A Group-theoretic Approach to Covering Systems
In this article, we show how group actions can be used to examine the set of all covering systems of the integers with a fixed set of distinct moduli.
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Інститут прикладної математики і механіки НАН України
2015
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| Цитувати: | A Group-theoretic Approach to Covering Systems / L. Jones, D. White // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 250-262. — Бібліогр.: 49 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859831592471494656 |
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| author | Jones, L. White, D. |
| author_facet | Jones, L. White, D. |
| citation_txt | A Group-theoretic Approach to Covering Systems / L. Jones, D. White // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 250-262. — Бібліогр.: 49 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | In this article, we show how group actions can be used to examine the set of all covering systems of the integers with a fixed set of distinct moduli.
|
| first_indexed | 2025-12-07T15:32:21Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 250–262
© Journal “Algebra and Discrete Mathematics”
A group-theoretic approach to covering systems
Lenny Jones and Daniel White
Communicated by L. A. Kurdachenko
Abstract. In this article, we show how group actions can
be used to examine the set of all covering systems of the integers
with a fixed set of distinct moduli.
1. Introduction
A (finite) covering system C, or simply a covering, of the integers is a
system of t congruences x ≡ ri (mod mi), with mi > 1 for all 1 6 i 6 t,
such that every integer n satisfies at least one of these congruences.
The concept of a covering was introduced by Paul Erdős in a paper in
1950 [8], where he used a covering to find an arithmetic progression of
counterexamples to Polignac’s conjecture that every positive integer can
be written in the form 2k + p, where p is a prime. Since then, numerous
authors have used covering systems to investigate and solve various
problems [1–4,4–7,9–16,18–21,23–30,32–35,37–41,43–49].
Under the restriction that all moduli in a covering are distinct, Erdős
made the following statement in [8]:
“It seems likely that for every c there exists such a system all
the moduli of which are > c."
This conjecture, known as the minimum modulus problem, remained
unresolved until recently when Bob Hough [20] showed that it is false.
Since the minimum modulus in a covering is now known to be bounded
2010 MSC: Primary 11B25; Secondary 05E18, 11A07.
Key words and phrases: covering system, group action, congruence.
L. Jones, D. White 251
above, one can naively speculate as to whether a categorization of all
covering systems with a fixed minimum modulus might be possible in
some way. Admittedly, such a notion seems intractable, if not impossible.
But perhaps, a less ambitious task is possible. For example, could an
enumeration be given of all coverings with a fixed set of moduli or a fixed
least common multiple of the moduli? Recently [31], we have accomplished
this goal for a very specific situation involving primitive covering numbers–
a notion introduced by Zhi-Wei Sun [47] in 2007. While the methods in
[31] are purely combinatorial, we show in this article how certain group
actions can be used to examine the set of all covering systems of the
integers with a fixed set of distinct moduli.
2. Preliminaries
It will be convenient on occasion to write any covering C = {(ri, mi)},
where x ≡ ri (mod mi) is a congruence in the covering, simply as C =
[r1, r2, . . . , rt], when the moduli are written as a list [m1, m2, . . . , mt]. We
write lcm(M) to denote the least common multiple of the elements in a
set or list of moduli M . We let ΓM , or simply Γ, if there is no ambiguity,
denote the set of all coverings having moduli M . We define a covering C
to be minimal if no proper subset of C is a covering . We also define a
set, or list, of distinct moduli M to be minimal if every possible covering
using all the elements of M is minimal. A positive integer L is called
a covering number if there exists a covering of the integers where the
moduli are distinct divisors of L greater than 1. A covering number L is
called a primitive covering number if no proper divisor of L is a covering
number. The following two theorems concerning covering numbers, which
we state without proof, are due to Zhi-Wei Sun [47].
Theorem 2.1. Let p1, p2, . . . , pr be distinct primes, and let a1, a2, . . . , ar
be positive integers. Suppose that
∏
0<t<s
(at + 1) > ps − 1 + δr,s, for all s = 1, 2, · · · , r, (1)
where δr,s is Kronecker’s delta, and the empty product
∏
0<t<1(at + 1) is
defined to be 1. Then pa1
1 pa2
2 · · · par
r is a covering number.
Infinitely many primitive covering numbers can be constructed using
Theorem 2.1. We let ⌊x⌋ denote the greatest integer less than or equal
to x.
252 A group-theoretic approach to covering systems
Theorem 2.2. Let r > 1 and let 2 = p1 < p2 < · · · < pr be primes.
