A covering system is a finite collection of congruences 
with 
whose residue classes together contain every integer.
It is distinct when its moduli 
are pairwise different (Erdős 1950). A complete
residue system for a single modulus gives a covering, but its moduli repeat.
Filaseta et al. (2026) proved that a distinct covering system with smallest modulus a prime 
satisfies
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Filaseta et al. (2026) also proved that there is no covering system with distinct moduli that are all multiples of such a prime. Filaseta et al. (2026) also proved that at most nine distinct covering systems can have pairwise disjoint sets of moduli.
and Some Related Problems." Summa Brasil. Math. 2,
113-123, 1950.Filaseta, M.; Kalogirou, A.; and Klein, J. "Results
for Distinct Covering Systems with a Prime Minimum Modulus 
." 25 Oct 2026.