Gow--McGuire Conjecture 1. Let q≠ 13 be any odd prime power. Let alpha∈F_q^2setminusF_q be fixed. Then there exists lambda∈F_q such that f_lambda(x)=x^2+x+lambda-alpha is a primitive quadratic polynomial in F_q^2[x].

Gow–McGuire conjecture on primitive quartic polynomials

The Gow–McGuire conjectures assert the existence of primitive polynomials in specified quartic-related families over finite fields. Recent preprints claim asymptotic resolution and, in a stronger version, all cases except explicitly listed small-field exceptions.

August 2026 developments

Gow and McGuire claim Conjecture 1 for all sufficiently large odd prime powers $q$ , using a Fu–Wan character-sum estimate; small- $q$ cases remain untreated. A separate preprint claims Conjecture 1 for every odd prime power $q\ne13$ , and Conjectures 2 and 3 for $q>43$ , with exact finite verification. Both claims are unrefereed; the stronger result supersedes its earlier $q>204931$ version. The asymptotic paper says its proof was assisted by Aristotle.

Current status (as of August 2026): The conjecture is claimed proved asymptotically, and a separate unrefereed preprint claims the sharper ranges $q\ne13$ and $q>43$ ; these proofs remain unverified and the exceptional small- $q$ cases are not settled.

Gow–McGuire conjecture on primitive quartic polynomials

Progress summary

Primary source

Additional references

Solutions 0