On the Frobenius’ Problem of three numbers

Abstract : Given $k$ natural numbers $\{a_1, \ldots ,a_k\} \subset \mathbb{N}$ with $1 \leq a_1 < a_2 < \ldots < a_k$ and $\mathrm{gcd} (a_1, \ldots ,a_k)=1$, let be $R(a_1, \ldots ,a_k) = \{ \lambda_1 a_1+ \cdots + \lambda_k a_k | \space \lambda_i \in \mathbb{N}, i=1 \div k\}$ and $\overline{R}(a_1, \ldots ,a_k) = \mathbb{N} \backslash R (a_1, \ldots ,a_k)$. It is easy to see that $| \overline{R}(a_1, \ldots ,a_k)| < \infty$. The $\textit{Frobenius Problem}$ related to the set $\{a_1, \ldots ,a_k\}$ consists on the computation of $f(a_1, \ldots ,a_k)=\max \overline{R} (a_1, \ldots ,a_k)$, also called the $\textit{Frobenius number}$, and the cardinal $| \overline{R}(a_1, \ldots ,a_k)|$. The solution of the Frobenius Problem is the explicit computation of the set $\overline{R} (a_1,\ldots ,a_k)$. In some cases it is known a sharp upper bound for the Frobenius number. When $k=3$ this bound is known to be $$F(N)=\max\limits_{\substack{0 \lt a \lt b \lt N \\ \mathrm{gcd}(a,b,N)=1}} f(a,b,N)= \begin{cases} 2(\lfloor N/2 \rfloor -1)^2-1 & \textrm{if } N \equiv 0 (\mod 2),\\ 2 \lfloor N/2 \rfloor (\lfloor N/2 \rfloor -1) -1 & \textrm{if } N \equiv 1 (\mod 2).\\ \end{cases}$$ This bound is given in [Dixmier1990]. In this work we give a geometrical proof of this bound which allows us to give the solution of the Frobenius problem for all the sets $\{\alpha ,\beta ,N\}$ such that $f(\alpha ,\beta ,N)=F(N)$.
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Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.317-322, 2005, DMTCS Proceedings
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Francesc Aguiló, Alícia Miralles. On the Frobenius’ Problem of three numbers. Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.317-322, 2005, DMTCS Proceedings. 〈hal-01184451〉

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