A Group-theoretic Approach to Covering Systems

Lenny Jones, Daniel White

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.

Keywords


covering system; group action; congruence

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References


B. Banks, C. Finch, F. Luca, C. Pomerance and P. Stu{a}nicu{a}, emph{Sierpi'{n}ski and Carmichael Numbers}, Trans. Amer. Math. Soc. (to appear)

Y.G. Chen, emph{On integers of the form $2^npm p_1^{alpha_1}cdots p_r^{alpha_r}$}, Proc. Amer. Math.

Soc. {bf 128} (2000), 1613--1616.

Y.G. Chen, emph{On integers of the form $k2^n+1$},

{Proc. Amer. Math. Soc.} {bf 129} (2001) 355--361.

Y.G. Chen, emph{On integers of the forms $k-2sp

n$ and $k2sp n+1$,} {J. Number Theory} {bf 89}

(2001) 121--125.

Y.G. Chen, emph{On integers of the forms $ksp

r-2sp n$ and $ksp r2sp n+1$,} {J. Number Theory} {bf 98} (2003) 310--319

Y.G. Chen, emph{On integers of the forms $kpm

^n$ and $k2^npm 1$}, {J. Number Theory}

{bf 125} (2007) 14--25.

F. Cohen and J. L. Selfridge, emph{Not every number is the sum or difference of two prime powers},

Math. Comput. {bf 29} (1975), 79-–81.

P.~ErdH{o}s, emph{On integers of the form $2sp

k+p$ and some related problems}, Summa Brasil. Math., (1950), 113--123.

M.~Filaseta, emph{Coverings of the integers

associated with an irreducibility theorem of A. Schinzel}, {Number theory for the millennium, II}

(2002) 1--24.

M.~Filaseta, C.~Finch and M.~Kozek, emph{On powers associated with

Sierpi'{n}ski numbers, Riesel numbers and Polignac's conjecture}, {J. Number Theory} {bf 128}

(2008) 1916--1940.

M. Filaseta, K. Ford and S. Konyagin, emph{On an irreducibility theorem

of A. Schinzel associated with coverings of the integers}, Illinois J. Math. {bf 44} (2000) 633--643.

M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu, emph{Sieving by large integers and covering systems of congruences}, J. Amer. Math. Soc. {bf 20} (2007), no. 2, 495-–517.

M.~Filaseta and J. Harrington, emph{A polynomial investigation inspired by work of Schinzel and Sierpi'{n}ski}, Acta Arith. {bf 155} (2012), no. 2, 149–-161.

M. Filaseta, M. Kozek, C. Nicol and John Selfridge, emph{Composites that remain composite after changing a digit}, J. Comb. Number Theory {bf 2} (2010), no. 1, 25-–36.

C. Finch, J. Harrington and L. Jones, emph{Nonlinear Sierpi'{n}ski and Riesel numbers}, {J. Number Theory} {bf 133} (2013), no. 2, 534-–544.

C. Finch and L. Jones, emph{Perfect Power Riesel Numbers} (submitted).

P.-H. Fuss, emph{Correspondance Math'{e}matique et Physique de Quelques C'{e}l`{e}bres

G'{e}om`{e}tres du XVIII `{E}me Si`{e}cle}, textbf{I, II}, Johnson Reprint, New York, 1968.

J. Grantham, W. Jarnicki, J. Rickert and S. Wagon, emph{Repeatedly appending any digit to generate composite numbers}, {Amer. Math. Monthly} (to appear).

Song Guo and Zhi-Wei Sun, emph{On odd covering systems with distinct moduli}, Adv. in Appl. Math. {bf 35} (2005), no. 2, 182-–187.

B. Hough, emph{Solution of the minimum modulus problem for covering systems}, Ann. of Math. (2) {bf 181} (2015), no. 1, 361--382.

A.S. Izotov, emph{A note on Sierpi'{n}ski

numbers}, {Fibonacci Quart.} {bf 33} (1995) 206--207.

Scott Jenkin and Jamie Simpson, emph{Composite covering systems of minimum cardinality}, Integers {bf 3} (2003), A13, 11 pp.

L.~Jones, emph{Fibonacci variations of a conjecture of Polignac}, Integers {bf 12} (2012), no. 4, 659-–667.

L.~Jones, emph{Polynomial variations on a theme of

Sierpi'{n}ski}, {Int. J. Number Theory} {bf 5} (2009) 1--17.

