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ON A LINEAR DIOPHANTINE PROBLEM OF FROBENIUS
 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A14
, 2006
"... Let a1, a2,..., ak be positive and pairwise coprime integers with product P. For each i, 1 ≤ i ≤ k, set Ai = P/ai. We find closed form expressions for the functions g(A1, A2,..., Ak) and n(A1, A2,..., Ak) that denote the largest (respectively, the number of) N such that the equation A1x1 + A2x2 + · ..."
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Let a1, a2,..., ak be positive and pairwise coprime integers with product P. For each i, 1 ≤ i ≤ k, set Ai = P/ai. We find closed form expressions for the functions g(A1, A2,..., Ak) and n(A1, A2,..., Ak) that denote the largest (respectively, the number of) N such that the equation A1x1 + A2x2 + · · · + Akxk = N has no solution in nonnegative integers xi. This is a special case of the wellknown Coin Exchange Problem of Frobenius.
ON INTEGERS NONREPRESENTABLE BY A GENERALIZED ARITHMETIC PROGRESSION
"... We consider those positive integers that are not representable as linear combinations of terms of a generalized arithmetic progression with nonnegative integer coefficients. To do this, we make use of the numerical semigroup generated by a generalized arithmetic progression. The number of integers n ..."
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We consider those positive integers that are not representable as linear combinations of terms of a generalized arithmetic progression with nonnegative integer coefficients. To do this, we make use of the numerical semigroup generated by a generalized arithmetic progression. The number of integers nonrepresentable by such a numerical semigroup is determined as well as that of its dual. In addition, we find the number and the sum of those integers representable by the dual of the semigroup that are not representable by the semigroup itself. 1.
A characterization of the Frobenius problem and its application to arithmetic progressions
, 2006
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On a Generalization of the Coin Exchange Problem for Three Variables
"... Given relatively prime and positive integers a1, a2,...,ak, let Γ denote the set of nonnegative integers representable by the form a1x1 + a2x2 + · · · + akxk, and let Γ ⋆ denote the positive integers in Γ. Let S ⋆ (a1, a2,...,ak) denote the set of all positive integers n not in Γ for which n + Γ ⋆ ..."
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Given relatively prime and positive integers a1, a2,...,ak, let Γ denote the set of nonnegative integers representable by the form a1x1 + a2x2 + · · · + akxk, and let Γ ⋆ denote the positive integers in Γ. Let S ⋆ (a1, a2,...,ak) denote the set of all positive integers n not in Γ for which n + Γ ⋆ is contained in Γ ⋆. The purpose of this article is to determine an algorithm which can be used to obtain the set S ⋆ in the three variable case. In particular, we show that the set S ⋆ (a1, a2, a3) has at most two elements. We also obtain a formula for g(a1, a2, a3), the largest integer not representable by the form a1x1 + a2x2 + a3x3 with the xi’s nonnegative integers. 1
The Frobenius Problem for Modified Arithmetic Progressions
"... For a set of positive and relatively prime integers A, let Γ(A) denote the set of integers obtained by taking all nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to Γ(A). For the modified arithmetic progression A = {a,ha+d,h ..."
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For a set of positive and relatively prime integers A, let Γ(A) denote the set of integers obtained by taking all nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to Γ(A). For the modified arithmetic progression A = {a,ha+d,ha+2d,...,ha+kd}, gcd(a,d) = 1, we determine the largest integer g(A) that does not belong to Γ(A), and the number of integers n(A) that do not belong to Γ(A). We also determine the set of integers S ⋆ (A) that do not belong to Γ(A) which, when added to any positive integer in Γ(A), result in an integer in Γ(A). Our results generalize the corresponding results for arithmetic progressions. 1