2026/07/30 by Juncheng Zhou, Hongfeng Wu
Mathematics · #math.NT
arxiv created 2026/07/30 · arxiv updated 2026/07/31
Let \(q\) be an odd prime power, let \(μ∈\Fq^×\), and let \(α∈\Fq2∖\Fq\). We study primitive polynomials in the family \(x2+μx+λ-α\), where \(λ∈\Fq\). 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.