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The Ramanujan Journal

, Volume 48, Issue 3, pp 545–565 | Cite as

Some problems concerning the Frobenius number for extensions of an arithmetic progression

  • Sanjit Singh Batra
  • Nikhil Kumar
  • Amitabha TripathiEmail author
Article
  • 111 Downloads

Abstract

For positive and relative prime set of integers \(A=\{a_1,\ldots ,a_k\}\), let \({\varGamma }(A)\) denote the set of integers of the form \(a_1x_1+\cdots +a_kx_k\) with each \(x_i \ge 0\). It is well known that \({\varGamma }^c(A)=\mathbb {N}\setminus {\varGamma }(A)\) is a finite set, so that \({\texttt {g}}(A)\), which denotes the largest integer in \({\varGamma }^c(A)\), is well defined. Let \(A=AP(a,d,k)\) denote the set \(\{a,a+d,\ldots ,a+(k-1)d\}\) of integers in arithmetic progression, and let \(\gcd (a,d)=1\). We (i) determine the set \(A^+=\left\{ b \in {\varGamma }^c(A): {\texttt {g}}(A \cup \{b\})={\texttt {g}}(A) \right\} \); (ii) determine a subset \(\overline{A^+}\) of \({\varGamma }^c(A)\) of largest cardinality such that \(A \cup \overline{A^+}\) is an independent set and \({\texttt {g}}(A\,\cup \,\overline{A^+})={\texttt {g}}(A)\); and (iii) determine \({\texttt {g}}(A \cup \{b\})\) for some class of values of b that includes results of some recent work.

Keywords

Basis Independent basis Representable Frobenius number 

Mathematics Subject Classification

11D07 

Notes

Acknowledgements

The authors are grateful for the comments from an anonymous referee.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Sanjit Singh Batra
    • 1
    • 2
    • 3
  • Nikhil Kumar
    • 2
  • Amitabha Tripathi
    • 4
    Email author
  1. 1.Department of Molecular & Human GeneticsBaylor College of MedicineHoustonUSA
  2. 2.Department of Computer Science & EngineeringIndian Institute of Technology, Hauz KhasNew DelhiIndia
  3. 3.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  4. 4.Department of MathematicsIndian Institute of Technology, Hauz KhasNew DelhiIndia

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