Student Solves a Long-Standing Problem About the Limits of Addition

Original version one This story appeared How many magazines.

The simplest ideas in mathematics can be the most confused.

Add it. It is a correct operation: one of the first mathematical truths we learn is 1 plus 1. However, mathematicians do not have unanswered questions regarding the types of specimens that can still be added. “It’s one of the most basic things you can do,” he said Benjamin BedertA graduate student at Oxford University. “Somehow it’s still very mysterious on the way.”

In trying this secret, mathematicians hope to understand the extra power limit. Since the beginning of the 20th century, they studied the nature of the “total free” sets, which were not added to a third of the two number in the set. For example, add any two odd numbers and you will receive an equal number. Therefore the sum of a single number of numbers.

In 1965, in a paper, prolific mathematician Paul Erdős, made a simple question about how the sets without the total amount. But for decades, the progress of the problem was insignificant.

“The shock is a very basic voice of understanding,” said Julian SahasrabudheA mathematician at the University of Cambridge.

This is until February. After sixty years after taking the problem of Erdős, Bedert solved it. He showed that there are positive and negative counting numbers in any set consisting of integers The bottom of a number of numbers that are not sumuz. Its proof, the depths of mathematics, the difference in the areas, which opens the secret structure not only in total facilities, but other samples.

“It’s a fantastic achievement,” said Sahasrabude.

Stuck in the middle

Erdős knew that any integers should be smaller, in total. Review the set of non-silent {1, 2, 3}. It contains five different sums of five different places like {1} and {2, 3}.

Erdős wanted to know how much this phenomenon was. If you have a set with a million integers, how big is the bottom of the largest amount without the largest amount?

In many cases large. If you choose a million integers randomly, half of them will be the only one that gives a submarine with an amount of about 500,000 elements.

Paul Erdős was famous for getting acquainted with the deep grounds that continue to direct math research today.

Photography: George CHICORY

In 1965, in his paper, Erdős, a few lines showed an argument that was shining by a long proof and other mathematicians N The integers have at least one amount of subset N/ 3 elements.

Was not satisfied again. His proof was engaged in the average: The sum of the total free sub-parts found and calculated that they were average size N/ 3. However, the largest sub-lower sections in such a collection were usually larger than average.

Erdős, he wanted to measure the size of the lower lower parts of the bottom.

Mathematicians will soon become bigger than your set, the largest amount of undersecrific N/ 3. In fact, the deviation will grow endlessly. This forecast – the size of the bottom of the greatest total N/ 3 plus some deviations to infinity NNow the sum is known as free sets.