P. Bui

Directed graphs defined by prime divisors of polynomial values

Let L be the class of all nonconstant polynomials f(x) == Ax2 + Bx + C ∈ Z[x], which are not of the forms Ax2, Bx. We consider the directed graph Gf whose vertex set is P and p → q iff q | f(p), p ̸= q. Let Lf (p) be the set of simple directed paths starting at p and ℓf (p) = sup{|L| : L ∈ Lf (p)}, where |L| is the num...