| author: | Francesc Aguiló and Alícia Miralles |
|---|---|
| title: | Frobenius' Problem |
| keywords: | Frobenius problem, L-shaped tile, Smith normal form, Minimum Distance Diagram |
| abstract: |
Given
k
natural numbers
{a
with
1
,...,a
k
}⊂ℕ
1≤a
and
1
<a
2
<..<a
k
gcd
(a
1
,...,a
k
)=1
R(a
and
1
,...,a
k
)={λ
1
a
1
+⋯+λ
k
a
k
| λ
i
∈ℕ, i=1÷ k}
R
(a
1
,...,a
k
)=ℕ \ R(a
1
,...,a
k
)
|
. The Frobenius Problem related to the set
R
(a
1
,...,a
k
)|<∞
{a
consists on the computation of
1
,...,a
k
}
f(a
, also called the Frobenius number, and the
cardinal
1
,...,a
k
)=
max
R
(a
1
,...,a
k
)
|
. The solution of the Frobenius Problem is the
explicit computation of the set
R
(a
1
,...,a
k
)|
R
(a
1
,...,a
k
)
k=3
this bound is known to be
F(N)=
This bound is given in [Dixmier1990]. In this work we
give a geometrical proof of this bound which allows us to
give the solution of the Frobenius problem for all the sets
max
0<a<b<N, gcd(a,b,N)=1
f(a,b,N)=
[begin cases]
2(⌊N/2⌋-1)
2
-1
if
N≡0 (2),2⌊N/2⌋(⌊N/2⌋-1)-1
if
N≡1 (2).[end cases]
{α,β,N}
such that
f(α,β,N)=F(N)
.
|
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| reference: | Francesc Aguiló and Alícia Miralles (2005), Frobenius' Problem, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 317-322 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dmAE0162.pdf (170 K) |
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