On the Gow--McGuire Conjecture for Primitive Quadratic Polynomials

Let $q$ be an odd prime power, let $\mu\in\F_q^\times$, and let $\alpha\in\F_{q^2}\setminus\F_q$. We study primitive polynomials in the family $x^2+\mu x+\lambda-\alpha$, where $\lambda\in\F_q$. A root parametrization reduces the problem to finding primitive values of a rational function on $q+1$ points. Combining character-sum estimates, refined prime sieves, and an exact finite computation, we prove Conjecture~3 of Gow and McGuire for every odd prime power $q>204931$. Their Conjectures~1 and~2 follow in the same range.