本論文主要探討馬可夫型的數論問題。更詳細的說，給定整
數k，我們研究同時滿足x|y^2 + k 和y|x^2 + k的正整數x 和y的性
質。更進一步地，我們證明了當k 是正整數的時候，這樣的正整數
x 和y有無窮多組解。值得一提的是，這個問題等價於決定正整數
a的值，使得丟翻圖方程x^2 - axy + y^2 + k = 0 有正整數解x和y。
我們證明了當這個方程對應的基本解(x_0,y_0) 滿足d = min{x_0, y_0}
時，a := a(k) = (k+r^2)/(d(d+r)) + 2，其中r 為滿足(k+r^2)/(d(d+r))
∈ N 的所有正整數。

In this thesis, we investigate positive integer x and y satisfying x|y^2+
k and y|x^2 + k for each k ∈ Z. We show that such positive integer solutions
have infinitely many numbers. In particular, via this issue, we
can further study the positive integer solutions to Diophantine equation
x^2 -axy +y^2 +k = 0. Moreover, if (x_0, y_0) is the corresponding fundamental
solution and d = min{x_0, y_0}, then a := a(k) = (k+r^2)/(d(d+r)) + 2 where
r runs all positive integers satisfying k+r^2)/(d(d+r))∈ N .

摘要 ii
Abstract iii
Acknowledgements iv
List of Tables vi
1 Introduction 1
1.1 Pell equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.2 Continued fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2 Main results 8
3 Other Results 25
4 Appendix 29
4.1 Open problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References 34

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