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系統識別號 U0026-0102201714221000
論文名稱(中文) 三元二次齊次式
論文名稱(英文) The representation of the ternary quadratic forms
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 104
學期 2
出版年 105
研究生(中文) 張睿恩
研究生(英文) Jui-En Chang
學號 l16031051
學位類別 碩士
語文別 英文
論文頁數 35頁
口試委員 指導教授-黃柏嶧
口試委員-黃世昌
口試委員-蕭仁傑
中文關鍵字 三個平方數的和  混合三角數與平方數之和  Universal  Asymptotically Universal  Almost Universal 
英文關鍵字 Sums of Three Squares  Mixed Sums of Squares and Triangular Numbers  Universal Forms  Asymptotically Universal  Almost Universal 
學科別分類
中文摘要 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.
英文摘要 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.
論文目次 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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19. L. Panaitopol. On the representation of natural numbers as sums of squares. Amer.Math. Monthly, (112):168{171, 2005.
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21. S. Ramanujan. On the expression of a number in the form ax2 + by2 + cz2 + du2.Proc. Camb. Philo. Soc, (19):11{21, 1916.
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23. Z. W. Sun. On sums of primes and triangular numbers. J. Combin. Number Theorey, (1):65{76, 2009.
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