和整树相关的一个联立不定方程

A simultaneous diophantine equation related to integral trees

文[1]在研究直径为4的整树时提出了求一个联立不定方程a2+b2=m+r+1及a2b2=mr+1(其中b>a)的全部解的问题.本文运用Pell方程的基本性质,获得了这一联立不定方程在m=r时的全部正整数解.

[ 1 ] proposed to solve a simultaneous diophantine equatioin a2 ＋ b2 = m ＋ r ＋ 1 and a2b2 = mr ＋ 1 with b 〉 a while studying integral trees of diameter 4. In this paper, using the basic eqality of Pell equation, we obtain all postive integer solutions to this simultaneous diophantine equation when m = r.