Here's an obscure sequence for Neil, from the Number Theory list. Rich rcs@cs.arizona.edu ---------------------- Date: Thu, 12 Dec 2002 23:16:36 -0500 Sender: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU> From: Ismael Jimenez Calvo <ltqij04@pinar2.csic.es> Subject: Ten new "good examples of Hall's conjecture" M. Hall [Hall] conjectured that the nonzero difference k = x^3 - y^2 can not be less than Cx^{1/2}, for a constant C. His original conjecture, probably false, has been reformulated in the following way: "For any exponent e < 1/2, a constant K_e > 0 exists such that |x^3 - y^2| > K_ex^e" As much as , at present, this conjecture is neither proved nor disproved, it is interesting to enumerate the known cases when k < x^{1/2}, what I address as "good examples of Hall's conjecture" borrowing notation used for the related and more general ABC conjecture. A detailed theoretical and computational account on this subject can be found in [Elkies] where he published a table with the 25 items known. Simplifying, we can attribute 13 of them to Gebel Petho and Zimmer [GPZ], who developed an algorithm to search for all integer points in the elliptic curves y^2 = x^3 + k for |k| < 100 000. The other 11 items where found by N.D. Elkies using base reduction algorithms. German Saez, Javier Herranz and I have developed a new algorithm based in methods related to those used in [JS] that have found ten new "good examples of Hall's conjecture". Very briefly, the method starts from the fact that for x=t^2 and t integer, k is zero. If we consider rational values of t instead, and the integer values near t^2, we find that the points (x,k) correspond to the integer points of a set of cubic polynomials. Knowing the polynomials, we can select those which may contain a integer point with a small k . The algorithm is probabilistic in the sense that it seeks for "good examples of Hall's conjecture" where they can be found with higher probability. Nevertheless, the algorithm has found all the items known and 10 more which are displayed in the table bellow. One of them, the item #16, unnoticed for any reason in past computations, has been found using the new algorithm. The table contains the parameter r = sqrt(x)/k for each x value. The values of y and k are not listed because they can be easily computed from x taking y as the nearest integer to x^{3/2}. The algorithm was programmed as a PARI script that was translated to C and compiled with the utility GP2C. The executable was run in a Pentium III at 1000 Mhz during 30 days. For more information, you can see the web page: http://www.terra.es/personal9/ismaeljc/hall.html ---- Ismael Jimenez Calvo Good Examples of Hall's Conjecture ======================================== # x r ======================================== 1 2 1.41 2 5234 4.26 GPZ 3 8158 3.76 GPZ 4 93844 1.03 GPZ 5 367806 2.93 GPZ 6 421351 1.05 GPZ 7 720114 3.77 GPZ 8 939787 3.16 GPZ 9 28187351 4.38 GPZ 10 110781386 1.23 GPZ 11 154319269 1.08 GPZ 12 384242766 1.34 GPZ 13 390620082 1.33 GPZ 14 3790689201 2.20 GPZ 15 65589428378 2.19 E 16 952764389446 1.15 JHS 17 12438517260105 1.27 E 18 35495694227489 1.15 E 19 53197086958290 1.66 E 20 5853886516781223 46.60 E 21 12813608766102806 1.30 E 22 23415546067124892 1.46 E 23 38115991067861271 6.50 E 24 322001299796379844 1.04 E 25 471477085999389882 1.38 E 26 810574762403977064 4.66 E 27 9870884617163518770 1.90 JHS 28 42532374580189966073 3.47 JHS 29 51698891432429706382 1.75 JHS 30 44648329463517920535 1.79 JHS 31 231411667627225650649 3.71 JHS 32 601724682280310364065 1.88 JHS 33 4996798823245299750533 2.17 JHS 34 14038790674256691230847 1.27 JHS 35 372193377967238474960883 1.33 JHS GPZ - J. Gebel, A. Petho and H.G.Zimmer. E - Noam D. Elkies JHS - I. Jimenez, J. Herranz and G. Saez. References: [Elkies] Elkies, N.D.: Rational points near curves and small nonzero |x3 - y2| via lattice reduction. Pages 33--63 in Algorithmic Number Theory. Proceedings of ANTS-IV;W. Bosma, ed.; Berlin. Springer, 2000; LNCS 1838. [Hall] Hall, M.: The Diophantine equation x^3 - y^2 = k. Pages 173-198 in Computers in Number Theory (A. Atkin, B. Birch, eds.; Academic Press, 1971). [GPZ] Gebel, J., Petho, A., and Zimmer, H.G.: On Mordell's equation, Compositio Math. 110 (1998), 335-367. [JS] Jimenez Calvo, I. and Saez Moreno, G.: Approximate Power roots in Z_m. pages 310-323 in ISC 2001; G.I. Davida and Y. Frankel, eds.; Berlin: Springer, 2001; LNCS 2200.