Publication Date

1968

Document Type

Dissertation/Thesis

First Advisor

McKenzie, Harvey C.||Beach, James W.

Degree Name

M.S. (Master of Science)

Department

Department of Mathematics

LCSH

Diophantine analysis

Abstract

The Diophantine equation, x² - Dy² = N, where D and N are known integers, is called the Pell equation. If D is a positive integer not a perfect square, solutions of the Pell equation are closely connected with the convergents to the continued fraction expansion of D. If N = ± 1, methods of finding all integral solutions are known. It is the purpose of this study to take specific forms of √D and to determine those values of N for which the convergents pₙ/qₙ of the continued fraction expansion of √D provide solutions, x = pₙ, y = qₙ, to the Pell equation. The continued fraction expansion of the quadratic irrational √D is periodic. This investigation uses those forms of D for which the continued fraction expansion of √D has a repeating part of length r = 1, 2, or 3. If r = 1, √D has a continued fraction expansion given by √D = [aₒ,̅2̅aₒ] where aₒ denotes the greatest integer in √D and the bar over 2aₒ indicates the repeating part of the continued fraction expansion. It is proved that if r = 1, then D = aₒ² + 1 and the Pell equation, x² - Dy² = N, has solutions x = pₙ, y = qₙ for N = ± 1, where pₙ/qₙ denotes the n-th convergent to the continued fraction expansion of √D. Also, if r = 1, then x² - Dy² = ± 1 has all positive integral solutions given by the convergents Pₙ/qₙ to √D. With aₒ and Pₙ/qₙ as defined above and r = 2, √D = [aₒ,̅a₁,̅2̅aₒ], it is proved that D = aₒ² + 2aₒ/a₁ where a₁ is a positive integer which divides 2aₒ. Also, the Pell equation has solutions x = pₙ, y = qₙ for N = + 1 and N = aₒ² - D. In this case, the convergents pₙ/qₙ to √D provide all positive integral solutions of x² - Dy² = 1 and x² - Dy² = aₒ² - D. For r = 3, a special form of √D is considered, namely √D = [aₒ,̅2,̅2,̅2̅aₒ]. for this form of √D, it is proved that D = aₒ² + (4aₒ + 1)/5 where a ≡ 1 mod 5. Also, it is shown that x = pₙ, y = qₙ constitute all positive integral solutions of the Pell equation, x² - Dy² = N, for N = ± 1 or N = ±(D - aₒ²). Following the proofs of the preceding statements, a brief history of the Pell equation is given as well as some of its mathematical applications.

Comments

Includes bibliographical references.

Extent

57 pages

Language

eng

Publisher

Northern Illinois University

Rights Statement

In Copyright

Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

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Text

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