Covering systems with restricted divisibility

Abstract

We prove that every distinct covering system has a modulus divisible by either 2 or 3.

An upper bound for the minimum modulus in a covering system with squarefree moduli

“…The computations for (10), in particular for the upper bound on w N (B N ) where N = 10 6 , provide us with all but the last factor (where j = N ) of the product for M 0 in (6). Including that factor and applying Lemma 4.7, we obtain…”

“…Before leaving this introduction, we mention some related literature. Some other recent work on this subject include [7][8][9][10][11][12]. More information related to covering systems with distinct squarefree moduli can be found in Kruckenberg's dissertation [13].…”

University Digitalization - A Fashionable Trend Or Strategic Factor Of Regional Development?

“…We show an example of a covering system with distict moduli in [4,59] and an extra modulus 180. It is (4,3), (8,5), (6,2), (12,1), (24,17), (16,9), (48,33), (9,7), (18,4), (36,10), (5,4), (10,8), (15,0), (20,12), (30,6), (40,22), (45,10), (7,6), (14,12), (21,18), (28,10), (35,2), (42,36), (56,42), and (180, 136).…”

“…Since 2 5 · 3 > 59, 3 3 · 4 > 59, 5 2 · 6 > 59, and 11 · 12 > 59, using Corollary 1 we get L(C)|2 4 · 3 2 · 5 · 7. So, the moduli of C are in {4, 8,16,6,12,24,48,9,18,36,5,10,20,40,15,30,45,7,14,21,28,35, 42, 56}. Since there are exactly 7 multiples of 7, by Lemma 1, we can exchange them for LCM [1, 2, 3, 4, 5, 6, 8] = 120.…”

Covering systems with odd moduli

“…If the answer to this question is yes, then such a covering system would be called an odd covering system of the integers. Hough and Nielsen [12] showed in 2019 that if the moduli of a covering system are distinct and greater than 1, then at least one of the moduli of the system must be divisible by either 2 or 3.…”