朱樺臺灣大學：數學研究所張憶婷Chang, Yi-TingYi-TingChang2007-11-282018-06-282007-11-282018-06-282007http://ntur.lib.ntu.edu.tw//handle/246246/59499令 $a,b,c,d$ 為一組獨立的正整數。若一個非負整數可表為 $c_1a+c_2b+c_3c+c_4d$ 的形式，其中 $c_i$ 均為非負整數，則稱它可被 $a,b,c,d$ 表示。 我們將給出在特殊情形中，不能由 $a,b,c,d$ 表出的非負整數個數 $n(a,b,c,d)$，及最大不可表的整數 $g(a,b,c,d)$。最後並討論由 $a,b,c,d$ 生成的半群對稱性。Let $a,b,c,d$ be independent positive integers. A nonnegative integer is said to be represented by $a,b,c,d$ if it can be represented as the form $c_1a+c_2b+c_3c+c_4d$, where the $c_i$'s are nonnegative integers. We will find the number $n(a,b,c,d)$ of nonnegative integers cannot be represented by $a,b,c,d$, and the number $g(a,b,c,d)$ which is the largest integer cannot be represented by $a,b,c,d$ in some special cases. Finally we discuss the symmetry property of the semigroup generated by $a,b,c,d$.Acknowledgements i Abstract in Chinese ii Abstract iii Contents iv List of Figures vi 1 Introduction 1 2 Results on Three Elements 5 2.1 R‥odseth’s Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Another Viewpoint on Three Elements Case . . . . . . . . . . . . . . 7 3 Results on Four Elements 15 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 n(a, b, c, d) for m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 g(a, b, c, d) for m = 1 and n = 1, 2 . . . . . . . . . . . . . . . . . . . . 21 3.4 Symmetry for m = 1 and n = 1 . . . . . . . . . . . . . . . . . . . . . 33 3.5 Almost Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . 40 References 45491505 bytesapplication/pdfen-USFrobenius問題半群對稱性Frobenius problemsemigroupsymmetrythesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/59499/1/ntu-96-R94221025-1.pdf