Frobenius Problem For A Class Of Numerical Semigroups

The so-called Frobenius number of a numerical semigroup refers to the largest positive integer that does not belong to a numerical semigroup,and the famous Frobenius problem is that the given Frobenius number is only dependent on the calculating formula of the generator system of the numerical semigroups.At present,the Frobenius problem of arbitrary numerical semigroups with embedding dimension not less than three is known to be an N-P problem.Therefore,the Frobenius problem of numerical semigroups generated by some special sequence has been studied instead.In recent years,the Frobenius problem of numerical semigroups generated by some sequence,such as Thabit sequence,has been successfully solved by means of the relationship between the Apéry sets and Frobenius numbers of numerical semigroups.Compared with the more general sequence of Thabit sequence,the related Frobenius problems of numerical semigroup S?k,n?generated by it has been solved at present.In this paper,on the basis of numerical semigroupsS?k,n?,we study the properties of new numerical semigroups T?k,m,n?in a more general form.By induction reasoning and theoretic proof,we first determine the minimal generators system of the new numerical semigroupT?k,m,n?,and according to the minimal generators system,the embedding dimension of it and the related properties of the Apéry set about multiplicity ₀s are determined.Finally,according to the relationship between the Apéry set and the Frobenius number,the formulas for calculating the Frobenius number in two special cases are determined.The whole content of the article is divided into four chapters:In Chapter 1,the background and significance of this study are introduced firstly,and then the research progress and main conclusions of this paper are introduced.In Chapter 2,some basic concepts and theorem related to numerical semigroups and Frobenius numbers are introduced.In Chapter 3,the embedding dimension and its minimal generator system of new numerical semigroupsT?k,m,n?are determined by exploring,and discussed the related properties of Apéry set about multiplicity ₀s.In Chapter 4,the Frobenius number formulas of new numerical semigroupsT?k,m,n?under two special conditions,m=2^k-1-1 and m=2 ^t-1?t?1?,are explored anddetermined,that is,the Frobenius problem of T?k,m,n?is solved.