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Finiteness of Disjoint Covering Systems with Precisely One Repeated Modulus

Yu Hashimoto

Abstract

We prove that for each fixed $m \ge 2$, there are only finitely many disjoint covering systems with minimum modulus at least $3$ in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly $m$ times.

Finiteness of Disjoint Covering Systems with Precisely One Repeated Modulus

Abstract

We prove that for each fixed , there are only finitely many disjoint covering systems with minimum modulus at least in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly times.

Paper Structure

This paper contains 12 sections, 14 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Parallelotope partitions and cell inequalities
  3. Parallelotopes, cells, and cell partitions
  4. Cell inequalities
  5. Two immediate corollaries
  6. Coset-partition formulation of the cell inequalities
  7. Main Result
  8. Notation for the proof
  9. A divisor-sum inequality
  10. Auxiliary Lemmas
  11. Proof of the theorem
  12. Discussion

Key Result

Theorem 1.1

For each fixed $m\ge 2$, there exist only finitely many DCSs whose moduli satisfy eq:basicas.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.4
  • Corollary 2.5
  • Corollary 2.6
  • ...and 17 more