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On the greatest prime factor of $(ab+1)(ac+1)$

Author(s): P. Corvaja; U. Zannier
Journal: Proc. Amer. Math. Soc. 131 (2003), 1705-1709.
MSC (2000): Primary 11J25
Posted: November 4, 2002
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Abstract: We prove that for integers $a>b>c>0$, the greatest prime factor of $(ab+1)(ac+1)$ tends to infinity with $a$. In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product $(ab+1)(ac+1)(bc+1)$.

References:

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Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of $a^{n}-1$ and $b^{n}-1$, to appear in Math. Zeit.

[CZ]
P. Corvaja, U. Zannier, Diophantine equations with power sums and universal Hilbert sets, Indagationes Math. 9 (1998), 317-332. MR 2000j:11045

[GS]
K. Györy, A. Sarkozy, On prime factors of integers of the form $(ab+1)(ac+1)(bc+1)$, Acta Arith. 79 (1997). MR 98b:11030

[GSS]
K. Györy, A. Sarkozy, C.L. Stewart, On the number of prime factors of integers of the form $ab+1$, Acta Arith. 74 (1996), 365-385. MR 97c:11091

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S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983. MR 85j:11005

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[ST]
C.L. Stewart, R. Tijdeman, On the greatest prime factor of $(ab+1)(ac+1)(bc+1)$, Acta Arith. 79 (1997). MR 98f:11101

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Additional Information:

P. Corvaja
Affiliation: Dipartimento di Matematica e Informatica, via delle Scienze, 206, 33100 Udine, Italy
Email: corvaja@dimi.uniud.it

U. Zannier
Affiliation: Istituto Universitario di Architettura di Venezia - DCA, S. Croce, 191, 30135 Venezia, Italy
Email: zannier@iuav.it

DOI: 10.1090/S0002-9939-02-06771-0
PII: S 0002-9939(02)06771-0
Received by editor(s): February 7, 2002
Posted: November 4, 2002
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2002, American Mathematical Society

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