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Beynon, Meurig On Raney's binary encoding for continued fractions, generalisations of Pell's Equation, and the theory of factorisation. University of Warwick. Department of Computer Science.
Selenius first obtains the regular continued fraction RCF and replaces all 1's by a process of singularisation. Contrary to a disparaging review by D. Lehmer of A. Ayyangar's paper below, NSCF has some nice properties. Jackson, K. Matthews, On Shanks' algorithm for computing the continued fraction of log b a , Journal of Integer Sequences 5 article See the online BCMath program. Here are the first partial quotients of the continued fraction expansion of log 2 3. Matthews, R.
This used the 2x2 matrix point of view that R. Walters, Alternate derivation of some regular continued fractions , J. Soc 8 , We used a result of D. Lehmer Arithmetical theory of certain Hurwitzian continued fractions , Amer. Also see online BCMath program. Some continued fraction identities pdf. A unimodular matrix and Pell's equation pdf. Reduced quadratic irrationals and Pell's equation pdf.
Latexed version of Theory of the nearest square continued fraction , A. Krishnaswami Ayyangar, J. Mysore Univ. A, 1 , The original is somewhat indistinct in some places, so I retyped it in Latex, expanding and changing the author's proofs in some places when they were hard to follow.
We give a variant of his proof. On the definition of nearest integer reduced quadratic surd with John Robertson. Hurwitz gave a definition of reduced quadratic surd for his nearest integer continued fraction expansion which characterises purely periodic surds.
We give a more accessible proof. Matthews, John P. Robertson, Jim White, Midpoint criteria for solving Pell's equation using the nearest square continued fraction , Math. 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.
Slidetalk John P. Robertson, Keith R. Keith R. Williams and P. Matthews, Unisequences and nearest integer continued fraction midpoint criteria for Pell's equation , Journal of Integer Sequences, 12 , Article Williams using singular continued fractions. We derive these criteria without the use of singular continued fractions. This proves a conjecture of John Robertson. Continuants and half-regular continued fractions updated 2nd August Testing a quadratic surd for being NSCF reduced.
It was superceded by the more elegant and efficient test of item 21 below. Bosma and C. We present a test for determining whether a real quadratic irrational has a purely periodic nearest square continued fraction expansion. This test is somewhat more explicit than the standard test and simplifies the programming of the algorithm Keith R.
Matthews, On the optimal continued fraction expansion of a quadratic surd , J. Math Soc. We describe the period structure of the optimal continued fraction expansion of a quadratic surd, in terms of the period of the nearest square continued fraction expansion.
The analysis results in a faster algorithm for determining the optimal continued fraction expansion of a quadratic surd and has been implemented in a BCMATH program. On the convergents of semi-regular continued fractions revised 25th November Matthews, J. Robertson, J.
White, On a diophantine equation of Andrej Dujella , Math. Glasnik, vol. It has been conjectured by Andrej Dujella in that there is at most one such solution for each k. Improved version of slidetalk given at the ANU on 30th November Keith Matthews Last modified 8th December
In mathematics , a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. The numerical value of an infinite continued fraction is irrational ; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions.
A special case of the quadratic Diophantine equation having the form. The equation. While Fermat deserves credit for being the first to extensively study the equation, the erroneous attribution to Pell was perpetrated by none other than Euler himself Nagell , p. The Pell equation was also solved by the Indian mathematician Bhaskara. Pell equations are extremely important in number theory , and arise in the investigation of numbers which are figurate in more than one way, for example, simultaneously square and triangular. However, several different techniques are required to solve this equation for arbitrary values of , , and. Pell equations of the form 1 , as well as certain cases of the analogous equation with a minus sign on the right,.
D. We can now prove the standard recursive formulas for pn and qn. Proposition 8. Let β = [a0,a1,a2, ] be a continued fraction. The numerator and denomina-.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Authors: Martin Bridgeman.
Now, we use the Euclidean algorithm to generate a finite simple continued fraction representation for any rational. Show that if x, y and x1,y1 are solutions, and the. Navigation menu In mathematicsa continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In contrast, an infinite continued fraction is an infinite expression.
Чья смена? - громко спросил он, пробегая глазами список. Согласно расписанию, в полночь должен был заступить на двойную смену новый сотрудник по имени Зейденберг. Чатрукьян еще раз обвел глазами пустую лабораторию и нахмурился. - Где же он, черт возьми. Глядя на оживающий монитор, он подумал, известно ли Стратмору, что в лаборатории систем безопасности нет ни души.
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Джабба открыл рот. - Но, директор, ведь это… - Риск, - прервал его Фонтейн. - Однако мы можем выиграть. - Он взял у Джаббы мобильный телефон и нажал несколько кнопок. - Мидж, - сказал .
Стратмор задумался. - Должно быть, где-то замыкание. Желтый сигнал тревоги вспыхнул над шифровалкой, и свет, пульсируя, прерывистыми пятнами упал налицо коммандера.
Solution to Pell's Equation. 9. References. 1. Continued Fractions. This rather long section gives several crucial tools for solving Pell's.
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ReplySelenius first obtains the regular continued fraction RCF and replaces all 1's by a process of singularisation.
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