Seminar Event Detail


Colloquium Series

Date:  Tuesday, January 11, 2022
Location:  1360 East Hall (4:00 PM to 5:00 PM)

Title:  Covering systems of congruences

Abstract:   A distinct covering system of congruences is a list of congruences
\[
a_i \bmod m_i, \qquad i = 1, 2, ..., k
\]
whose union is the integers. Erd\H{o}s asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollob\'{a}s, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.

Files:


Speaker:  Robert Hough
Institution:  Stony Brook University

Event Organizer:     

 

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