Integer Translates of a Monic Cubic: Resultants, Finite-Field GCDs, and a General Shift-Counting Dichotomy

We study integer translates of a monic cubic through resultants and finite-field GCDs, deriving an explicit translate-resultant identity. When the associated auxiliary cubic F is irreducible with Galois group S₃, we show that, for all but finitely many primes p, the number of nonzero shifts giving a nontrivial GCD modulo p is always 0 or 6, with 6 occurring exactly when F splits completely. Via the Spearman–Williams theorem, this splitting condition translates into representation by binary quadratic forms, illustrated at discriminants −23 and −87.