Document Type

Article

Publication Date

2003

Abstract

Finding polynomial solutions to Pell’s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In this paper, for each triple of positive integers (c, h, f) satisfying c 2 − f h2 = 1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t), h(t), f(t)) which satisfy c(t) 2 − f(t) h(t) 2 = 1 and (c(0), h(0), f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t), h(t)) constitute the fundamental polynomial solution to the Pell’s equation above. The continued fraction expansion of p f(t) is given in certain general cases (for example, when the continued fraction expansion of √ f has odd period length, or even period length or has period length ≡ 2 mod 4 and the middle quotient has a particular form etc). Some applications to determining the fundamental unit in real quadratic fields is also discussed.

Publication Title

Journal of the London Mathematical Society

ISSN

0024-6107

Publisher

London Mathematical Society

Volume

67

Issue

1

First Page

16

Last Page

28

Comments

Preprint is available here.

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