Some extensions and refinements of a theorem of Sylvester (Q2781458)

Let \(d>1\), \(k\geq 3\), \(n\geq 1\) be integers with \(\gcd(n,d)= 1\). Further, let NEWLINE\[NEWLINE\Delta_0= n(n+d)\dots (n+(k-1)d).NEWLINE\]NEWLINE For any integer \(\nu> 1\), let \(\omega(\nu)\) be the number of distinct prime factors of \(\nu\) and \(P(\nu)\) its greatest prime factor. Sylvester proved that \(P(\Delta_0)> k\) if \(n\geq d+k\), which was improved by \textit{T. N. Shorey} and \textit{R. Tijdeman} [A tribute to Paul Erdős, Cambridge Univ. Press, 385-389 (1990; Zbl 0709.11004)] to: \(P(\Delta_0)> k\) unless \((n,d,k)= (2,7,3)\). \textit{T. N. Shorey} and \textit{R. Tijdeman} also proved [J. Sichuan Univ., Nat. Sci. Ed. 26, Spec. Iss., 72-74 (1989; Zbl 0705.11052)] that \(\omega(\Delta_0)\geq \pi(k)\), which was sharpened by \textit{P. Moree} [Acta Arith. 70, 295-312 (1995; Zbl 0821.11044)] to: \(\omega(\Delta_0)> \pi(k)\) if \(k\geq 4\) and \((n,d,k)\neq (1,2,5)\). Because of a conjecture of Schinzel, known as Hypothesis H, Shorey and Tijdeman's \(\omega\) result is likely to be best possible for \(k=3\). For \(k=4\) or 5 Hypothesis H implies that \(\omega(\Delta_0)= \pi(k)+1\) for infinitely many \(d\). NEWLINENEWLINENEWLINEThe main result of the present paper considerably improves Moree's result for \(k\geq 6\). In fact it is shown that \(\omega(\Delta_0)> \frac 65\pi(k)+1\) if \(k\geq 6\), except for 14 explicitly given triples \((n,d,k)\). This result includes all four results mentioned above for \(k\geq 6\). The proofs of this and related results are elementary in nature, but the computational effort required is substantial.