一类Pell方程组正整数解的存在性

设a和b是无平方因子正整数，k是适合k≡1或2(mod 4)的正整数.根据Pell方程解的性质，运用初等数论的方法给出了方程组x2－ay2＝1和y2－bz2＝4在a＝k(k＋1)且b是偶数时有正整数解的充要条件.

Existence of solution for the system of Pell equations

Let a and b be positive integers with square free， and let k be a positive integer with k≡1 or 2(mod4). In the case that a＝k(k＋1) and b is even，by elementary number theory methods and the properties of solutions of Pell equations， a sufficient and necessary condition that the system of equations x2－ay2＝1and y2－bz2＝4 have positive integer solutions is given.