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研究生： 林志穎
Chih-Ying Lin
論文名稱： On the Diophantine Equations x^2+y^2+z^2=kxyz
指導教授： 洪有情
Hung, Yu-Ching
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2007
畢業學年度： 95
語文別： 英文
論文頁數： 16
中文關鍵詞： 馬可夫方程式
英文關鍵詞： Markoff Equation
論文種類： 學術論文
相關次數： 點閱：124  下載：8 
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    • 這篇論文中，我們就k值來探討丟番圖方程x^2+y^2+z^2=kxyz之解的情形：
    (1)當k不為1和3時，此方程式無正整數解。
    (2)當k=1時，有無限多組正整數解。若將解表為(a,b,c),a小於或等於b小
    於或等於c,則
    ① 當c=3p^n或c=6p^n時，有解必唯一。
    ② 若c為奇數，當c-2=p^n或c+2=p^n時，有解必唯一。
    ③ 若c為偶數，當c-2=4p^n或c+2=8p^n時，有解必唯一。
    (3)當k=3時，即為大家熟知的馬可夫方程式。

    In this paper, we discuss the positive integers solutions of the Diophantine equations x^2+y^2+z^2=kxyz.
    (1)When k doesn't equal to 1 and 3, the equations have no
    positive integers solutions.
    (2)When k=1, the equation has infitely many positive
    integers solutions. We can let (a,b,c) be the solution
    and arrange its entries in ascending order.
    ①The solution is determined uniquely by c when c=3p^n
    or c=6p^n.
    ②If c is odd, the solution is determined uniquely by c
    when c-2=p^n or c+2=p^n .
    ③If c is even, the solution is determined uniquely
    by c when c-2=4p^n or c+2=8p^n.
    (3)When k=3, it is the well known Markoff equation.

    Contents 1 Introduction.....1 2 The case for $k\geq{4}$.....3 3 The case for k=2.....4 4 The case for k=1.....5 4.1 The tree structure of solutions.....5 4.2 Markoff Matrix.....5 4.3 Proof of Theorem 1.1.....11 4.4 Properties of the solutions.....12 4.5 Proof of Theorem 1.2.....13 5 The case for k=3.....16

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