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The Diophantine Frobenius Problem$
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Jorge L. Ramírez Alfonsín

Print publication date: 2005

Print ISBN-13: 9780198568209

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198568209.001.0001

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The Frobenius number for small n

The Frobenius number for small n

(p.31) 2 The Frobenius number for small n
The Diophantine Frobenius Problem

J. L. Ramírez Alfonsín

Oxford University Press

It is easy to solve FP when n = 2. Indeed, g(a1, a2) = a1a2 - a1 - a2. However, the computation of a (simple) formula when n = 3 is much more difficult and has been the subject of numerous research papers over a long period. F. Curtis has proved that the search for such a formula is, in some sense, doomed to failure since the Frobenius number cannot be given by ‘closed’ formulas of a certain type. Recently, an explicit formula for computing g(a1, a2, a3) has been found. After presenting four different proofs of equality (1), one of which uses the well-known Pick's theorem, this chapter presents the result of Curtis, the general formula, and summarizes the known upper bounds for g(a1, a2, a3) as well as exact formulas for particular triples.

Keywords:   Frobenius number, arithmetic sequences, Pick's theorem, F. Curtis

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