Calculation of the Regulator of Q(√D) by use of the Nearest Integer Continued Fraction Algorithm

Abstract. There are some minor errors in one of the algorithms and two of the tables in a paper by Williams and Buhr. These errors do not affect the major conclusions of the paper.We present corrections to one of the algorithms and two of the tables in [1]. These corrections do not affect the major conclusions of the paper.In the algorithm for computing the NICF of √ D on the bottom half of page 373, when Q k < 0, T k should be defined asR k+1 should be defined as• Otherwise, R k+1 is, as in [1], the remainder on dividing P k+1 + T k+1 by Q k+1 .Note that the formula for R k+1 has to be used with k = −1 in order to set the value of R 0 . In the other formulas in this algorithm k ≥ 0. Also, P k+1 = T k − R k .

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confidence: 68%

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confidence: 99%

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Corrigenda to ``Calculation of the regulator of $\mathbf {Q}(\sqrt {D})$ by use of the nearest integer continued fraction algorithm''

Matthews

Robertson²,

White

2009

“…Later Hurwitz [3] gave further results concerning the NICF and more recently other results have been obtained by Williams [7], Williams and Buhr [6], and Adams [1]. For example, in [6], the following theorem is proved.…”

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confidence: 89%

An analogue of the nearest integer continued fraction for certain cubic irrationalities

Williams¹,

Dueck²

1984

Self Cite

Abstract. Let 8 be any irrational and define Ne(9) to be that integer such that \8 -Ne{6)\ < \. Put p0 = 6, r0 = Ne(p0), pk+ | = \/(rk -pk), rk+x = Ne(pk+I). Then the r's here are the partial quotients of the nearest integer continued fraction (NICF) expansion of 9. When D is a positive nonsquare integer, and 6 = {D, this expansion is periodic. It can be used to find the regulator of 2(/JD ) in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field íwith negative discriminant A. This new algorithm (NIVCF) is periodic and can be used to find the regulator of §.If / < ^/|A j/148, the NIVCF algorithm can be used to find any algebraic integer a of € such that N(a) = I. Numerical results suggest that the NIVCF algorithm finds the regulator of f = 2(/ö ) in about 80 percent of the time needed by Voronoi's algorithm.

“…Buhr [13] gave six midpoint criteria for the nearest integer continued fraction of B. Minnegerode [7] and A. Hurwitz [5]. In our paper, we give three midpoint criteria in terms of the nearest square continued fraction.…”

Section: Introductionmentioning

confidence: 96%

Midpoint criteria for solving Pell’s equation using the nearest square continued fraction

Matthews

Robertson²,

White

2010

Abstract. We derive midpoint criteria for solving Pell's equation x 2 − Dy 2 = ±1, using the nearest square continued fraction expansion of √ D. The period of the expansion is on average 70% that of the regular continued fraction. We derive similar criteria for the diophantine equation y 2 = ±1, where D ≡ 1 (mod 4). We also present some numerical results and conclude with a comparison of the computational performance of the regular, nearest square and nearest integer continued fraction algorithms.