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Refined upper bounds for the linear Diophantine problem of Frobenius

Matthias Beck·Shelemyahu Zacks
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Abstract

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d. We introduce a method to compute upper bounds for g(a_1,a_2,a_3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.

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Matthias Beck, Shelemyahu Zacks. 2005-01-02. Refined upper bounds for the linear Diophantine problem of Frobenius. https://arxiv.org/abs/math/0305420

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