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Covering systems of congruences
Probability| Speaker: | Robert Hough, Stony Brook University |
| Location: | 2112 MSB |
| Start time: | Thu, May 28 2026, 1:10PM |
Description
A distinct covering system of congruences is a list of congruences \[ a_i \bmod m_i, \qquad 1 < m_1 < m_2 < ... < m_k \] whose union is the integers. Erdos asked if the least modulus $m_1$ can be arbitrarily large ($\$1000$ minimum modulus problem) and if all of the moduli can be odd ($\$25$ odd modulus problem). I will discuss my solution of the minimum modulus problem using techniques from probabilistic combinatorics, and my proof with Pace Nielsen that any distinct covering system of congruences has a modulus divisible by 2 or 3. I'll also discuss recent results of other authors on the problems.