# Covering systems of congruences

#### Abstract

In 1950 Erdos made the following conjecture ($1000 reward for a solution): Given an integer {\it m\/} there exists a collection of congruences {\it C\/} such that: (i)~Every integer satisfies at least one member of {\it C}. (ii)~No modulus appears more than once in {\it C}. (iii)~The least modulus in {\it C} is greater than {\it m}. Collections of arithmetic progressions satisfying (i) are now known as Covering Systems. Since Erdos made this conjecture the study of covering systems has encompassed a growing range of problems. We deal with three areas in particular. First, we show that every disjoint covering system corresponds to a combinatorial identity, we show how to determine the identity and we give a new identity discovered using this method. We also give necessary and sufficient conditions on a disjoint covering system for this identity to be nice (single terms on the left and right hand sides). Second, we develop an analysis of disjoint and incongruent covering systems that allows us to prove the nonexistence of finite and infinite covering systems which are both incongruent and disjoint. Third, we examine some functions related to Beatty Sequences (covering systems with irrational moduli). We give a simple characterization of $m\in \rm{I\!N}$ such that the fractional part of $m\sqrt{2}$ (written (($m\sqrt{2})))$ is less than $1/\sqrt{2}.$ We use this characterization to find easily computed formulas for F(1,N) = $\vert\{m\ :\ 1\le m\le N,\ ((m\sqrt{2})) < {1\over\sqrt{2}}\}\vert$ and $\Delta (1,N)$ = F$(1,N)-{N\over\sqrt{2}}.$

#### Subject Area

Mathematics

#### Recommended Citation

Lewis, Ethan L, "Covering systems of congruences" (1994). *Dissertations available from ProQuest*. AAI9427566.

https://repository.upenn.edu/dissertations/AAI9427566