A New Solution To An Age Old Problem.

 

Recently a mathematician working at the University of Oxford, proposed a new solution to a problem that has been around for centuries.  The problem involves unit fractions or fractions with 1 in the numerator such as 1/7 and was first considered back in Egyptian Times.  The reason they worked with unit fractions is because it was the only type of fraction found in their number system. 

In order to express larger fractions they added two smaller ones together.  For instance, to represent 3/4, they'd add 1/2 + 1/4.  In the 1970's, two mathematicians suggested that if you had a sufficiently large proportion of whole numbers, there would be a subset of reciprocals that add up to one. Even if the numbers were chosen to make it difficult to find a subset, the subset still exists.

One paper, from about 20 years ago, used a color method to find the solution.  The idea is that the whole numbers are sorted into different colored buckets and at least one bucket is likely to have a subset that adds up to one.  

This mathematician used principles from harmonic analysis which is a branch closely related to calculus and applied them to the problem. He relied on an exponential sum which is used to determine the number of integer solutions to a problem.  It was used to determine the number of subsets that contained a sum of unit fractions that equaled one. He was able to confirm that that there did indeed exist the subset but not predict the density or how many subsets existed because the harmonic analysis uses the bucket sorting idea, not the density one.

When the methods from harmonic analysis were used, he wanted to avoid composite numbers with lots of factors because they tend to make large denominators larger. So he focused on proving that if there are lots of numbers with relatively small prime numbers, there will be a subset whose reciprocals add up to 1. In addition, he was able to show that there was always at least one bucket that had enough numbers for this to occur.

Unfortunately, this criteria cannot be applied to the density.  One cannot just choose a convent bucket. If they choose a bucket with nothing in it, this doesn't work.  As another mathematician was preparing to read this particular paper to others, he was able to figure out how to use this idea with numbers that had a large number of prime numbers. 

The problem with using the exponential sum is that you cannot get a precise number as it will always be an approximation or estimation. Rather than looking for fractions that added up to one, he looked for subsets whose reciprocals added up to smaller fractions and used those smaller fractions to find one.  Although there are still questions regarding this out there, this solution was one of elegance and beauty.  Let me know what you think, I'd love to hear.