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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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