Abstract
Let \(N \geq2\) and let \(1 < a_1 < \cdots < a_N\) be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as \(\sum_{i=1}^N a_i x_i {\rm where\ } x_1,\ldots,x_N\) are non-negative integers. The condition that \(\gcd(a_1,\ldots,a_N)=1\) implies that such a number exists. The general problem of determining the Frobenius number given N and \(a_1,\ldots,a_N\) is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.
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Fukshansky, L., Robins, S. Frobenius Problem and the Covering Radius of a Lattice. Discrete Comput Geom 37, 471–483 (2007). https://doi.org/10.1007/s00454-006-1295-2
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DOI: https://doi.org/10.1007/s00454-006-1295-2