On the diophantine Frobenius problem. (English) Zbl 0923.11044

Hamidoune, Y. O.

Port. Math. 55, No. 4, 425-449 (1998).

Let \(X\subset\mathbb{N}\) be a finite subset such that \(\text{gcd}(X) =1\) and \(g(X)\) the Frobenius number of \(X\). Then it is defined \[ f(n,M)= \max \bigl\{g(X); \text{gcd} (X)=1,\;| X|= n,\;\max(X)=M\bigr\}. \] Furthermore a family \({\mathcal F}_{n,M}\) is defined being an extension of known families having a large Frobenius number; this definition is rather complicated and cannot be given here. Let \(A\) be a set with cardinality \(n\) and maximal element \(M\). The main results then are \[ g(A)\leq \Bigl(M- {n\over 2}\Bigr)^2/n-1 \] for \(A\not\in {\mathcal F}_{n,M}\), the value of \(f(n,M)\) for \(M\geq n(n-1)+2\) and a precise description of the sets \(A\) with \(g(A)=f(n,M)\).

Reviewer: J.Piehler (Merseburg)

MSC:

11D04

Linear Diophantine equations

Keywords:

diophantine Frobenius problem; Frobenius number