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Products of factorials which are products of factorials

Saša Novaković

Abstract

In this note, we look at the diophantine equation $$ \prod_{i=1}^ta_i!=\prod_{j=1}^sn_i!, \quad n_1\geq \cdots \geq n_s\geq 2 \quad \textnormal{and}\quad n_1>a_1\geq a_2\geq\cdots \geq a_t\geq2. $$ \noindent Let $s<t$. Under the (explicit) abc conjecture, we show that it has only finitely many nontrivial solutions in a certain subset of $\mathbb{N}^{t+s}$ of positive asymptotic density.

Products of factorials which are products of factorials

Abstract

In this note, we look at the diophantine equation \noindent Let . Under the (explicit) abc conjecture, we show that it has only finitely many nontrivial solutions in a certain subset of of positive asymptotic density.
Paper Structure (4 sections, 7 theorems, 49 equations)

This paper contains 4 sections, 7 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorems 1.1 and 1.2
  4. Proof of Theorem 1.3

Key Result

Theorem 1.1

The explicit abc conjecture implies that (1) has only finitely many solutions $(a_1,...,a_t,n_1,n_2)\in \mathcal{N}(c)_i$. Hence the explicit abc conjecture implies that has finitely many solutions $(a_1,...,a_t,n_1,n_2)\in \mathcal{N}(c)$.

Theorems & Definitions (12)

  • Conjecture : classical abc conjecture
  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 2 more