Suppose further that pt+1 ≡ 1 (mod pt − 1) for all 0 < t < r − 1, and
pr > (pr−1 − 2)(pr−1 − 3). Then
p
p2−1
p1−1
−1
1 . . . p
pr−1−1
pr−2−1
−1
r−2 p
⌊
pr−1
pr−1−1
⌋
r−1 pr
is a primitive covering number.
It is straightforward to see that Theorem 2.2 produces an infinite set
L of primitive covering numbers, and that every element of L satisfies (1).
In [47], Sun conjectured that every primitive covering number pα1
1 · · · pαr
r ,
where p1, . . . , pr are distinct primes, satisfies (1). However, this conjecture
is now known to be false [31].
Unless stated otherwise, we assume throughout this article that the
moduli in all coverings are distinct, and that all sets of moduli are minimal.
3. Counting the number of coverings
without group theory
While it is the main goal of this paper to use group-theoretic techniques
to impose some structure on, and examine, the set of all coverings with
a fixed list of distinct moduli, there are certain situations when some
information can be obtained without the use of group theory. In particular,
using a combinatorial approach, a formula was given in [31] for |ΓM |,
when L ∈ L and M is minimal with lcm(M) = L. The following theorem
illustrates another situation when |ΓM | can be determined without the
use of group theory.
Theorem 3.1. For k > 2, let
Mk = [2, 22, . . . , 2k, 3, 2k−1 · 3, 2k · 3].
For brevity of notation, let Γk denote the set of all coverings using the
moduli Mk. Then
|Γk| = 2k+1 · 3.
Proof. The proof is by induction on k. First let k = 2. The set Γ2 of all
possible coverings using the moduli M2 = [2, 4, 3, 6, 12] is easy to generate
L. Jones, D. White 253
using a computer. We get that
Γ2 = {[0, 1, 0, 1, 11], [0, 1, 0, 5, 7], [0, 1, 1, 3, 11], [0, 1, 1, 5, 3],
[0, 1, 2, 1, 3], [0, 1, 2, 3, 7], [1, 2, 0, 2, 4], [1, 0, 0, 2, 10],
[1, 0, 0, 4, 2], [1, 0, 1, 0, 2], [1, 2, 0, 4, 8], [1, 0, 1, 2, 6],
[1, 0, 2, 0, 10], [1, 2, 1, 0, 8], [1, 0, 2, 4, 6], [1, 2, 1, 2, 0],
[1, 2, 2, 0, 4], [1, 2, 2, 4, 0], [0, 3, 0, 1, 5], [0, 3, 0, 5, 1],
[0, 3, 1, 3, 5], [0, 3, 1, 5, 9], [0, 3, 2, 1, 9], [0, 3, 2, 3, 1]} .
(2)
Observe that |Γ2| = 24, so that the base case is verified. Let Lk = 2k · 3.
Assume, by induction, that |Γk| = 2k+1·3. Let M̂k = {2, 22, . . . , 2k, 3, 2k ·3}.
Let R̂k be a list of residues in a covering in Γk corresponding to the moduli
M̂k. There is just one hole modulo Lk left to fill to complete a covering in
Γk, and this can be done in exactly one way using a residue r (mod 2k−1·3).
Thus, there are exactly two holes modulo Lk+1 that need to be filled to
complete a covering in Γk. These two holes can be filled in exactly two
ways using the two moduli 2k+1 and 2k+1 · 3 in the following way. We can
use either
r (mod 2k+1) and r + 2k · 3 (mod 2k+1 · 3),
or
r + 2k · 3 (mod 2k+1) and r (mod 2k+1 · 3).
Thus, we have shown that |Γk+1| = 2 |Γk| = 2k+2 · 3, and the proof is
complete.
Remark 3.2. Note that when k = 2 in Theorem 3.1, we have L = 12 ∈ L,
and so this is a special case addressed in [31].
4. Group theory and covering systems
In this section, we develop a group-theoretic approach to describe
a relationship among the elements in Γ, and to help determine |Γ|. In
particular, we investigate when there exist finite groups that act on Γ and
we exploit this action to enumerate and categorize the elements of Γ. We
let orbG(C) and stabG(C) denote, respectively, the orbit and stabilizer of
C ∈ Γ under the action of some group G. We begin by providing a brief
analysis, without general proofs, in the situation when L ∈ L and M is
minimal.
