Products of \(p+a\) and the graphs of I. Katai (Q1198493)

The study of additive functions on the set \(p+a\) leads to the following graph: we draw an edge from a prime \(p\) to another prime \(q\) if \(q\mid p+a\). \textit{R. M. Pollack}, \textit{H. N. Shapiro} and \textit{G. H. Sparer} [Commun. Pure Appl. Math. 27, 669-713 (1974; Zbl 0302.10006)] conjectured that (besides the singular loops \(p\to p\) for prime divisors of a) there is only one connected component in this graph and it contains the smallest prime not dividing \(a\). The paper formulates two equivalent conjectures and quotes some numerical investigations that verify it for \(a\leq 3000\).