On prime factors of integers of the form \(ab+1\) (Q2707226)

Let \(N\) be a positive integer and let \(A\) and \(B\) be subsets of \(\{1, \ldots ,N\}\). In this paper the authors discuss estimates for the least and the greatest prime factors of integers of the form \(ab+1\), where \(a\in A\) and \(b\in B\). Denoting by \(p(n)\) and \(P(n)\) the least and the greatest prime factor of \(n\), respectively, the following theorem is proved: For each \(\varepsilon >0\) there are numbers \(N_0=N(\varepsilon)\) and \(C=C(\varepsilon)\), which are effectively computable in terms of \(\varepsilon\), such that if \(N>N_0, \;A, B \subset \{1,2,\ldots ,N\}\) and \(\min(|A|,|B|)>C\cdot (N/ \log N)\), then there are \(a\) in \(A\) and \(b\) in \(B\) such that NEWLINE\[NEWLINEP(a\cdot b+1)>(1-\varepsilon)\min (|A|,|B|)\log N.NEWLINE\]NEWLINE An estimation for the primes \(p(ab+1)\) is also shown.