On the largest prime factor of

Authors:
Florian Luca and Filip Najman

Journal:
Math. Comp. **80** (2011), 429-435

MSC (2010):
Primary 11D09, 11Y50

DOI:
https://doi.org/10.1090/S0025-5718-2010-02381-6

Published electronically:
July 20, 2010

Erratum:
Math. Comp. 83 (2014), 337.

Table supplement:
supplement

MathSciNet review:
2728988

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we find all integers such that has only prime factors smaller than . This gives some interesting numerical corollaries. For example, for any positive integer we can find the largest positive integer such that all prime factors of each of are less than 100.

**1.**Yu. Bilu, G. Hanrot, P. M. Voutier, `Existence of primitive divisors of Lucas and Lehmer numbers. With an appendix by M. Mignotte',*J. Reine Angew. Math.***539**(2001), 75-122. MR**1863855 (2002j:11027)****2.**J. Buchmann,*A subexponential algorithm for the determination of class groups and regulators of algebraic number fields*, Seminaire de Theorie des Nombres (1990), 27-41. MR**1104698 (92g:11125)****3.**R. D. Carmichael,*On the numerical factors of arithmetic forms*, Ann. of Math.**15**(1913), 30-70. MR**1502458****4.**A. Dabrowski,*On the Brocard-Ramanujan problem and generalizations*, Preprint, 2009.**5.**M. J. Jacobson Jr., H. C . Williams,*Solving the Pell Equation*, Springer, 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.**F. Luca, `Primitive divisors of Lucas sequences and prime factors of and ',*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.

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
11D09,
11Y50

Retrieve articles in all journals with MSC (2010): 11D09, 11Y50

Additional Information

**Florian Luca**

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

Email:
fluca@matmor.unam.mx

**Filip Najman**

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

Email:
fnajman@math.hr

DOI:
https://doi.org/10.1090/S0025-5718-2010-02381-6

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