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The Frobenius Problem and Its Generalizations

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Book cover Developments in Language Theory (DLT 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5257))

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Abstract

Let x 1, x 2, ..., x n be positive integers. It is well-known that every sufficiently large integer can be represented as a non-negative integer linear combination of the x i if and only if \(\gcd(x_1, x_2, \ldots, x_n) = 1\). The Frobenius problem is the following: given positive integers x 1, x 2, ..., x n with \(\gcd(x_1, x_2, \ldots, x_n) = 1 \), compute the largest integer not representable as a non-negative integer linear combination of the x i . This largest integer is sometimes denoted g(x 1,..., x n ).

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  1. School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada

    Jeffrey Shallit

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  1. Jeffrey Shallit
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Masami Ito Masafumi Toyama

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© 2008 Springer-Verlag Berlin Heidelberg

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Shallit, J. (2008). The Frobenius Problem and Its Generalizations. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2008. Lecture Notes in Computer Science, vol 5257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85780-8_5

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  • DOI: https://doi.org/10.1007/978-3-540-85780-8_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85779-2

  • Online ISBN: 978-3-540-85780-8

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