Talk:Covering system

Merger proposal
It seems to me that Covering set and Covering system are the simply the same thing. Richard Pinch (talk) 11:50, 3 August 2008 (UTC)

I'd say that these deserve different articles. Technically, you could consider a covering set, P, to be a covering system for the sequence, S, it covers; you just have to change the rule "every member of S is divisible by a member of P" to "every member of S is in the arithmetic progression of multiples of some p in P". Even then, though, you have greatly expanded the notion of covering system, since you are now including "covering systems" that cover subsets of the integers, rather than the entire set of integers. In other words, covering sets might be called a generalization of covering systems (taking away the restriction that the set of numbers covered must be all of Z). I say "might be called a generalization" because they are not quite so; we add the restrictions, in considering a covering set of primes to be the covering system of arithmetic multiples of those primes, that the moduli in our covering system must be prime and that the remainders must all be 0. So neither idea fully contains the other one. Charlesfahringer (talk) 19:11, 27 September 2008 (UTC)

"Unsolved problem" definitely solved
The preprint of Bob Hough mentioned in the article did appear meanwhile http://annals.math.princeton.edu/2015/181-1/p06 and thus this problem is no longer open.

One should update this (there is also an explicit bound 10^16).

And instead one should put into that box "List of unsolved problems in mathematics" the other famous open problem regarding covering systems, the problem of odd moduli, mentioned as "famous unsolved conjecture from Erdős and Selfridge".


 * The article (as I found it today) has already been updated to reflect the suggestions in the preceding unsigned comments. yoyo (talk) 23:01, 20 March 2018 (UTC)

Tone and accessibility
The tone of the present article is suitable for people who use mathematics professionally, but somewhat too remote to be readily accessible by the lay reader. Articles in number theory have these distinct and identifiable audiences:
 * Readers with little exposure to maths, brought here by curiosity
 * Recreational mathematicians
 * Readers with professions in the mathematical sciences who:
 * Have no special knowledge of the topic, or
 * Are topic specialists

Although any of these readers may seek to improve the article in various ways, those in the first two categories — the lay readers — may struggle with a very succinct presentation. They may need better motivation to understand (or bother reading!) a rather dry recitation of facts.

Accordingly, I've added a couple of very simple examples and facts to the end of the section on Examples. However, this is just a first step towards making the article better reading for lay readers, and the material I've just added really belongs up front. Most Wikipedia readers stumbling on the present lead would probably go no further. Right now, it's a rigorous mathematical definition — and nothing else! We should be able to manage a gentler introduction to this fascinating topic. I've put a rewrite on my "To do" list, but if you get here before I can, please have at it! yoyo (talk) 23:23, 20 March 2018 (UTC)