On a generalization of the Brocard--Ramanujan Diophantine equation

TL;DR

This work advances a generalization of the Brocard–Ramanujan Diophantine equation by studying $\prod_{i=1}^r Q_i(A_i^{n_i} n_i!) = f(x,y)$ with $Q_i$ vanishing at $0$ and $f$ a homogeneous polynomial of degree $d\ge2$. It develops an algebraic-number-theoretic framework (splitting fields, Galois groups, Chebotarev density) together with valuation arguments to prove finiteness of solutions under a key degree-multiplicity constraint $d>l_1+\cdots+l_r$ (or its variants for irreducible factors). Conditional results using the abc conjecture extend finiteness to broader cases, and the methods are extended to multi-variable and two-factor specializations, where Stirling-type analytic estimates yield explicit finiteness via $N(F(n_1,\dots,n_r))^{1+\varepsilon}/F(n_1,\dots,n_r)\to0$. Collectively, the paper connects factorial-based Diophantine representations with algebraic geometry and analytic number theory to widen the landscape of finiteness results for structured polynomial equations.

Abstract

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous polynomials of degree $d_i\geq 2$. Fix some positive integers $A_1,...,A_r$. We show that under certain conditions, the diophantine equation $\prod_{i=1}^rQ_i(A_i^{n_i}n_i!)=f(x,y)$ has finitely many integer solutions.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture : abc conjecture
• Theorem 1.5
• Remark 1.6
• Proposition 1.7
• Lemma 2.1: WT, Lemma 3.1
• Lemma 2.2: WT, Lemma 3.2