Minimal relations and the Diophantine Frobenius problem in embedding dimension three

The Diophantine Frobenius Problem

An Estimate for Frobenius’ Diophantine Problem in Three Dimensions

We give upper and lower bounds for the largest integer not representable as a positive linear combination of three given integers, disproving an upper bound conjectured by Beck, Einstein and Zacks.

The Frobenius problem for numerical semigroups with embedding dimension equal to three

If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = 〈a, b, cb − da〉 with a, b, c, d positive integers such that gcd(a, b) = gcd(a, c) = gcd(b, d) = 1, c ∈ {2, . . . , a− 1}, and a < b < cb− da. In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseu...

On a Linear Diophantine Problem of Frobenius

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 diophantine problem of Frobenius: A close bound

The conductor of n positive integer numbers a l,a2, ... ,an' whose greatest coII1mon divisor is equal'to I, is'defmed as the th.e minimal K, such that for every m ~K , the equation a1x 1+a2X 2+ ... +an Xn=m, h!ls a solution over the nOI}negative integers. In this notewe give a polYIlomial aigorithm computing'a close bound ~ for the conductor K o( n given positive integers, when n is fixed. The ...