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https://doi.org/10.3336/gm.60.1.02

A note on some polynomial-factorial Diophantine equations

Saša Novaković ; Hochschule Fresenius University of applied Sciences, 40476 Düsseldorf, Germany

Preuzmi JATS datoteku

Sažetak

In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation
\(n!=x^2-1\). 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 \(P(x)\in\mathbb{Z}[x]\) of
degree \(d\geq 2\) have only finitely many integer solutions with \(n\gt 0\). 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 \(g(x_1,\ldots,x_r)=P(x)\) where \(P(x)\in\mathbb{Z}[x]\) of degree \(d\geq 2\) and \(g(x_1,\ldots,x_r)\in \mathbb{Z}[x_1,\ldots,x_r]\)
where for the \(x_i\) one may also plug in \(A^{n}\) 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 \(g(x_1,\ldots,x_r)=f(x,y)\) where \(f(x,y)\in\mathbb{Z}[x,y]\) is
a homogeneous polynomial of degree \(\geq2\).

Ključne riječi

Diophantine equations, factorials, polynomials

Hrčak ID:

332475

URI

https://hrcak.srce.hr/332475

Datum izdavanja:

6.3.2026.

Posjeta: 403 *