2004 IRISH SCIENTIST YEAR BOOK

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Synge Street Christian Brothers' School, Dublin
Ronan Larkin
Generalised Continued Fractions

Ronan Larkin at the Exhibition

In January 2003, I entered a project for the EsatBT Young Scientist Competition on the topic of continued fractions. Using algorithms implemented on spreadsheets, I discovered many new continued fraction expansions. But all through my first project the same question kept coming up. Why was it so difficult to represent most algebraic and transcendental numbers as traditional continued fractions that have regular patterns? This was not a new question � most mathematicians who worked with continued fractions over the last 300 years would have asked themselves this question and then put it quietly to one side since the question seemed hopelessly difficult and impossible to answer.

In 1997 a Venezuelan mathematician called Domingo Gomez invented a new type of continued fraction which he called a "Generalised Continued Fraction" (GCF). Using this new concept Gomez solved a very old and notorious problem in mathematics � how can we represent 3 v2 as a continued fraction which has a simple pattern? The answer given by Gomez is that you need a Generalised Continued Fraction to do this.

Gupta and Mittal (2000) devised an algorithm that reveals that, if we have a suitable pair of irrational numbers (I refer to this pair of numbers as "Gupta�Mittal conjugates"), the Gupta�Mittal GCF of each is periodic after a certain point.

This year's project is therefore a very natural development of last year's. I have extended the work of Gomez and of Gupta and Mittal in a number of directions:

  • �I have proved a number of theorems that are generalisations of the formulas stated by Gomez.

  • �Using spreadsheets, I can generate and display GCFs of both the Gomez and the Gupta-Mittal types and study their convergence properties.

  • �I have discovered many new examples of Gupta�Mittal GCFs.

  • �I have proved a theorem that identifies an infinite set of Gupta�Mittal conjugate pairs. Each of these pairs generates a Gupta�Mittal GCF with a particularly simple form of periodicity.

  • �A detailed study and comparison of the convergence of the different types of GCFs generated in this project has been made. I have identified types of convergence behaviour never seen before in continued fractions.

  • �Neither the Gomez method nor the Gupta�Mittal method seems to have any relevance to the problem of generating GCFs with regular patterns for transcendental numbers. This project reports the discovery of algorithms that generate GCFs for transcendental numbers. One of the outcomes of this discovery is the possibility of new and useful rational approximations for transcendental numbers.

Ronan Larkin entered his project in the Senior Individual Section in the Chemical, Physical and Mathematical Sciences Category at the EsatBT Young Scientist & Technology Exhibition in January 2004. He won the top prize � Young Scientist of the Year. His teacher was Mr Jim Cooke.


This article was sponsored by Oldbury Publishing