Positive Frobenius Numbers of Three Arguments
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Let ,
,
be three positive integers with
. It is well-known that all sufficiently large integers are representable as positive linear combinations of
,
,
. Consider
, the positive Frobenius number of
,
,
, defined to be the largest integer not representable as a positive linear combination of
,
,
. Then
is the usual Frobenius number, that is, the largest integer not representable as a non-negative linear combination
,
,
. (
differs from the positive Frobenius number
in that multipliers for linear combinations of larger integers are allowed to be zero.) The function
corresponds to the Mathematica built-in function FrobeniusNumber[a,b,c].
Contributed by: Abdelwaheb Miled (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
S. M. Johnson, "A Linear Diophantine Problem," Canadian Journal of Mathematics, 12, 1960 pp. 390–398.
A. Miled, "On a Problem of Frobenius in Three Numbers," arXiv.org, 2009.
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