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


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.


Basis Independent basis Representable Frobenius number 

Mathematics Subject Classification




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


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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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