254 A group-theoretic approach to covering systems
4.1. A group action in Sun’s primitive covering number
situation
A formula was given in [31] for |ΓM | when L ∈ L and M is minimal
with lcm(M) = L. From this formula, a finite group G can be constructed
that acts transitively on ΓM . This formula, and consequently the group
G, are quite complicated in general. However, in special situations, G can
be described fairly easily. Let
L = pα1
1 pα2
2 · · · p
αr−1
r−1 pr ∈ L.
Under certain restrictions, the formula in [31] for |ΓM | reduces to
|ΓM | =
r∏
i=1
(pi!)
αi . (3)
Remark 4.1. Formula 3 also holds for values of L 6∈ L. See Table 1.
A consequence of (3) is the existence of a finite group
G ≃ (Sp1
)a1 × · · · × (Spr )ar , (4)
where
(Spi
)ai = Spi
× · · · × Spi︸ ︷︷ ︸
ai−factors
and Spi
is the symmetric group on pi letters, that acts transitively on Γ by
appropriately permuting the residues. The following example illustrates
this process.
An example: L=12 with M =[m1,m2,m3,m4,m5]=[2,4,3,6,12]
We see easily that L = 12 is a primitive covering number satisfying (1).
• p1 = 2
We seek a group H1 ≃ S2 × S2. We start with the element h =
(12)(34). To construct the other three nontrivial elements of H1, we
conjugate h by the elements (24) and (23) to get
H1 = {(1), (12)(34), (14)(23), (13)(24)}
• p2 = 3
We seek a group H2 ≃ S3. Let
H2 = {(1), (12), (23), (13), (123), (132)} .
L. Jones, D. White 255
Therefore, G = H1 ×H2. We write a covering C as [r1, r2, r3, r4, r5], where
ri (mod mi) is a congruence in C. We illustrate the action on the set
Γ of all 24 coverings given in (2). As an example, let C = [1, 2, 1, 0, 8].
We use the Chinese remainder theorem to decompose the residues on
the composite moduli into prime power moduli, and we substitute pk
i
(mod pk
i ) for 0 (mod pk
i ). We also place subscripts on the residues in these
decompositions to remind us of the prime power moduli. Thus,
C = [1, 2, 1, [22, 33], [44, 23]] .
Let g = ((14)(23), (123)). Then
g.C = [4, 3, 2, [32, 13], [14, 33]] = [0, 3, 2, 1, 9] ∈ Γ,
and it is easy to verify that orbG(C) = Γ.
If it is the desire to navigate explicitly among the coverings C ∈ Γ via
this action of G, we see from the previous example that the process is
somewhat cumbersome. We show in the next section that, for any value
of L, there is a more easily-described group that acts on the set of all
coverings. The disadvantage is that the action is not always transitive.
4.2. A group action in the general situation
In this section, we lift the restriction that L must satisfy (1). Let ZL
be the additive group of integers modulo L. We define the holomorph of
ZL to be
Hol (ZL) = Aut (ZL) ⋉ ZL ≃ Z
∗
L ⋉ ZL, (5)
where Z
∗
L is the group of units in the ring ZL of integers modulo L. Note
that |Hol (ZL)| = φ(L)L. For brevity of notation, we let G = Hol (ZL).
Remark 4.2. More typically, a semidirect product is written using the
notation A⋊B. However, it is more convenient here to use the isomorphic
group B ⋉ A.
Theorem 4.3. There is a natural (left) action of G on Γ.
Proof. Let g = (a, x) ∈ G and C =
{
(ri, mi)
∣∣∣ 1 6 i 6 t
}
∈ Γ. Define
g.C :=
{
(ari + x, mi)
∣∣∣∣ 1 6 i 6 t
}
.
256 A group-theoretic approach to covering systems
We first show that g.C is indeed a covering. Let n be any integer. Since
C is a covering, there exists j such that
a−1(n − x) ≡ rj (mod mj).
Hence,
n ≡ arj + x (mod mj),
so that n is covered by g.C.
Note that (1, 0) ∈ G is the identity element in G, and that (1, 0).C = C.
Next, let h ∈ G with h = (b, y). By the definition of the operation in G,
we have that
gh = (a, x)(b, y) = (ab, ay + x).
Thus,
(gh).C =
{
(abri + ay + x, mi)
∣∣∣∣ 1 6 i 6 t
}
=
{
(a(bri + y) + x, mi)
∣∣∣∣ 1 6 i 6 t
}
= g.
{
(bri + y, mi)
∣∣∣∣ 1 6 i 6 t
}
= g. (h.C) ,
which completes the proof.
