Analysis Of The Integers And The Set And The Frobenius Number Of New Valuation Formula

In this paper,we use addition theory on the Abelian group to study two problems on the Abelian group, the direct problem ,that is, let A and B, what are properties and the structure of the sumset A+B? , and the inverse problem- when |A+B| is as small as possible, what are properties and the structure of the set A and the set B?The main work of this paper has three aspects. Firstly, we introduce a new result of the direct and inverse problems on the Abelian group; secendly, we deduce computation on |2A| in some special cases; lastly, we useaddition theory on the Abelian group (z_a0,+) to improve Vitek's bound andShen Jian's bound in two special cases, and obtain some new estimated formulas for the Frobenius numbers.This paper is organized as follows:Chapter 1 Introduction. We introduce the main problem of study on additive number theory and the main result of this paper.Chapter 2 The sumset on integers. We introduce mainly a new result on the direct and inverse problems of the sumset on integers, and we deduce computation on |2A| in four special cases.Chapter 3 Some new estimated formulas for the Frobenius numbers. We use addition theory on the Abelian group (z_a0,+) to improve Vitek'sbound and Shen Jian's bound in two special cases, and obtain some new estimated formulas for the Frobenius numbers.