# On integer covering systems with all moduli distinct

Hi all. There is a conjecture by Erdős that states "There doesn't exist an integer covering system with all moduli odd AND distinct." The link is http://en.wikipedia.org/wiki/Covering_system

I think that, if a covering exists with all moduli distinct, not only the lcm of the moduli must be even, it must also be abundant. In this sense, the odd perfect number conjecture and this conjecture are related in my opinion.

Is my conjecture true or any counterexample exists? Has the same conjecture been made before? Any help would be appreciated. Thanks in advance.

Each modulus $m$ covers a proportion $1/m$ of the integers. To cover all the integers, we need $\sum 1/m\ge1$, where the sum is over all the moduli. Multiply by the lcm, $L$, to get $\sum(L/m)\ge L$. But all the terms in the sum are distinct, proper divisors of $L$, so $L$ is abundant (unless we have equality, but it's known that a distinct cover can't be exact, so in fact all the inequalities are strict).