ON A CONJECTURE OF CRITTENDEN AND VANDEN-EYNDEN CONCERNING COVERINGS BY ARITHMETIC PROGRESSIONS

Crittenden and Vanden Eynden conjectured that if n arithmetic progress ions, each having modulus at least k, include all the integers from 1 to k2(n-k+1), then they include all the integers. They proved this for the cases k = 1 and k = 2. We give various necessary conditions for a counterexample to the conjecture; in particular we show that ifa coun terexample exists for some value of k, then one exists for that k and a value of n less than an explicit function of k.