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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.

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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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