Integer Translates of a Monic Cubic: Resultants and Common Factors over Finite Fields

We study integer translates f(x) and f(x+k) of a monic cubic polynomial over the integers. We derive an explicit formula for their resultant and use it to classify the possible degrees of their greatest common divisor after reduction modulo a prime. We then specialize to P(x) = x^3 - x^2 + 1, obtaining a shift-counting criterion and analyzing the associated cubic polynomial and its Galois group. The resulting conditions lead to a density statement for primes admitting nontrivial shifts.