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# Colloquium Series Seminar

Covering systems of congruences
Tuesday, January 11, 2022
4:00-5:00 PM
1360 East Hall Map
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. Speaker(s): Robert Hough (Stony Brook University)
Building: East Hall Workshop / Seminar Mathematics Happening @ Michigan from Department of Mathematics