Dick Ahlstrom,
Science Editor
A fifth-year student from CBS Synge Street, Dublin, has completed a stunning piece of mathematical detective work with his research into "continued fractions".
He improved on formulas produced by two other research groups, but more importantly he developed an entirely new way to study continued fractions produced by concepts such as pi.
Everyone is familiar with ongoing numbers that don't reach an end, for example the decimal for one third or the value of pi. These decimals can also be expressed as continuing fractions and this is the research area that took hold of Ronan Larkin (16).
It is an exotic form of pure maths that doesn't have many applications, he admits. It is also an area of particular interest to maths experts, however, who are ever on the lookout for decimals or fractions that repeat as they progress.
"If it is repeating, a pattern is involved," explained Ronan. "When mathematicians see a pattern they become interested." He started to study research announced in 1997 by a Venezuelan, Domingo Gomez, who claimed he had developed formulas for continued fractions arising from irrational numbers such as the cubed root of two minus one. Gomez provided no information, only clues as to how he had produced the formulas, so Ronan developed his own methods, first using one hinted at by Gomez and then an entirely new one.
He did something similar with another way to tackle continued fractions, the "Gupta-Mittal Algorithm". Again he improved on it and also proved his work using a standard computer programme. "I implemented it on a spreadsheet which hasn't been done before." Most importantly, he tackled the continued fractions produced by what are termed transcendental numbers, for example pi or e, the base of natural logs. He developed a generalised formula for transcendental numbers.
"I was able to come up with continued fractions for transcendental numbers, which has never been done before," Ronan explained.