Every positive integer is the Frobenius number of a numerical semigroup with three generators

The multidimensional Frobenius problem

“…This generalizes a one-dimensional construction found in [Rosales et al 2004]. If we allow m = 2, then it is an open problem to determine whether all G are possible.…”

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“…Next, we characterize possible G in our context for the special case m = 1. This generalizes a one-dimensional construction found in [Rosales et al 2004]. If we allow m = 2, then it is an open problem to determine whether all G are possible.…”

The set of solutions of a proportionally modular Diophantine inequality

“…In our setting these elements will be either of the form an 1 or bn 3 for some a, b ∈ N. Thus it is important to know when an 1 and bn 3 are congruent modulo n 2 . This is solved in the first of the following two lemmas, which can be obtained from [11,Lemmas 7 and 9]. The second result describes the shape of the set Ap(S, n 2 ).…”