×

zbMATH — the first resource for mathematics

On a linear diophantine problem for geometrical type sequences. (English) Zbl 0615.10021
Let \(A=\{a_ 0,a_ 1,...,a_ n\}\) be a set of relatively prime positive integers, g(A) the Frobenius number, i.e. the greatest b for which \(b=\sum^{n}_{i=0}a_ ix_ i\) has no solution in nonnegative integers \(x_ i\), and n(A) the number of all positive b for which there is no such solution.
In this paper the author gives the exact values of g(A) and n(A) in the case where the set A forms an almost geometric progression \(a_ j=h_ ja_ 0+\prod^{j-1}_{i=0}q_ i\) for \(j=1,...,n\), \(g.c.d.(a_ 0,q_ 0)=0\) with \(a_ 0\), \(h_ j\), \(q_{j-1}\), \(j=1,...,n\) given positive integers.
Reviewer: J.Piehler

MSC:
11D04 Linear Diophantine equations
90C10 Integer programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boros, E., Subadditive approach to a linear Diophantine problem of Frobenius, ()
[2] Brauer, A., On a problem of partitions, Amer. J. math., 64, 299-312, (1942) · Zbl 0061.06801
[3] Brauer, A.; Seelbinder, B.M., On a problem of partitions, II, Amer. J. math., 76, 343-346, (1954) · Zbl 0056.26901
[4] Erdös, P.; Graham, R.L., On a linear Diophantine problem of Frobenius, Acta arithmetica, 21, 399-408, (1972) · Zbl 0246.10010
[5] Goldberg, E.L., On a linear Diophantine equation, Acta arithmetica, 31, 239-246, (1976) · Zbl 0312.10007
[6] Hofmeister, G.R., Zu einem problem von Frobenius, Norske vid. selk. skrifter, 5, 1-37, (1966)
[7] Hujter, M., On a sharp upper and lower bound for the Frobenius problem, () · Zbl 0877.65028
[8] Hujter, M., On a problem of Frobenius: a survey, () · Zbl 0877.65028
[9] Lewin, M., A bound for a solution of a linear Diophantine problem, J. London math. soc., 6, 61-69, (1972) · Zbl 0246.10009
[10] Lewin, M., On a linear Diophantine problem, Bull. London math. soc., 5, 75-78, (1973) · Zbl 0261.10012
[11] Selmer, E.S., On the linear Diophantine problem of Frobenius, J. reine angewandte math., 293/294, 1-17, (1977) · Zbl 0349.10009
[12] Selmer, E.S.; Beyer, O., On the linear Diophantine problem of Frobenius in three variables, J. reine angewandte math., 301, 161-170, (1978) · Zbl 0374.10010
[13] Sylvester, J., Mathematical questions with their solutions, Educational times, 4, 21, (1884)
[14] Vitek, Y., Bounds for a linear Diophantine problem of Frobenius, J. London math. soc., 10, 79-85, (1975) · Zbl 0301.10020
[15] Vizvari, B., On the connection of the Frobenius problem and the knapsack problem, (), 799-819
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.