as asserted > in the beal conjecture. Now that it's on record that Granville posed the problem in 1992 before the Beal conjecture, you are seizing on minor differences, any differences, between your version and his. It is not a major discovery that 8 + 8 = 16 is a solution if you allow common factors. Less obvious, but still child's play for professionals (I remember the trick from high school math competitions), is that there are similar trivial solutions for all exponents. So the need for co-prime bases was certainly clear to Granville and/or his audience of number theorists, and whether or not it was specifically spelled out in the problem statement is not a significant difference. In any case he clarified later that indeed, primitive solutions were intended. So you noticed that there are solutions with common factors? Nice, but no big deal. You also ran a computer search for primitive solutions, and from that surmised that none exist? Even if the guess turns out correct, it's at the level of a school science project, not the "significant discovery" you claim. > I additionally actually tested the assertion to > confirm some probability of the concept. How far did you test? There are known solutions with "co-prime bases" in the tens of millions, for exponents 2,3,7 and 2,3,8. > Granville publicly stated in 1998 that he doubted the beal conjecture > was true - Even if true, so what? He raised the question before you did, only without presupposing the answer. Your subsequent guess as to what the answer is does not add anything, since it follows from the standard arguments (which you were unaware of) coupled to a computer search. If Granville has deeper reasons for disbelieving the heuristics, that would be much more interesting, though it doesn't touch the priority issue. Again, I doubt Granville or any other number theorist would seriously claim the problem as their own; his 1992 proposal is more like public domain prior art invalidating a patent claim. > hardly evidence of common prior knowledge Common prior knowledge was that the equation should have few or no solutions -- certainly a finite number of primitive solutions for any given exponents. Finiteness was expected based on probabilistic arguments, the ABC conjecture and other considerations, and was proved in a 1995 paper coauthored by....Granville. > Vanderpoorten stated in the first edition of his 1997 book that he > was "proposing" that co-prime bases were impossible (ie: the beal > conjecture). --hardly consistent with prior common knowledge On what page of what book does he propose this? I checked his book on FLT without finding any support for your claims, just the opposite. > Mathematicians wrote me in 1994 after I disseminated my assertions and > research agreeing that the concept was unknown and calling it "quite > remarkable". Quote in full any mathematician who endorses your claim of a new significant discovery. Just one will suffice. ============================================================================== From: andybeal@my-deja.com Subject: Re: Against the term "Beal Conjecture". Date: Wed, 23 Aug 2000 08:48:52 GMT Newsgroups: sci.math >Granville himself publicized (and published) by 1993 a problem that >includes yours as a special case. Even that was not original, i.e. he >was apparently drawing attention to an already well-known circle of >ideas and saying that this was a concrete problem that deserved work. >Granville by the way is well-versed in number theory and (I presume >from the contents of some of his articles) much of the relevant >history. I believe his advisor was Ribenboim who wrote numerous >articles and at least two books on FLT. Granville never hypothesized that co-prime bases were impossible and certaintly never impirically tested the concept. Granville is hardly well versed with regard to these issues. Granville stated publicly in 1998 that he doubted the beal conjecture was true. So much for it being common knowledge. I did not simply pose the problem, I reasoned that co- prime bases might be impossible and impirically tested a reasonable range before announcing the assertion. > I do expect that there was computational work published, on your > equations or similar ones, and that if you look at the publications >they will make statements from which your conjecture can be easily >inferred. I am also confident that I could find references predating >your conjecture where such ideas are spelled out in print. Your confidnece and expectations appear to be misplaced. Talk and bullshit are cheap -- cite your reference(s) ---andrew bael Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Hull Loss Incident Subject: Re: Against the term "Beal Conjecture". Date: Thu, 24 Aug 2000 03:36:01 -0400 Newsgroups: sci.math andybeal@my-deja.com wrote: > Granville never hypothesized that co-prime bases were impossible Concerning "coprime", he was aware that 8+8=16. Concerning "impossible", he was aware that short-range computer searches don't count for much. > and certaintly never impirically tested the concept. He listed some solutions in the question, presumably as a result of someone having looked for them. His formulation, unlike yours, would not be invalidated by further searching. On the contrary, it was obviously meant to prompt more serious empirical tests. > Granville stated publicly in > 1998 that he doubted the beal conjecture was true. Given your (in)accuracy so far, I'll believe it when i see a complete quote, preferably one that i can check. > So much for it being > common knowledge. I did not simply pose the problem, I reasoned that co- > prime bases might be impossible and impirically tested a reasonable > range before announcing the assertion. You ran a computer search and the world should genuflect? Ludicrous. ============================================================================== From: andybeal@my-deja.com Subject: Re: Against the term "Beal Conjecture". Date: Wed, 23 Aug 2000 08:21:11 GMT Newsgroups: sci.math I certaintly hope that Andrew Bremner is a better mathematician than he is researcher. The fact is that I appear to have been the first to reason that a^x +b^y = c^z might be impossible with co-prime bases, impirically test a reasonable range of variables, and announce the conjecture. The fact is that Andrew Granville said publicly in 1998 that he doubted the conjecture was true, hardly consistent with it being commonm knowledge. The fact is that Vanderpoorten claimed credit in his 1997 book for "proposing" the statement, hardly consistent with it being common knowledge. The fact is that Bremners "review" below is bullshit and begs the question of who inspired the review and for what purposes. The reality is that there are so few number theorists out there that few are able to see through bremners "review". The reasoned conclusion that co-prime bases are impossible is hardly evident in any of Brun's work, hardly implicit in any of Granvilles work prior to 1996, and is hardly many decades old. Where are the truth seekers out there among mathematicians??? ---andrew beal [HLI article of 21 Aug 2000 quoted in toto, here deleted --djr] Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Against the term "Beal Conjecture". Date: 23 Aug 2000 13:44:59 GMT Newsgroups: sci.math In article <8o01hc$qds$1@nnrp1.deja.com>, wrote: >I certaintly hope that Andrew Bremner is a better mathematician than he >is researcher. > >The fact is that I appear ...not to know what constitutes mathematical work. (a) Your first sentence contrasts activity A with activity B when they are synonymous. (b) Your multiple postings suggest there is tremendous value in stating guesses about things. There is not. We have, for example, an enthusiastic amateur who regularly posts guesses and puzzles and so on to sci.math, and they often make for interesting diversions. We also see occasional puzzles by people with significant mathematical training (e.g. Noam Elkies posts to rec.puzzles, and Charles Nicol will ask here what look like simple-minded challenges). Not wanting to appear too elitist, I have to tell you that the first person's guesses are not worth recording for posterity but those of the other people are, simply because the result of decades of training is that a person can make a conjecture based on some _structure_ of the situation, and not a simple check of examples. If it should happen that the amateur and the professional both, independently, come to ask the same question, I would take it as an indication that the conjecture is a natural one. Fine. But I have to stress there is no "credit" for making guesses in mathematics. Our primary interest is in providing proofs for assertions, or at least indications of plausibility. Let me comment that I'm happy that there are amateurs interested in mathematical questions, and that in particular I'm happy that Beal has offered money to draw attention to a question in mathematics. It may be a silly way to attract young people to mathematics, but since talented youth are drawn away from mathematics by similar silly tactics, I figure this is a fine counterweight. The only thing I am objecting to is the claim that Beal has advanced our understanding of mathematics in any way by offering this prize. For comparison I would like to remind readers about the "Nobel Prize in Economics". There are disputes about the choice of name for that award, since the prize has nothing whatever to do with Alfred Nobel. It's just a slightly underhanded way to bestow some additional gravitas on the subject of Economics. Well... so what? I figure if a person with money wants to offer money to people who are doing research (many of them mathematicians!) then I don't much care what they call it. Whatever celebrity accompanies the award belongs to person who advances the discipline, not the person with the money. dave