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研究生： 張睿恩
Chang, Jui-En
論文名稱： 三元二次齊次式
The representation of the ternary quadratic forms
指導教授： 黃柏嶧
Huang, Po-Yi
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2016
畢業學年度： 104
語文別： 英文
論文頁數： 35
中文關鍵詞： 三個平方數的和 、混合三角數與平方數之和 、Universal 、Asymptotically Universal 、Almost Universal
外文關鍵詞： Sums of Three Squares, Mixed Sums of Squares and Triangular Numbers, Universal Forms, Asymptotically Universal, Almost Universal
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    • 一開始我們將會證明沒有任何一個三個平方和可以表示所有正整數，接著我們
    會介紹Gauss and Legendre 定理並且給予一個判斷三個平方和為規則型或不規則形的方法。再來，我們會證明不論哪一種形式的三種平方數與三角數的和，只有有限多個可以表示所有正整數。最後，我們會介紹一些方法去判斷這些三元二次式是否為asymptotically universal或almost universal.

    We will show that all sums of three squares are not universal, and introduce the Gauss and Legendre theorems. We will also give a method to check if a triple sum of squares
    is regular or irregular. Then, we will study the ternary quadratic forms of mixed sums of squares and triangular numbers. We will show that there are only nitely ternary
    forms of mixed sums of squares and triangular numbers in each three types which is universal. Finally, we will introduce some methods to determine the ternary quadratic forms of mixed sums of squares and triangular numbers is asymptotically universal or almost universal.

    1 Introduction 1 2 Sums of three squares for universal forms 3 2.1 Universal forms for sums of three squares . . . . . . . . . . . . . . . . 3 2.2 Gauss and Legendre Theorem . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Numbers represented by various ternary forms . . . . . . . . . . . . . . 6 2.4 Regular and irregular forms . . . . . . . . . . . . . . . . . . . . . . . . 10 3 The universal forms for mixed Sums of squares and triangular num- bers 13 3.1 Some useful lemma and theorem . . . . . . . . . . . . . . . . . . . . . 13 3.2 The type of ax2 + by2 + cTz . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 The type of ax2 + bTy + cTz . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Almost and Asymptotically universal forms of mixed sums of squares and triangular numbers 23 4.1 Background and some important de nitions . . . . . . . . . . . . . . . 23 4.2 Asymptotically universal mixed sums of squares and triangular numbers 24 4.3 Almost universal mixed sums of squares and triangular numbers . . . . 27 Bibliography 34

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