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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)



On the largest prime factor of $ x^2-1$

Authors: Florian Luca and Filip Najman
Journal: Math. Comp. 80 (2011), 429-435
MSC (2010): Primary 11D09, 11Y50
Published electronically: July 20, 2010
Erratum: Math. Comp. 83 (2014), 337.
Table supplement: supplement
MathSciNet review: 2728988
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we find all integers $ x$ such that $ x^{2}-1$ has only prime factors smaller than $ 100$. This gives some interesting numerical corollaries. For example, for any positive integer $ n$ we can find the largest positive integer $ x$ such that all prime factors of each of $ x, x+1,\ldots, x+n$ are less than 100.

References [Enhancements On Off] (What's this?)

  • 1. Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122. With an appendix by M. Mignotte. MR 1863855 (2002j:11027), 10.1515/crll.2001.080
  • 2. Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 27–41. MR 1104698 (92g:11125)
  • 3. R. D. Carmichael, On the numerical factors of the arithmetic forms 𝛼ⁿ±𝛽ⁿ, Ann. of Math. (2) 15 (1913/14), no. 1-4, 30–48. MR 1502458, 10.2307/1967797
  • 4. A.  Dabrowski, On the Brocard-Ramanujan problem and generalizations, Preprint, 2009.
  • 5. Michael J. Jacobson Jr. and Hugh C. Williams, Solving the Pell equation, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2009. MR 2466979 (2009i:11003)
  • 6. D. H. Lehmer, On a problem of Störmer, Illinois J. Math. 8 (1964), 57–79. MR 0158849 (28 #2072)
  • 7. Florian Luca, Primitive divisors of Lucas sequences and prime factors of 𝑥²+1 and 𝑥⁴+1, Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.) 31 (2004), 19–24. MR 2125596 (2005k:11022)
  • 8. M. Maurer, Regulator approximation and fundamental unit computation for real quadratic orders, PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik, Darmstadt, Germany, 2000.
  • 9. F. Najman, Compact representation of quadratic integers and integer points on some elliptic curves, Rocky Mountain J. Math., to appear.
  • 10. T. N.  Shorey and R. Tijdeman, Generalizations of some irreducibility results by Schur, Acta Arith., to appear.

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Additional Information

Florian Luca
Affiliation: Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico

Filip Najman
Affiliation: Department of Mathematics, University of Zagreb, Bijenička Cesta 30, 10000 Zagreb, Croatia

Keywords: Pell equation, compact representation, Lucas sequence.
Received by editor(s): July 16, 2009
Received by editor(s) in revised form: October 27, 2009
Published electronically: July 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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