Exact coverings of the integers by arithmetic progressions
A system of arithmetic progressions is said to be exactly covering (ECS) if each integer belongs to exactly one progression of that system. An ECS in which every modulus (difference) occurs at most M times is denoted by ECS(M). Main theorem: Let \(p_ 1<...<p_ t\) be the distinct prime divisors of the moduli in an ECS(M), then \(p_ t\leq M\prod^{t-1}_{i=1}p_ i/(p_ i-1).\) This is a generalization of a theorem by N. Burshtein, who proved similar results for a special case of so called natural exactly covering systems. Corollary (4.1). The greatest prime \(p_ t\) dividing the modulus of any arithmetic progression in an ECS(M) is at most equal to the greatest prime \(p^*\) satisfying \(M\prod_{p<p^*}p/(p-1)\geq p^*.\) This represents for the case \(M=2\) the proof of a conjecture by Burshtein: all moduli of an ECS(2) are of the form \(2^ a 3^ b 5^ c 7^ d.\)
upper and lower bounds on prime divisors of moduli
scientific article; zbMATH DE number 3948406