Theorem 4.4. Let C =
{
(ri, mi)
∣∣∣ 1 6 i 6 t
}
∈ Γ. Then
|orbG(C)| > κ(L)φ(L), (6)
where κ(L) denotes the square-free kernel of L, and φ is Euler’s totient
function. Moreover, equality holds in (6) if
κ(L) (ri − rj) ≡ 0 (mod gcd (mi, mj)) for all i and j. (7)
Proof. Let g = (a, x) ∈ stab(C). Then g.C = C and hence
(a − 1)ri + x ≡ 0 (mod mi), (8)
for all (ri, mi) ∈ C. Let p be a prime such that L ≡ 0 (mod p), and let
Cp =
{
(ri, mi) ∈ C
∣∣∣∣ mi ≡ 0 (mod p)
}
.
L. Jones, D. White 257
Since C is a covering, there exist i and j, with i 6=j and (ri, mi), (rj , mj)∈Cp,
such that ri 6≡ rj (mod p). For this particular pair of congruences in Cp,
we have by (8) that
(a − 1)ri + x ≡ (a − 1)rj + x (mod p). (9)
Rearranging (9) and using the fact that ri 6≡ rj (mod p), we get that
a ≡ 1 (mod p). Thus,
a ≡ 1 (mod κ(L)). (10)
There are exactly φ(L)/φ(κ(L)) = L/κ(L) distinct values of a ∈ Z
∗
L that
satisfy (10). For each such value of a, we claim that there is at most one
value of x ∈ ZL that satisfies all congruences in (8). To see this, we fix
a and write a − 1 = zκ(L) for some integer z with 0 6 z 6 L/κ(L) − 1.
Then the system of congruences (8) can be rewritten as the following
system of congruences in the variable x:
x ≡ −zκ(L)ri (mod mi), for all (ri, mi) ∈ C. (11)
By the generalized Chinese remainder theorem, the system (11) has a
solution x ∈ ZL, and it is unique, if and only if
zκ(L) (ri − rj) ≡ 0 (mod gcd (mi, mj))
for all i and j. Thus, we have shown that
|stabG(C)| 6
L
κ(L)
.
Consequently, since |G| = φ(L)L, we have that
|orbG(C)| = [G : stabG(C)] > κ(L)φ(L).
Moreover, if
κ(L) (ri − rj) ≡ 0 (mod gcd (mi, mj))
for all i and j, then
zκ(L) (ri − rj) ≡ 0 (mod gcd (mi, mj))
for any fixed z and all i and j. Thus, in this case, the system (11) has a
unique solution, and so equality holds in (6).
258 A group-theoretic approach to covering systems
The following corollary is immediate from Theorem 4.4.
Corollary 4.5. Let C =
{
(ri, mi)
∣∣∣ 1 6 i 6 t
}
∈ Γ. If (7) holds and G
acts transitively on Γ, then
|Γ| = |orbG(C)| = κ(L)φ(L). (12)
Condition (7) alone is not sufficient to deduce (12). For example, let
L = 36 and M = [2, 3, 4, 6, 9, 18, 36]. Then, computer computations show
that |Γ| = 144 and each C ∈ Γ satisfies (7). Also, there are two orbits of
size κ(L)φ(L) = 72, so that G does not act transitively on Γ. Thus, in
this case, we see that |Γ| = 2κ(L)φ(L).
Corollary 4.6. If L is square-free, then equality holds in (6) for all
C ∈ Γ.
Proof. Since |orbG(C)| divides |G| = Lφ(L), we have that |orbG(C)| 6
Lφ(L). Since L is square-free, κ(L) = L, and therefore by Theorem 4.4,
we deduce that
Lφ(L) > |orbG(C)| > κ(L)φ(L) = Lφ(L).
If we want to utilize Theorem 4.4 to determine |Γ|, then the question
of transitivity of the action of G on Γ is a main concern. Unfortunately,
we have been unable to find a way to determine when this occurs in
general. For certain values of L and certain lists M , we used a computer
to determine |Γ| and |orbG(C)|. This information is given in Table 1. We
denote the number of orbits as #. A complete set of orbit representatives
for each example given in Table 1 is available upon request. Note that, in
Table 1, L ∈ S only for L = 80 and L = 90.
