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...

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as ∑ N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1, ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-hard...

The generalized Frobenius number is the largest integer represented in at most p ways by a linear combination of nonnegative integers given positive 1 ,a 2 ,⋯,a k . When p=0, it reduces to classical number. In this paper, we give when j =(b n+j-1 -1)/(b-1) (b≥2) as generalization result p=0 [16].

The famous linear diophantine problem of Frobenius is the to determine largest integer (Frobenius number) whose number representations in terms $a_1,\dots,a_k$ at most zero, that not representable. In other words, all integers greater than this can be represented for least one way. One natural generalizations find (generalized a given nonnegative $p$. It easy explicit form case two variables. H...

New family of flat potential (Darboux-Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius manifolds on Hurwitz spaces found by B.Dubrovin more than 10 years ago. The branch points of a ramified...

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as P N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1 , ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-har...

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the met...