On the finiteness of solutions for polynomial-factorial Diophantine equations

Abstract

Abstract We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that l ! {l!} is represented by N A ⁢ ( x ) {N_{A}(x)} , where N A {N_{A}} is a norm form constructed from the field norm of a field extension K / 𝐐 {K/\mathbf{Q}} . We also deal with the equation N A ⁢ ( x ) = l ! S {N_{A}(x)=l!_{S}} , where l ! S {l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of 𝐙 {\mathbf{Z}} and for any polynomial P ⁢ ( x ) ∈ 𝐙 ⁢ [ x ] {P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation P ⁢ ( x ) = l ! S {P(x)=l!_{S}} has only finitely many solutions ( x , l ) {(x,l)} . For some special infinite subsets S of 𝐙 {\mathbf{Z}} , we can show the finiteness of solutions for the equation P ⁢ ( x ) = l ! S {P(x)=l!_{S}} unconditionally.