Simple applications of continued fractions and an elementary result on Heron’s algorithm

Antonino Leonardis
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 4, Pages 59—69
DOI: 10.7546/nntdm.2018.24.4.59-69
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Authors and affiliations

Antonino Leonardis
Department of Mathematics and Computer Science, Universit della Calabria
Arcavacata di Rende, Italy

Abstract

The paper is a continuation of the author’s previous works on continued fractions, and has been presented at the AMS special session on Continued Fraction during the Joint Mathematical Meetings 2017, Atlanta GA. The first part is more introductory/educational, explaining the importance of matricial and Diophantine methods in the topic of continued fractions. We will begin this part discussing geometrical illusions which can arise from properties of continued fractions and associated matrices, proving thoroughly the mathematical reasons of this fact. After this, we will deal with the Pythagorean problem of the right-angled isosceles triangles finding all solutions to the simple Diophantine equation l2 + (l + 1)2 = d2, which will give a “Pseudo-Pythagorean” triangle. In the second part, recalling all the methods introduced in the first one, we will prove a theorem (the main result), which relates continued fractions with Heron’s algorithm, giving some examples. This theorem is proved in a complete form, which considers all possibilities and its vice-versa, unlike all other minor results that can be found in the literature (see [2, 3]).

Keywords

  • Continued fractions
  • Heron’s algorithm
  • Matrices
  • Fibonacci sequence
  • Pythagorean triples
  • Diophantine equations

2010 Mathematics Subject Classification

  • 11A55
  • 11C20
  • 11D09

References

  1. Cassels, J.W. S. (1957) An Introduction to Diophantine Approximation, Cambridge University Press, New York.
  2. Garver, R. (1932) A Square Root Method and Continued Fractions, The American Mathematical Monthly, 39 (9), 533–535.
  3. O’Dorney, E. (2015) Continued fractions and linear fractional transformations, Integers, 15, 1–23.
  4. Perron, O. (1913) Die Lehre von den Kettenbrchen, B. G. Teubner, Lepizig, Berlin.

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Cite this paper

APA

Antonino, L. (2018). Simple applications of continued fractions and an elementary result on Heron’s algorithm. Notes on Number Theory and Discrete Mathematics, 24(4), 59-69, doi: 10.7546/nntdm.2018.24.4.59-69.

Chicago

Leonardis, Antonino. “Simple Applications of Continued Fractions and an Elementary Result on Heron’s Algorithm.” Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 59-69, doi: 10.7546/nntdm.2018.24.4.59-69.

MLA

Leonardis, Antonino. “Simple Applications of Continued Fractions and an Elementary Result on Heron’s Algorithm.” Notes on Number Theory and Discrete Mathematics 24.4 (2018): 59-69. Print, doi: 10.7546/nntdm.2018.24.4.59-69.

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