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The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$

Saša Novaković

TL;DR

This work investigates the Diophantine equation $P(x)=\prod_{i=1}^r H_{n_i}$ with $P\in\mathbb{Z}[x]$ and $H_{n_i}$ drawn from divisible, factorial-like sequences. It establishes finiteness of integer solutions in several regimes: (i) for irreducible $P$ with $\deg P\ge2$ and $H_n$ among $A^n n!$, $A^n n!!$, $A^n [1,...,n]$, or $A^n n\#$ (Theorem 1); (ii) under the ABC conjecture for broad classes of $H_n$ (Theorem 2); and (iii) under Szpiro’s weak conjecture for additional cases (Theorem 3). It further proves finiteness for restricted solution sets when the variables are constrained to $\mathcal{F}_k$ or $S_{a,R}$ (Theorems 4 and 5) and finally handles the case $d>r$ via prime-valuation methods to exclude infinitely many solutions (Theorem 6). Across these results, the authors develop a toolkit based on radical bounds, prime divisors, and growth estimates for factorial-like sequences, connecting Diophantine finiteness with deep conjectures in number theory and extending Brocard–Ramanujan-type finiteness phenomena to products of structured divisible sequences.

Abstract

Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for equations of the form $P(x)=\prod_{i=1}^rH_{n_i}$, where $P(x)$ is a polynomial and $H_{n_i}$ are divisible sequences.

The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$

TL;DR

This work investigates the Diophantine equation with and drawn from divisible, factorial-like sequences. It establishes finiteness of integer solutions in several regimes: (i) for irreducible with and among , , , or (Theorem 1); (ii) under the ABC conjecture for broad classes of (Theorem 2); and (iii) under Szpiro’s weak conjecture for additional cases (Theorem 3). It further proves finiteness for restricted solution sets when the variables are constrained to or (Theorems 4 and 5) and finally handles the case via prime-valuation methods to exclude infinitely many solutions (Theorem 6). Across these results, the authors develop a toolkit based on radical bounds, prime divisors, and growth estimates for factorial-like sequences, connecting Diophantine finiteness with deep conjectures in number theory and extending Brocard–Ramanujan-type finiteness phenomena to products of structured divisible sequences.

Abstract

Naciri proved that for any integer , the Brocard--Ramanujan equation has only finitely many integer solutions, assuming is a -free integer or a prime power. In the present paper we prove similar statements for equations of the form , where is a polynomial and are divisible sequences.
Paper Structure (7 sections, 12 theorems, 42 equations)

This paper contains 7 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Proof of Theorem 1
  4. Proof of Theorem 2
  5. Proof of Theorem 3
  6. Proof of Theorems 4 and 5
  7. Proof of Theorem 6

Key Result

Theorem 1

Let $P(x)\in \mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$ of degree $d\geq 2$. Let $H_{n_i}$ be either $A_i^{n_i}[1,...,n_i], A_i^{n_i}n_i!, A_i^{n_i}n_i!!$ or $A_i^{p_i}p_{n_i}\#$ or $A_i^{n_i}p_i\#$ or $A_i^{n_i}n_i\#$. Here $A_i$ are fixed positive integers. Then the diophantine

Theorems & Definitions (18)

  • Theorem 1
  • Conjecture : ABC conjecture
  • Theorem 2
  • Conjecture : Weak form of Szpiro's conjecture
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 2.1
  • proof
  • ...and 8 more