A new algorithm to search for small nonzero $|x^3-y^2|$ values
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- by I. Jiménez Calvo, J. Herranz and G. Sáez PDF
- Math. Comp. 78 (2009), 2435-2444 Request permission
Abstract:
In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero $k=\lvert x^3-y^2\rvert$ values. Seventeen new values of $k<x^{1/2}$ are reported.References
- A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser. A 263 (1967/68), 173–191. MR 228424, DOI 10.1098/rsta.1968.0010
- B. J. Birch, S. Chowla, Marshall Hall Jr., and A. Schinzel, On the difference $x^{3}-y^{2}$, Norske Vid. Selsk. Forh. (Trondheim) 38 (1965), 65–69. MR 186620
- L. V. Danilov, The Diophantine equation $x^{3}-y^{2}=k$ and a conjecture of M. Hall, Mat. Zametki 32 (1982), no. 3, 273–275, 425 (Russian). MR 677595
- H. Davenport, The diophantine equation $y^2 - k = x^3$. Norske Vid. Selsk. Forh. 38 (1965), 86–87.
- Noam D. Elkies, Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 33–63. MR 1850598, DOI 10.1007/10722028_{2}
- J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell’s equation, Compositio Math. 110 (1998), no. 3, 335–367. MR 1602064, DOI 10.1023/A:1000281602647
- Marshall Hall Jr., The Diophantine equation $x^{3}-y^{2}=k$, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 173–198. MR 0323705
- Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoret. Comput. Sci. 3 (1976/77), no. 3, 321–348. MR 498355, DOI 10.1016/0304-3975(76)90050-5
- Serge Lang, Conjectured Diophantine estimates on elliptic curves, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171. MR 717593
- Satya Mohit and M. Ram Murty, Wieferich primes and Hall’s conjecture, C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 1, 29–32 (English, with French summary). MR 1618973
- Joseph Oesterlé, Nouvelles approches du “théorème” de Fermat, Astérisque 161-162 (1988), Exp. No. 694, 4, 165–186 (1989) (French). Séminaire Bourbaki, Vol. 1987/88. MR 992208
- C. Padró and G. Sáez, Taking cube roots in $\Bbb Z_m$, Appl. Math. Lett. 15 (2002), no. 6, 703–708. MR 1913273, DOI 10.1016/S0893-9659(02)00031-9
- The PARI Group. PARI/GP, Version 2.1.0, 2002, Bordeaux. Available from http://www.parigp-home.de/.
- H. M. Stark, Effective estimates of solutions of some Diophantine equations, Acta Arith. 24 (1973), 251–259. MR 340175, DOI 10.4064/aa-24-3-251-259
Additional Information
- I. Jiménez Calvo
- Affiliation: C/Virgen de las Viñas 11, 28031–Madrid, Spain
- Email: ijcalvo@gmail.com
- J. Herranz
- Affiliation: IIIA-CSIC, Campus de la UAB, E-08193 Bellaterra, Catalonia, Spain
- Email: jherranz@iiia.csic.es
- G. Sáez
- Affiliation: Dept. de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, c/Jordi Girona, 1-3, 08034-Barcelona, Spain
- Email: german@ma4.upc.es
- Received by editor(s): April 18, 2005
- Received by editor(s) in revised form: November 11, 2008
- Published electronically: February 13, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 2435-2444
- MSC (2000): Primary 11Y50, 65A05; Secondary 11D25, 14H52
- DOI: https://doi.org/10.1090/S0025-5718-09-02240-6
- MathSciNet review: 2521296