Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a 1 < ⋯ < a N be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as ∑ i = 1 N a i x i where x 1 , … , x N are nonnegative integers. More generally, the s -Frobenius number is defi...

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Bibliographic Details
Published in:European journal of combinatorics 2011-04, Vol.32 (3), p.361-368
Main Authors: Fukshansky, Lenny, Schürmann, Achill
Format: Article
Language:English
Subjects:
Combinatorial analysis
Integers
Lower bounds
Representations
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Summary:Let N ≥ 2 and let 1 < a 1 < ⋯ < a N be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as ∑ i = 1 N a i x i where x 1 , … , x N are nonnegative integers. More generally, the s -Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques from the geometry of numbers to give upper and lower bounds on the s -Frobenius number for any nonnegative integer s .
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2010.11.001