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- W168537323 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 $$F(N)=maxlimits_{substack{0 lt a lt b lt N mathrm{gcd}(a,b,N)=1}} f(a,b,N)= begin{cases} 2(lfloor N/2 rfloor -1)^2-1 & textrm{if } N equiv 0 (mod 2), 2 lfloor N/2 rfloor (lfloor N/2 rfloor -1) -1 & textrm{if } N equiv 1 (mod 2). end{cases}$$ 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 $f(alpha ,beta ,N)=F(N)$." @default.
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- W168537323 date "2005-01-01" @default.
- W168537323 modified "2023-09-25" @default.
- W168537323 title "On the Frobenius’ Problem of three numbers" @default.
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- W168537323 doi "https://doi.org/10.46298/dmtcs.3462" @default.
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