The so-called Frobenius number in the famous linear Diophantine problem of is largest integer such that equation [Formula: see text] ([Formula: are given positive integers with text]) does not have a non-negative solution text]. generalized (called text]-Frobenius number) this has at most solutions. That is, when text], original number. In paper, we introduce and discuss text]-numerical semigro...

The Money Changing Problem is as follows: Let a1 < a2 < · · · < ak be fixed positive integers with gcd(a1, . . . , ak) = 1. Given some integer n, are there non-negative integers x1, . . . , xk such that ∑ i aixi = n? The Frobenius number g(a1, . . . , ak) is the largest integer n such that the above problem has no decomposition x1, . . . , xk. There exist algorithms that, for fixed k, compute t...

We study the extended Frobenius problem for sequences of form $\{f_a+f_n\}_{n\in\mathbb{N}}$, where $\{f_n\}_{n\in\mathbb{N}}$ is Fibonacci sequence and $f_a$ a number. As consequence, we show that family numerical semigroups associated to these satisfies Wilf's conjecture.

The numerical semigroup generated by relatively prime positive integers a1, . . . , an is the set S of all linear combinations of a1, . . . , an with nonnegative integral coefficients. The largest integer which is not an element of S is called the Frobenius number of S. Recently, J. M. Maŕın, J. L. Ramı́rez Alfonśın, and M. P. Revuelta determined the Frobenius number of a Fibonacci semigroup, th...

Let R ∈ Z be a vector with strictly positive integers entries. We denote its transpose by R = (r1, . . . , rn). In this article, unless specified otherwise, any integer vector denoted R will be assumed to have gcd(r1, . . . , rn) = 1. Let D ∈ Z. We define the degree of D as degR(D) := D ·R. When the context makes the reference to R unnecessary, we will denote degR simply by deg. The kernel of t...

We study the structure of family numerical semigroups with fixed multiplicity and Frobenius number. give an algorithmic method to compute all in this family. As application we set given genus.

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable nondegenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theo...

Given a set of positive integers A = {a1, . . . , ad} with gcd(a1, . . . , ad) = 1, we call an integer n representable if there exist nonnegative integers m1, . . . ,md such that n = m1a1 + · · ·+mdad . In this paper, we discuss the linear diophantine problem of Frobenius: namely, find the largest integer which is not representable. We call this largest integer the Frobenius number g(a1, . . . ...