# Unbounded Discrepancy in Frobenius Numbers

@inproceedings{Shallit2011UnboundedDI,
title={Unbounded Discrepancy in Frobenius Numbers},
author={Jeffrey Shallit and James Stankewicz},
booktitle={Integers},
year={2011}
}
• Published in Integers 26 February 2010
• Mathematics, Computer Science
Abstract Let gj denote the largest integer that is represented exactly j times as a non-negative integer linear combination of {x 1, . . . , xn }. We show that for any k > 0, and n = 5, the quantity g 0 – gk is unbounded. Furthermore, we provide examples with g 0 > gk for n ≥ 6 and g 0 > g 1 for n ≥ 4.
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## References

SHOWING 1-5 OF 5 REFERENCES
An Extreme Family of Generalized Frobenius Numbers
• Mathematics, Computer Science
Integers
• 2011
A family of integers is constructed, based on a recent paper by Tripathi, whose generalized Frobenius numbers g 0, g 1, g 2, . . . exhibit unnatural jumps, and any integer larger than has at least representations.
On a Generalization of the Frobenius Number
We consider a generalization of the Frobenius problem, where the object of interest is the greatest integer having exactly j representations by a collection of positive relatively prime integers. We
The diophantine frobenius problem
• Mathematics
• 2005
Preface Acknowledgements 1. Algorithmic Aspects 2. The Frobenius Number for Small n 3. The General Problem 4. Sylvester Denumerant 5. Integers without Representation 6. Generalizations and Related
Ramı́rez Alfonśın. The Diophantine Frobenius Problem
• 2005