臺灣博碩士論文加值系統

在這篇論文中，我們要討論下列兩個方程式：當p＞n＞1時， 或-1和 。我們發現在某些情形之下，若方程式 或-1有整數解時，則p不能以 的形式表示，其中a和b是整數。

Abstract
Let p > n > 1 be squarefree positive odd integers. In this
thesis, we will study the relation between the following two
Diophantine equations: px 2
− ny 2 = 1 or − 1 and p = x 2 +ny 2 .
We find under some situations, one of equations px 2
− ny 2 = 1
and − 1 has integral solutions implies that p can not be of the
form a 2 + nb 2 for some integers a and b.

Contents
1 Introduction 1
2 Preliminaries 2
2.1 Continued Fractions . . . . . . . . . . . . . . . 2
2.2 Legendre Symbol . . . . . . . . . . . . . . . . . 7
2.3 Some Well-known Results: . . . . . . . . . . . . 9
3 Main Theorem 16
Appendix 29
References 30

References
[1] P.Chowla and S. Chowla, Problems on periodic simple con-tinued
fractions, Proc. Nat. Acad. Sci. USA 69 (1972),
3745.
[2] David A. Cox, Primes of the form x 2 + nY 2 , A Wiley-Interscience
Publication,1989.
[3] A. Grelak and A. Grytczok, On the Diophantine equation
ax 2
− by 2 = c, Publ. Math. Debrecen 44/3-4 (1994), 291-
299.
[4] Ivan Niven, An introduction to the theory of numbers, John
Wiley & Sons, Inc,1991.
[5] A. J. van der Poorten and P. G. Walsh, A note on Jacobi
symbols and Continued Fractions, Amer. Math Monthly
106 (1999), no.1, 52-56.
[6] O. Perron, Die Lehre von den Kettenbruchen, Chelsea Pub-lishing
Company, New York, 1950.
[7] A. Schinzel, On two conjectures of P. Chowla and S.
Chowla concering continued fractions, Ann. Mat. Pura
Appl.98 (1974), 111-117.
[8] Takashi Ono, An Introduction to Algebraic Number The-ory,
Publishing Company, New York, 1990.
30