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On the diophantine Frobenius problem. (English) Zbl 0923.11044
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 $$g(A)=f(n,M)$$.

##### MSC:
 11D04 Linear Diophantine equations
##### Keywords:
diophantine Frobenius problem; Frobenius number
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