On the equation Π_P Φ_3(P) = Π_P P^2 over F_2[x]

Abstract

We work a polynomial variant of an arithmetic problem solved by Steuerwald in 1937. More precisely, we prove, under a mild condition, that the equation on the title, has no solutions A := Π_P P²  ∈ F₂[x], with irreducible P, and ω(A) < 16. Unconditionally, we prove that deg(P) ≥ 40. This improves on known results.