The Frobenius Coin Problem Upper Bounds on The Frobenius Number
نویسنده
چکیده
In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal to 4 units can be obtained, but 1 or 3 units cannot, so the largest unobtainable amount is 3. What about using coins of 7 and 10 units? We need to figure out which positive integers n are of the form
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Generalized Frobenius Numbers: Bounds and Average Behavior
Let n ≥ 2 and s ≥ 1 be integers and a = (a1, . . . , an) be a relatively prime integer n-tuple. The s-Frobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1, ..., xn are non-negative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recen...
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