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On a linear diophantine problem of Frobenius. (English) Zbl 0246.10010
Let $$a_1, \ldots ,a_n$$ be a sequence of integers satisfying $$(a_1, \ldots ,a_n)=1$$. Denote by $$G(a_1, \ldots ,a_n)$$ the greatest integer $$N$$ for which $$N= \sum ^n_{i=1}c_ia_i$$, $$c_i \geq 0$$ integer, has no solution. The problem of determining or estimating $$G(a_1, \ldots ,a_n)$$ is due to Frobenius and the problem has a large literature. The authors prove among others $G(a_1, \ldots ,a_n) \leq 2a_{n-1} \left[{a_n \over n} \right] -a_n.$ Put $$g(n,t)= \max_{a_i} G(a_1, \ldots, a_n)$$ where the maximum is taken over all the $$a_i$$ satisfying $$0<a_1< \ldots <a_n \leq t$$, $$(a_1, \ldots ,a_n)=1$$. Several results are proved about $$g(n,t)$$ and some open problems are stated one of which has been settled in a recent paper of M. Lewin [cf. the preceding review, J. Lond. Math. Soc., II. Ser. 6, 61-69 (1972; Zbl 0246.10009)].

##### MSC:
 11D04 Linear Diophantine equations
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