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Divisibility rules for integers presented as permutations

Thomas Oliver, Alexei Vernitski

TL;DR

The paper develops a factorial-number-system framework using tame permutations to represent integers and derives linear divisibility criteria that depend only on a finite initial block of factoradic digits, via inversion sets of prefix permutations. The main theorem expresses $n$ modulo $k$ as $n \equiv \sum_{i<j} \overleftrightarrow{i,j} \cdot j! \bmod k$, enabling simple arithmetical tests for divisibility by small $k$. This approach connects classical modular checks to a permutation-based representation, highlighting finite, linear divisibility rules and the structure of factoradic digits. The work provides a bridge between combinatorial representations and modular arithmetic, with concrete corollaries for practical divisibility testing.

Abstract

In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the factoradic digits, and use this to deduce various divisibility rules.

Divisibility rules for integers presented as permutations

TL;DR

The paper develops a factorial-number-system framework using tame permutations to represent integers and derives linear divisibility criteria that depend only on a finite initial block of factoradic digits, via inversion sets of prefix permutations. The main theorem expresses modulo as , enabling simple arithmetical tests for divisibility by small . This approach connects classical modular checks to a permutation-based representation, highlighting finite, linear divisibility rules and the structure of factoradic digits. The work provides a bridge between combinatorial representations and modular arithmetic, with concrete corollaries for practical divisibility testing.

Abstract

In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the factoradic digits, and use this to deduce various divisibility rules.

Paper Structure

This paper contains 3 sections, 4 theorems, 5 equations, 1 table.

Table of Contents

  1. Introduction
  2. What is the factoradic form of a non-negative integer?
  3. Using inversions to describe divisibility

Key Result

Theorem 1.1

[Corollary c.cor] For $n\in\mathbb{Z}_{\geq0}$ and $k\in\mathbb{Z}_{>0}$, we have where the values $\overleftrightarrow{i, j}$ are considered in the inversion set of the $k$-prefix for the factoradic form for $n$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 3.1
  • ...and 4 more