Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

A new algorithm to search for small nonzero $\vert x^3-y^2\vert$ values

Author(s): I. Jiménez Calvo; J. Herranz; G. Sáez.
Journal: Math. Comp. 78 (2009), 2435-2444.
MSC (2000): Primary 11Y50, 65A05; Secondary 11D25, 14H52
Posted: February 13, 2009
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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:

1.: A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms. II. The Diophantine equation . Philos. Trans. Roy. Soc. London, Ser. A 263, (1967-1968), 173-191 and 193-208. MR 0228424 (37:4005); MR0228425 (37:4006)
2.: B.J. Birch, S. Chowla, M. Hall and A. Schinzel, On the difference . Norske Vid. Selsk. Forh. 38 (1965), 65-69. MR 0186620 (32:4079)
3.: L.V. Danilov, The Diophantine equation and Hall's conjecture. Math. Notes Acad. Sci. USSR 32 (1982), 617-618. MR 677595 (84c:10014)
4.: H. Davenport, The diophantine equation . Norske Vid. Selsk. Forh. 38 (1965), 86-87.
5.: N.D. Elkies, Rational points near curves and small nonzero $\vert x^3 - y^2\vert$ via lattice reduction. Pages 33-63 in Algorithmic Number Theory (Proceedings of ANTS-IV;W. Bosma, ed.; Berlin: Springer, 2000; Lecture Notes in Comput. Sci. 1838). MR 1850598 (2002g:11035)
6.: J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation. Compositio Math. 110 (1998), 335-367. MR 1602064 (98m:11049)
7.: M. Hall, The Diophantine equation . Computers in Number Theory (A. Atkin, B. Birch, eds.; Academic Press, 1971), pp. 173-198. MR 0323705 (48:2061)
8.: D. E. Knuth and L. Trabb Pardo, Analysis of a simple factorization algorithm. Theoretical Computer Sci. 3 (1976) pp. 321-348. MR 0498355 (58:16485)
9.: S. Lang, Conjectured Diophantine estimates on elliptic curves. Arithmetic and Geometry, Volume dedicated to Shafarevich, Vol I, edited by M. Artin and J. Tate, Birkhäuser, 1983, pp. 155-171. MR 717593 (85d:11024)
10.: S. Mohit and M. R. Murty, Wieferich primes and Hall's conjecture, Comptes Rendus de l'Acad. Sciences (Canada) 20 (1998), 29-32. MR 1618973 (98m:11004)
11.: J. Oesterlé, Nouvelles approaches du ``théorème'' de Fermat. Sém. Bourbaki 1987/88, Exposé No. 694, 165-186, 1989. MR 0992208 (90g:11038)
12.: C. Padró and G. Sáez, Taking cube roots on $\mathbb{Z}_m$ . Appl. Math. Lett. 15 (2002), 703-708. MR 1913273 (2003e:11139)
13.: The PARI Group. PARI/GP, Version 2.1.0, 2002, Bordeaux. Available from http://www.parigp-home.de/.
14.: H. M. Stark, Effective estimates of solutions of some Diophantine equations. Acta Arith. 24, (1973), 251-259. MR 0340175 (49:4931)

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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

DOI: 10.1090/S0025-5718-09-02240-6
PII: S 0025-5718(09)02240-6
Keywords: Hall's conjecture, Mordell's equation
Received by editor(s): April 18, 2005
Received by editor(s) in revised form: November 11, 2008
Posted: February 13, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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