L.~Jones, emph{Using Lucas sequences to generalize a theorem of

Sierpi'{n}ski}, Acta Arith. {bf 152} (2012), no. 3, 307-–322.

L.~Jones, emph{Variations on a theme of

Sierpi'{n}ski}, {J. Integer Seq.} {bf 10} (2007) Article 07.4.4, 15 pp. (electronic).

L. Jones, emph{When does appending the same digit repeatedly on

the right of a positive integer generate a sequence of composite

integers?}, {Amer. Math. Monthly}, textbf{118} (2011), 153--160.

L. Jones and A. Lamarche, emph{Generating lowercase{$d$}-composite sandwich numbers} (submitted).

L. Jones and M. Markovich, emph{Generating composite sequences by appending digits to special types of integers}, The Fibonacci Quarterly (to appear).

L. Jones and D. White, emph{Appending digits to generate an infinite sequence of composite numbers}, J. Integer Seq. {bf 14} (2011), no. 5, Article 11.5.7, 12 pp.

L. Jones and D. White, emph{On primitive covering numbers}, url{http://arxiv.org/abs/1406.6851}.

L. Jones and D. White, emph{Sierpinski numbers in imaginary quadratic fields}, Integers {bf 12} (2012), no. 6, 1265-–1278.

C. E. Krukenberg, emph{Covering sets of the integers}, Ph.D. thesis, University of Illinois, Urbana-Champaign, (1971).

P. Nielsen, emph{A covering system whose smallest modulus is 40}, J. Number Theory {bf 129} (2009), no. 3, 640-–666.

Hao Pan and Zhi-Wei Sun, emph{A sharp result on $m$-covers}, Proc. Amer. Math. Soc. {bf 135} (2007), no. 11, 3515-–3520 (electronic).

A. de Polignac, emph{Recherches nouvelles sur les nombres premiers}, C. R.

Acad. Sci. Paris Math., textbf{29} (1849) 397--401, 738--739.

v{S}. Porubsk'{y}, emph{Covering systems, Kubert identities and difference equations}, Math. Slovaca {bf 50} (2000), no. 4, 381-–413.

v{S}. Porubsk'{y} and J. Sch"{o}nheim, emph{Covering systems of Paul ErdH{o}s. Past, present and future}, Paul ErdH{o}s and his mathematics, I (Budapest, 1999), 581-–627, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002.

v{S}. Porubsk'{y} and J. Sch"{o}nheim, emph{New necessary and sufficient conditions on $(a_i,m_i)$ in order that $xequiv a_i pmod{m_i}$ be a covering system}, The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications, 5 pp. (electronic), Electron. Notes Discrete Math., 11, Elsevier, Amsterdam, 2002.

v{S}. Porubsk'{y} and J. Sch"{o}nheim, emph{Old and new necessary and sufficient conditions on $(a_i,m_i)$ in order that $nequiv a_i pmod{m_i}$ be a covering system} Math. Slovaca {bf 53} (2003), no. 4, 341-–349.

H.~Riesel, emph{Naa gra stora primtal}, {Elementa} {bf 39} (1956) 258--260.

N. P. Romanoff, emph{Über einige Sätze der additiven Zahlentheorie.} (German) Math. Ann. {bf 109} (1934), no. 1, 668–-678.

W. Sierpi'{n}ski, emph{Sur un probl`{e}me concernant les nombres $k.2^n+1$}, emph{Elem. d. Math.} {bf 15} (1960), 73--74.

Xue Gong Sun, emph{On the density of integers of the form 2k+p in arithmetic progressions}, Acta Math. Sin. (Engl. Ser.) {bf 26} (2010), no. 1, 155-–160.

Xue-Gong Sun and Jin-Hui Fang, emph{On the density of integers of the form $(p-1)2^{-n}$ in arithmetic progressions}, Bull. Aust. Math. Soc. {bf 78} (2008), no. 3, 431-–436.

Zhi-Wei Sun, emph{A connection between covers of the integers and unit fractions}, Adv. in Appl. Math. {bf 38} (2007), no. 2, 267-–274.

Zhi-Wei Sun, emph{On covering numbers}, Combinatorial number theory, 443-–453, de Gruyter, Berlin, 2007.

Zhi-Wei Sun, emph{On $m$-covers and $m$-systems}, Bull. Aust. Math. Soc. {bf 81} (2010), no. 2, 223–-235.

Zhi-Wei Sun and Siman Yang, emph{Covers with less than 10 moduli and their applications}, J. Southeast Univ. (English Ed.) {bf 14} (1998), no. 2, 106-–114.


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