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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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Francesc Aguiló, Alícia Miralles. On the Frobenius’ Problem of three numbers. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.317-322, ⟨10.46298/dmtcs.3462⟩. ⟨hal-01184451⟩

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