Continued Fractions

Hensley, Doug

doi:10.1142/5931

Cited by 70 publications

(69 citation statements)

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“…For example, the cyclic method known in India in the 12th century, and the slightly less efficient but more regular English method 17th century, produce all solutions of x 2 − dy 2 = 1 (see [4]). But the most efficient method for finding the fundamental solution is based on the simple finite continued fraction expansion of √ d (see [2,5,6,[10][11][12][13] 1 , p n = a n p n−1 + p n−2 and q n = a n q n−1 + q n−2 for n ≥ 2. If r is odd, then the fundamental solution is (x 1 , y 1 ) = (p r , q r ), where p r /q r is the rth convergent of √ d and if r is even, then the fundamental solution is (…”

Section: Preliminariesmentioning

confidence: 99%

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The Diophantine equation x 2−(t 2+t)y 2−(4t+2)x+(4t 2+4t)y=0

Tekcan

Özkoç

2009

Rev Mat Complut

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Section: Preliminariesmentioning

confidence: 99%

“…It can be proved as in same way that the previous assertion was proved. , (5,6), (6,5), (6,6), (8,1), (8,10), (10, 0) , (2,11), (6,5), (6,8), (7,5), (7,8), (11,2), (11,11), (12, 0) ,…”

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confidence: 99%

The Diophantine equation x 2−(t 2+t)y 2−(4t+2)x+(4t 2+4t)y=0

Tekcan

Özkoç

2009

Rev Mat Complut

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“…Surprisingly, Doug Hensley [2] found examples of complex numbers that are algebraic of degree 4 over Q(i) and have bounded complex partial quotients (in the Hurwitz expansion). This paper attempts to collect and tidy up the examples and proofs of Hensley, and to generalize them to obtain the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Complex Numbers With Bounded Partial quotients

Bosma

Gruenewald²

2012

J. Aust. Math. Soc.

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Abstract. Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the generalization of the nearest integer continued fraction for complex numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent. IntroductionReal numbers admit regular continued fraction expansions that are unique except for an ambiguity in the ultimate partial fraction of rational numbers. The same is true for nearest integer continued fraction expansions. The nearest integer expansion is easily obtained from the regular expansion, by applying a certain modification for partial quotients that equal 1, as a result of which some partial quotients are incremented by 1, and some minus signs are introduced. As a consequence, for questions concerning the boundedness of partial quotients of real numbers, the behaviour of regular and nearest integer continued fractions is alike.In both cases, finite expansions occur precisely for the rational numbers, and ultimately periodic expansions occur precisely for quadratic irrational numbers. Not much is known about the partial quotients for other algebraic irrationalities. There exist transcendental numbers with bounded partial quotients, and also transcendental numbers with unbounded partial quotients.Date: September 12, 2011.

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“…We also collect there some (basic) results about these systems that we need in the proof of Theorem 2.1. An interested reader is, however, strongly encouraged to consult [8] and/or [5] (especially Chapter 11) for a deeper understanding of the theory of conformal iterated function systems or, more generally, of conformal graph directed Markov systems, the theme of the book [8], where one can also find some applications of this theory. An updated list of its selected applications to the theory of iteration of transcendental meromorphic functions, number theory, Kleinian groups, and other areas can be found in [10].…”

Section: Introductionmentioning

confidence: 99%