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

Abstract

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.