Let a,b be fixed positive integers such that a is not a perfect square and b is squarefree, and let ω(b) denote the number of distinct prime divisors of b. Let (u1, v1) denote the least solution of Pell equation u2 − av2 = 1. Further, for any positive integer n, let un = αn+ᾱn 2 and vn = αn−ᾱn 2a, where α = u1 + v1a and ᾱ = u1 − v1a. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations (∗)x2 − ay2 = 1 and y2 − bz2 = v 12 to have positive integer solutions (x,y,z) is obtained. By this result, we prove that if (∗) has a positive integer solution (x,y,z) for ω(b) ≤ 2 or 3 according to 2 ∤ b or not, then 4u12 − δ = bg2 and (x,y,z) = (ur,vr,gv[r+1 2 ]), where g is a positive integer, δ = 1 or 2 and r = 2 or 3 according to 2 ∤ b or not, [r+1 2 ] is the integer part of r+1 2, except for (a,b,x,y,z) = (24, 2134, 47525, 9701, 210)

中文翻译：

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关于联立 Pell 方程 x2 - ay2 = 1 和 y2 - bz2 = v12 的可解性

让一种,b是固定的正整数，使得一种不是一个完美的正方形并且b是无平方的，并且让ω(b)表示不同的主要因数的数量b. 让(你1, v1)表示 Pell 方程的最小解你2 - 一种v2 = 1. 此外，对于任何正整数n， 让你n = αn+ᾱn 2和vn = αn-ᾱn 2一种， 在哪里α = 你1 + v1一种和ᾱ = 你1 - v1一种. 在本文中，利用 Pell 方程的基本性质和关于二元四次丢番图方程的一些已知结果，建立方程组的充要条件(*)X2 - 一种是的2 = 1和是的2 - bz2 = v 12有正整数解(X,是的,z)获得。通过这个结果，我们证明如果(*)有一个正整数解(X,是的,z)为了ω(b) ≤ 2要么3根据2 ∤ b与否，那么4你12 - δ = bG2和(X,是的,z) = (你r,vr,Gv[r+1 2 ])， 在哪里G是一个正整数，δ = 1要么2和r = 2要么3根据2 ∤ b或不，[r+1 2 ]是整数部分r+1 2， 除了(一种,b,X,是的,z) = (24, 2134, 47525, 9701, 210)