A Greek mathematician has found the answer to a mind boggling maths problem that has remained unanswered for 78 years – until now.
Associate Professor of Mathematics Dimitris Koukoulopoulos together with Oxford University research professor James Maynard, has solved the Duffin and Schaeffer Conjecture.
First expressed in 1941 by mathematicians R J Duffin and A C Schaeffer, the last time a mathematician showed promise in solving the problem was in 1990. But it wouldn’t be until 29 years later that it would be fully proven by Koukoulopoulos and Maynard – two relatively young mathematicians, both aged in their 30’s.
The premise behind the Duffin and Schaeffer Conjecture is that a criteria can be set to approximate numbers if certain denominators are excluded – however if some denominators are excluded, even a sparse subset of them, some numbers may never be approximated. This maths problem belongs to the field of number theory and is called a diophantic approach, named in honour of Diophantus who was an Alexandrian Greek known for the Diophantine equation, concerned with approximating fractional numbers.
“So there are these two worlds where we can get almost all the numbers in one and almost none in the other, but there is a simple criterion that decides when we stumble upon each case,” Koukoulopoulos told Athens – Macedonia News Agency.
“Most numbers, such as the number π, which is a mathematical constant defined as the ratio of the circumference to the diameter of a circle (π = P / d) and equals 3,14159265, occurs very often in mathematics, in physics, and if one sits down and writes the decimal numbers to give an approximation of that number, they will find that they never end,” he explains.
While people cannot work with such complex number, computers certainly can. But when it comes to practically solving a complex problem such as this, he says it’s important to simplify.
“If I write the decimal numbers of π and stop at 3.14 I get an approximation of the number with an error. I can write this number 3141/1000 which is a fraction approaching π, but in fact the ancient Greeks also knew that a very good approach of π using much smaller numbers is the fraction (22/7) that uses a denominator a lot smaller. Its denominator is only 7, while the denominator of the other fraction is 1,000. The second fraction has much less complexity. And the question is, if we can use denominators of a barrier up to 1 millionth, how good an approximation of a number can we get to? In such big questions, the ‘diophantic approach’ needs a simple fraction to find simple numbers approximations.”
While solving the riddle is a remarkable achievement, Koukoulopoulos says that he doesn’t know whether it will be applicable to practical matters in life.
“In theoretical mathematics it would be nice to see your work applied in real life, but the nature of theoretical mathematics is such that applying ideas can take many years until something is done or even an indirect contribution is made,” he said.
Despite this however, the mathematician says that funding in this area is vital, given that greater understanding of such equations can have a direct impact on research in the fields of applied mathematics, physics and engineering.
Hailing from Kozani, Koukoulopoulos studied mathematics at the Aristotle University of Thessaloniki before obtaining his post-graduate degree from the University of Illinois. He then went on to get his PhD at Princeton, and has continued on to his current post at the University of Montreal where he is apart of a post-PhD program.