Bounds on generalized Frobenius numbers

Abstract 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 .

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