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

• Sanjit Singh Batra
• Nikhil Kumar
• Amitabha Tripathi
Article

## 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

11D07

## Notes

### Acknowledgements

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