Problem Statement
Can the smallest modulus of a covering system be arbitrarily large?
Categories:
Number Theory Covering Systems
Progress
Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown (contrary to Erdős' expectations) that the answer is no: the smallest modulus must be at most $10^{16}$.An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the upper bound on the smallest modulus to $616000$.
The best known lower bound is a covering system whose minimum modulus is $42$, due to Owens [Ow14].
Source: erdosproblems.com/2 | Last verified: January 13, 2026