Irrationality and transcendence of infinite continued fractions of square roots

2018 ◽  
Vol 184 (1) ◽  
pp. 31-36
Author(s):  
Jaroslav Hančl ◽  
Ondřej Kolouch ◽  
Radhakrishnan Nair
Keyword(s):  
Continued Fractions ◽  
Square Roots

scholarly journals Continued Fractions of Square Roots of Natural Numbers

10.14311/1821 ◽  
2013 ◽  
Vol 53 (4) ◽  
Author(s):  
L'ubomíra Balková ◽  
Aranka Hrušková
Keyword(s):  
Natural Number ◽  
Continued Fractions ◽  
Natural Numbers ◽  
Precise Form ◽  
Square Roots

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author describes periods and sometimes the precise form of continued fractions of ?N, where N is a natural number. In cases where we have been able to find such results in the literature, we recall the original authors, however many results seem to be new.

2. The Researches of M. E. de Jonquières on Periodic Continued Fractions

1884 ◽  
Vol 12 ◽  
pp. 389-398
Author(s):  
Thomas Muir
Keyword(s):  
Continued Fractions ◽  
Mathematical Theory ◽  
General Theorem ◽  
Square Roots ◽  
French Academy ◽  
Special Character ◽  
The Subject ◽  
Periodic Continued Fractions

1. During the present year there has appeared at intervals, in the Comptes Rendus of the French Academy, quite a series of communications by M. E. de Jonquieres, on the subject of those periodic continued fractions which are the equivalents of the square roots of integers. These communications have attracted attention, both on account of the number of results given in them, and because, as a writer in the Bulletin des Sciences Mathématiques says, of their interesting and profound character. To any one really intimate with the bibliography of the subject, this cannot but be a little surprising. It is true that the number of so-called theorems is great; but the very special character of a number of them, the fact that they are just such theorems as may be obtained by experiment and induction, and the want of demonstrations of them as evidence that the author was in possession of a mathematical theory of the subject, are points that have been too much overlooked. Further, and what is more important, many of the theorems are not new, and there is a sense in which the epithet “new” cannot fairly be applied to any of the earlier ones, because of the existence of a widely general theorem in which they are directly included, or from which they may with readiness be deduced.

Expressions for rational approximations to square roots of integers using Pell's equation

2019 ◽  
Vol 103 (556) ◽  
pp. 101-110
Author(s):  
Ken Surendran ◽  
Desarazu Krishna Babu
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Integer Solution ◽  
Rational Approximations ◽  
Positive Integer Solution ◽  
Pell’S Equation ◽  
Square Roots ◽  
Integer Solutions

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation:p2 −Dq2 = 1, whereDis any positive non-square integer. With known positive integer solutionp1 andq1 we can compute, using these recursive expressions,pnandqnfor alln> 1. See Table in [2] for a list of smallest integer, orfundamental, solutionsp1 andq1 forD≤ 128. These (pn,qn) pairs also formrational approximationstothat, as noted in [3, Chapter 3], match with convergents (Cn=pn/qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for.

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