On integers that are uniquely representable by modified arithmetic progressions

Sarthak Chimni, Soumya Sankar and Amitabha Tripathi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 3, Pages 36—44
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Authors and affiliations

Sarthak Chimni
Department of Mathematics, Statistics, and Computer Science
The University of Illinois at Chicago
851 South Morgan Street, Chicago, IL 60607, USA

Soumya Sankar
Department of Mathematics, University of Wisconsin–Madison
Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, USA

Amitabha Tripathi
Department of Mathematics, Indian Institute of Technology
Hauz Khas, New Delhi – 110016, India

Abstract

For positive integers a, d, h, k, gcd(a, d) = 1, let A = {a, ha+d, ha+2d, …, ha+kd}. We characterize the set of nonnegative integers that are uniquely representable by nonnegative integer linear combinations of elements of A.

Keywords

  • m-representable
  • Frobenius number

AMS Classification

  • 11D04

References

    1. Beck, M., & Kifer, C. (2011) An extreme family of generalized Frobenius numbers, Integers, Vol. 11, Article A24, 6 pages.
    2. Beck, M., & Robins, S., A formula related to the Frobenius problem in two dimensions, Number Theory New York Seminar 2003, D. Chudnovsky, G. Chudnovsky, M. Nathanson (eds.), 17–23.
    3. Brown, A., Dannenberg, E., Fox, J., Hanna, J., Keck, K., Moore, A., Robbins, Z., Samples, B.,& Stankewicz, J. (2010) On a generalization of the Frobenius number, J. Integer Seq., Vol. 13, Article 10.1.4, 1–6.
    4. Popoviciu, T. (1953) Asupra unei probleme de patitie a numerelor, Acad. Republicii Populare Romane, Filiala Cluj, Studii si cercetari stiintifice, 4, 7–58 (Romanian).
    5. Tripathi, A. (2000) The number of solutions to ax + by = n, Fibonacci Quart., 38(4), 290–294.
    6. Tripathi, A. (2013) The Frobenius problem for modified arithmetic progressions, J. Integer Seq., 16, 1–6.

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    Cite this paper

    APA

    Chimni, S., Sankar, S. & Tripathi, A. (2016). On integers that are uniquely representable by modified arithmetic progressions, Notes on Number Theory and Discrete Mathematics, 22(3), 36-44.

    Chicago

    Chimni, Sarthak, Soumya Sankar and Amitabha Tripathi. “On Integers that are Uniquely Representable by Modified Arithmetic Progressions.” Notes on Number Theory and Discrete Mathematics 22, no. 3 (2016): 36-44.

    MLA

    Chimni, Sarthak, Soumya Sankar and Amitabha Tripathi. “On Integers that are Uniquely Representable by Modified Arithmetic Progressions.” Notes on Number Theory and Discrete Mathematics 22.3 (2016): 36-44. Print.

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