A sieve application (Q5947039)

The author proves the following Theorem. Suppose \(g\geq 3\) is fixed. There exist positive absolute constants \(A\) and \(B\), and a \(g\)-term arithmetic progression in the interval \((1, 1+ g^{Ag})\), so that the prime factors of all the terms of this progression are at least as large as \(g^B\). NEWLINENEWLINENEWLINEThe author reformulates this problem as a sieve result, and he completes the proof by applying a version of the ``fundamental lemma'' [see Theorem 7.5 of \textit{H. Halberstam} and \textit{H.-E. Richert}, Sieve Methods, Academic Press (1974; Zbl 0298.10026)]. While any version of the fundamental lemma will serve for the proof, the author chooses the Brun-Hooley ``almost pure'' sieve [\textit{K. Ford} and \textit{H. Halberstam}, J. Number Theory 81, 335-350 (2000; Zbl 0978.11049)]. This, combined with some explicit estimates due to Rosser and Schoenfeld, allow him to establish the theorem with explicit values of \(A\) and \(B\). For example, the theorem is true with \(A=438\), \(B=14\) and with \(A=163\), \(B=5\).