On the greatest prime factor of $(ab+1)(ac+1)$
HTML articles powered by AMS MathViewer
- by P. Corvaja and U. Zannier
- Proc. Amer. Math. Soc. 131 (2003), 1705-1709
- DOI: https://doi.org/10.1090/S0002-9939-02-06771-0
- Published electronically: November 4, 2002
- PDF | Request permission
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 Gyรถry, Sarkozy and Stewart, predicting the same conclusion for the product $(ab+1)(ac+1)(bc+1)$.References
- Yann Bugeaud, On the greatest prime factor of $(ab+1)(bc+1)(ca+1)$, Acta Arith. 86 (1998), no.ย 1, 45โ49. MR 1642613, DOI 10.4064/aa-86-1-45-49
- 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.
- P. Corvaja and U. Zannier, Diophantine equations with power sums and universal Hilbert sets, Indag. Math. (N.S.) 9 (1998), no.ย 3, 317โ332. MR 1692189, DOI 10.1016/S0019-3577(98)80001-3
- K. Gyลry and A. Sรกrkรถzy, On prime factors of integers of the form $(ab+1)(bc+1)(ca+1)$, Acta Arith. 79 (1997), no.ย 2, 163โ171. MR 1438599, DOI 10.4064/aa-79-2-163-171
- K. Gyลry, A. Sรกrkรถzy, and C. L. Stewart, On the number of prime factors of integers of the form $ab+1$, Acta Arith. 74 (1996), no.ย 4, 365โ385. MR 1378230, DOI 10.4064/aa-74-4-365-385
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
- Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR 1176315, DOI 10.1007/BFb0098246
- C. L. Stewart and R. Tijdeman, On the greatest prime factor of $(ab+1)(ac+1)(bc+1)$, Acta Arith. 79 (1997), no.ย 1, 93โ101. MR 1438120, DOI 10.4064/aa-79-1-93-101
Bibliographic Information
- P. Corvaja
- Affiliation: Dipartimento di Matematica e Informatica, via delle Scienze, 206, 33100 Udine, Italy
- MR Author ID: 327308
- Email: corvaja@dimi.uniud.it
- U. Zannier
- Affiliation: Istituto Universitario di Architettura di Venezia - DCA, S. Croce, 191, 30135 Venezia, Italy
- MR Author ID: 186540
- Email: zannier@iuav.it
- Received by editor(s): February 7, 2002
- Published electronically: November 4, 2002
- Communicated by: David E. Rohrlich
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1705-1709
- MSC (2000): Primary 11J25
- DOI: https://doi.org/10.1090/S0002-9939-02-06771-0
- MathSciNet review: 1955256