On the greatest prime factor of (ab + 1) (ac + 1) (bc + 1)

Recently, Györy [2] has proved that (1) holds provided that at least one of P (a), P (b), P (c), P (a/b), P (a/c) and P (b/c) is bounded. While we have not been able to prove (1) we have been able to prove that if, a, b, c and d are positive integers with a 6= d and b 6= c then P ((ab+ 1)(ac+ 1)(bd+ 1)(cd+ 1))→∞ as the maximum of a, b, c and d tends to infinity. Notice, by symmetry, that there is no loss of generality in assuming that a ≥ b > c and that a > d. In fact, we are able to give an effective lower bound for the greatest prime factor of (ab+1)(ac+1)(bd+1)(cd+1) in terms of a.