L M # |orb(C)| |Γ|
36 [2, 3, 4, 6, 9, 18, 36] 2 72 144
60 [2, 3, 4, 5, 6, 10, 15, 20, 30] 6 480 2880
72 [2, 3, 4, 6, 9, 24, 36, 72] 2 144 288
80 [2, 4, 5, 8, 10, 16, 20, 40, 80] 6 320 1920
90 [2, 3, 9, 5, 6, 10, 15, 18, 30, 45] 12 720 8640
108 [2, 3, 4, 6, 9, 18, 27, 54, 108] 4 216 864
120 [2, 3, 4, 5, 8, 10, 12, 30, 40, 60] 6 960 5760
Table 1. Data concerning the action of G on Γ
The examples in Table 1 are all such that M is minimal, and the
cardinality of each orbit under the action of G is the same. However, there
L. Jones, D. White 259
are examples of lists of moduli such that the cardinalities of the orbits
are different. Although we cannot make it precise, there seems to be a
connection between this difference in the cardinalities of the orbits and
the following phenomenon.
Definition 4.7. Let M be a list of moduli such that ΓM 6= ∅ and, to
avoid a trivial situation, that some C ∈ ΓM is minimal. We say that M
is quasi-minimal if there exist C1, C2 ∈ ΓM such that C1 is minimal, but
C2 is not.
We give an example to illustrate that quasi-minimal M do exist.
Example 4.8. The list
M = [3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120]
is quasi-minimal since the covering
C1 ={(0, 3), (0, 4), (0, 5), (1, 6), (6, 8), (3, 10), (5, 12), (11, 15),
(7, 20), (10, 24), (2, 30), (34, 40), (59, 60), (98, 120)}
is minimal, but the covering
C2 ={(2, 3), (0, 4), (0, 5), (3, 6), (2, 8), (7, 10), (6, 12), (1, 15),
(19, 20), (22, 24), (13, 30), (0, 40), (49, 60), (0, 120)}
is not minimal. Note that the elements (0, 40) and (0, 120) can be removed
from C2 and the remaining set Ĉ2 is a covering; in fact, it is minimal.
Remark 4.9. The covering C1 in Example (4.8) is due to Erdős [8], while
the covering Ĉ2 is due to Krukenberg [33].
To illustrate the possible connection between quasi-minimality and
the difference in the cardinalities of the orbits, we give examples of two
coverings using M from Example 4.8 where the cardinalities of the orbits
under the action of G are different. The covering
C3 ={(1, 3), (2, 4), (0, 5), (3, 6), (4, 8), (1, 10), (0, 12), (8, 15),
(7, 20), (8, 24), (29, 30), (11, 40), (17, 60), (13, 120)}
is not minimal since removing the set of congruences {(11, 40), (13, 120)}
from C3 gives a covering. Examining the orbit of C3 under G, we see that
|orbG(C3)| = 3840. However, the covering
C4 ={(0, 3), (3, 4), (3, 5), (2, 6), (5, 8), (6, 10), (10, 12), (4, 15),
(0, 20), (17, 24), (22, 30), (25, 40), (37, 60), (1, 120)}
is minimal and |orbG(C4)| = 1920.
260 A group-theoretic approach to covering systems
5. Final comments
Until now, no attempt had been made to impose an algebraic structure
on the set of all coverings with a fixed list of moduli. While our results do
not provide an answer in the most general situation, they do indicate that
a rich and useful algebraic structure does indeed exist, and it is worthy
of further pursuit.
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Contact information
L. Jones,
D. White
Department of Mathematics,
Shippensburg University, Pennsylvania, USA
E-Mail(s): lkjone@ship.edu,
DWhite@ship.edu
Received by the editors: 27.06.2014
and in final form 24.01.2015.
|
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:32:21Z |
| publishDate | 2015 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Jones, L. White, D. 2019-06-16T10:08:49Z 2019-06-16T10:08:49Z 2015 A Group-theoretic Approach to Covering Systems / L. Jones, D. White // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 250-262. — Бібліогр.: 49 назв. — англ. 1726-3255 2010 MSC:Primary 11B25; Secondary 05E18, 11A07. https://nasplib.isofts.kiev.ua/handle/123456789/155167 In this article, we show how group actions can be used to examine the set of all covering systems of the integers with a fixed set of distinct moduli. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics A Group-theoretic Approach to Covering Systems Article published earlier |
| spellingShingle | A Group-theoretic Approach to Covering Systems Jones, L. White, D. |
| title | A Group-theoretic Approach to Covering Systems |
| title_full | A Group-theoretic Approach to Covering Systems |
| title_fullStr | A Group-theoretic Approach to Covering Systems |
| title_full_unstemmed | A Group-theoretic Approach to Covering Systems |
| title_short | A Group-theoretic Approach to Covering Systems |
| title_sort | group-theoretic approach to covering systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155167 |
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