The Frobenius Number of Geometric Sequences (original) (raw)

The Frobenius Number of Geometric

The Frobenius problem is about finding the largest integer that is not contained in the numerical semigroup generated by a given set of positive integers. In this paper, we derive a solution to the Frobenius problem for sets of the form {m k , m k−1 n, m k−2 n 2 ,. .. , n k }, where m, n are relatively prime positive integers.

Numerical semigroups with a given set of pseudo-Frobenius numbers

LMS Journal of Computation and Mathematics, 2016

The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of pseudo-Frobenius numbers of a numerical semigroup and, if so, to compute the set of all numerical semigroups having this set as set of pseudo-Frobenius numbers.

The frobenius problem for some numerical semigroups with embedding dimension equal to three

Hacettepe Journal of Mathematics and Statistics, 2015

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 pseudo-Frobenius numbers of a, b, cb − da in the case in which the interval a c , b d contains some integer.

FROBENIUS NUMBERS BY LATTICE POINT ENUMERATION

The Frobenius number g(A) of a set A = (a 1 , a 2 , . . . , a n ) of positive integers is the largest integer not representable as a nonnegative linear combination of the a i . We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in average time that is softly quadratic and we prove that this is the case for almost all of the steps. In practice, the algorithm is very fast: examples with n = 4 and the numbers in A having 100 digits take under one second. The running time increases with dimension and we can succeed up to n = 11.

Associated semigroups of numerical sets with fixed Frobenius number

arXiv: Combinatorics, 2019

A numerical set is a co-finite Subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its compliment. A numerical semigroup is a numerical set that is closed under addition. Each numerical set has an associated semigroup A(T)=t∣t+TsubseteqTA(T)=\{t|t+T\subseteq T\}A(T)=t∣t+TsubseteqT, which is a numerical semigroup with the same Frobenius number as that of TTT. For a fixed Frobenius number fff there are 2f−12^{f-1}2f−1 numerical sets. We give a complete asymptotic description of what percentage of these numerical sets are mapped to which semigroups. We also obtain parallel results for symmetric numerical sets.

Some contributions to the Frobenius' Problem

Electronic Notes in Discrete Mathematics, 2007

Given a set A = {a 1 , ..., a k } with 1 ≤ a 1 < ... < a k and gcd(a 1 , ..., a k) = 1, let us denote R(A) = {m ∈ N| ∃λ 1 , ..., λ k ∈ N : m = k i=1 λ i a i } and R(A) = N \R(A). The classical study of the Frobenius' Problem for a given set A is the computation of the number f (A) = max R(A) (also called the Frobenius