Abstract
We prove that whenever \( \mathcal{A} \) and \( \mathcal{B} \) are dense enough subsets of {1, ..., N}, there exist a ∈ \( \mathcal{A} \) and b ∈ \( \mathcal{B} \) such that the greatest prime factor of ab + 1 is at least \( N^{1 + |\mathcal{A}|/(9N)} \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Bombieri, J. B. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli II, Math. Ann., 227 (1987), 361–393.
G. Harman and K. Matomäki, Some problems of analytic number theory on arithmetic semigroups, Funct. Approx. Comment. Math., 38 (2008), 21–40.
C. Hooley, Applications of Sieve Methods, volume 70 of Cambridge Tracts in Mathematics, Cambridge University Press (Cambridge, 1976).
H. Iwaniec, Rosser’s sieve, Acta Arith., 36 (1980), 171–202.
H. Iwaniec and E. Kowalski, Analytic Number Theory, volume 53 of American Mathematical Society Colloquium Publications (Providence, Rhode Island, 2004).
A. Sárközy, Unsolved problems in number theory, Period. Math. Hungar., 42 (2001), 17–35.
A. Sárközy and C. L. Stewart, On prime factors of integers of the form ab + 1, Publ. Math. Debrecen, 56 (2000), 559–573.
C. L. Stewart, On the greatest prime factor of integers of the form ab + 1, Period. Math. Hungar., 43 (2001), 81–91.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Matomäki, K. On the greatest prime factor of ab + 1. Acta Math Hung 124, 115–123 (2009). https://doi.org/10.1007/s10474-009-8163-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-009-8163-5