Date: Wed, 23 Nov 94 02:01:29 GMT From: Chris Thompson To: Charles Blair , "Timothy Y. Chow" , Dave Rusin Subject: Re: H.E. Dudeney and elliptic cruves Thank you all for your e-mail responses to my sci.math posting "H.E. Dudeney and elliptic curves". Apologies for not replying before, and for combining them all together now. I haven't seen any responses posted in sci.math at all. Charles Blair writes: > Your post was very interesting, but I can't help with the details. > Martin Gardner edited an anthology of Dudeney stuff called 536 Puzzles > and Curious Problems with a long biographical introduction. My > impression was that Dudeney was almost entirely self-taught, so > it seems quite likely that his knowledge of the field was whatever > he saw in the book by Lucas. In fact, my e-conversation with Michel Gallant started because he had Gardner's book while I had Dudeney's! He couldn't find anything there that explained the background to this particular problem, though. Timothy Y. Chow writes: > In article <3a3ip3$gju@lyra.csx.cam.ac.uk> you write: > >The "formula" for deriving one solution from another is apparently > >equivalent to doubling in the group > > Just a guess, but I wouldn't assume that this is true. For one thing, > 2P is much more likely to have been derived by trial than 3P, and for > another (as you point out), this is inconsistent with the alternation > in sign that Dudeney mentions. > > >Was Dudeney simply unaware of modern developments in this area > > Another guess, but I would think that the answer to this one is yes. I agree that there are problems with trying to make the "formula" equivalent to doubling, but it has to have _something_ to do with the group structure. (Can't one mutter learnedly about isogenies to "prove" this?) What little I know about Fermat's work in this area is consistent with this, and one somehow has to get from P to 8P. I was rather hoping that some of the Fermat experts who came out of the woodwork to talk about Fermat's claims about 2^2^n+1 would have something to say about this (some of the other Diophantine equations that Fermat was claiming to have proved the insolubility of by infinite descent were definitely elliptic curves), but no luck so far. I don't think it is at all inconceivable that Dudeney could have found 3P=(919,-271,438) "by trial", as one can do a lot of filtering with congruences, and fixing z gives only a few possible pairs (x,y) with an efficient algorithm to find them. But admittedly, if one starts getting too sophisticated about this, one recovers group addition anyway! Dave Rusin writes: > I haven't a response to your question but I would appreciate copies of > an interesting responses you get. I will be teaching a course on > elliptic curves out of Cassels next semester and I am interested, of > course, in obtaining useful example, exercises, and motivating material. I wish I had something more useful to report. Cassels uses x^3+y^3+nz^3 = 0 in many of his examples and exercises, so there is certainly scope for making a connection. I have been meaning to ask Professor Cassels (who, although he retired several years ago, still attends the Number Theory seminars in Pure Maths here) whether he knows anything about the background to Dudeney's problem. Chris Thompson Cambridge University Computing Service Internet: cet1@phx.cam.ac.uk JANET: cet1@uk.ac.cam.phx ============================================================================== Date: Thu, 24 Nov 94 22:32:45 GMT From: Chris Thompson To: Charles Blair , "Timothy Y. Chow" , Dave Rusin Subject: Re: H.E. Dudeney and elliptic curves (cont) I asked Professor Cassels for his comments and have received the following informative reply. This was private e-mail: I haven't asked his permission to post it yet. Chris Thompson Cambridge University Computing Service Internet: cet1@phx.cam.ac.uk JANET: cet1@uk.ac.cam.phx ------ Dear Chris, No, Dudeney's arguments can hardly be said to be state of the art. Sylvester, or Lucas or P\'epin would have made a better job of it (or Mordell, who by 1915 had already written about other elliptic curves, though he was then more concerned with integral points. On the other hand, I think you expect unhistorically too much of him. The curve and tangent processes were certainly known in the 19th century. The tangent process goes back to Diophantos Himself and the tangent process to Newton [Cassels must have been dreaming when on p 24 of his `little book' he transposed their contributions]. I looked up Skolem's 1938 Ergebnisse volume `Diophantische Gleichungen'. The references he gives for the chord and tangent processes are Cauchy and an 1878 paper of Lucas. On the other hand, I do not think that Lucas or Sylvester understood that there was an underlying group: so they would not understand your notation P, 2P, 3P, 4P, ... : it must be 30 or 40 years since I looked at the Sylvester and Lucas material, so I may be mistaken. The references that Skolem gives for the group structure are papers of Poincar\'e (1901) and Hurwitz (1917). The Poincar\'e paper is very geometrical, and probably inaccessible to the number-theoretical amateur of the time. My interpretation of Dudeney's statement `there is an infinite number of fundamentals' seems to indicate that he knew about the tangent process, but only vaguely about the chord process. There seems to have been a great interest in various elliptic curves in the 19th century, much of it amateur and much of it rediscovery. For the equation X^3+Y^3=A, see Dickson, History of the theory of numbers, vol 2, pp.572-578. I expect you know the massive paper of Selmer, with its tables of generators (Acta Math 85(1951), 203-362 and 92(1954), 191-197: the latter one of the earliest applications of computers to number theory). Incidentally, the arguments of Desboves are very peculiar: I got the reference from Selmer. On your second question about Dudeney's assertion that the solutions obtained by the tangent process are alternately positive and negative, I suspect your guess is the correct one: he had observed that it happened in some cases and thought that it was a general phenomenon. All the best Ian PS I have only just transferred from PHOENIX to CUS and am enjoying the greater access to such things as Internet. ________________________________________ Prof. J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, CAMBRIDGE CB2 1SB UK