2023/08/21 by Saša Novaković, Novaković, Saša · 2 citations
Mathematics · Physics and Astronomy · #11D59 #11D61 #11D72 #14G05 #Advanced Mathematical Theories and Applications #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2308.11002
openalex publication_date 2023/08/21 · openalex created_date 2023/08/24 · openalex updated_date 2026/07/28
In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation n!=x2-1. It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed that a weak form of Szpiro's-conjecture implies that Brocard's equation has finitely many integer solutions. More generally, assuming the ABC-conjecture, Luca showed that equations of the form n!=P(x) where P(x)∈ℤ[x] of degree d≥ 2 have only finitely many integer solutions with n>0. And if P(x) is irreducible, Berend and Harmse proved unconditionally that P(x)=n! has only finitely many integer solutions. In this note we study diophantine equations of the form g(x1,...,xr)=P(x) where P(x)∈ℤ[x] of degree d≥ 2 and g(x1,...,xr)∈ ℤ[x1,...,xr] where for xi one may also plug in An or the Bhargava factorial n!S. We want to understand when there are finitely many or infinitely many integer solutions. Moreover, we study diophantine equations of the form g(x1,...,xr)=f(x,y) where f(x,y)∈ℤ[x,y] is a homogeneous polynomial of degree ≥2.