On the Diophantine Equations x^2+y^2+z^2=kxyz

On the Diophantine Equations x^2+y^2+z^2=kxyz

林志穎(Chih-Ying Lin)

指導教授 ： 洪有情

國立臺灣師範大學/理學院/數學系/碩士(2006年)

摘要

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。

關鍵字

馬可夫方程式

並列摘要

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.

並列關鍵字

Markoff Equation

參考文獻

[1] Arthur Baragar, On the unicity conjecture for Markoff numbers, Canad. Math. Bull. 39 (1996), no. 1. 3-9.

[2] J. O. Button, The uniqueness of the prime Markoff numbers, J. London Math. Soc. (2) 58 (1998), no. 1, 9-17.

[3] H. Cohn, Approach to Markoff's minimal forms through modular functions, Annals of Math. (2) 61 (1955), no. 1, 1-12.

[4] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and

[6] Hurwitz, A, Uber die Aufgabe der unbestimmten Analysis, Arch. Math. Phys. (3) 11 (1907), 185-196.