The student solves a long -term problem about the limits of addition

The original version to This story Appear Quanta magazine.

The simplest ideas in mathematics can be the most confused.

Take add. It is a clear process: one of the first mathematical facts that we learn is that 1 plus 1 is equal to 2. But mathematicians still have many questions that have not been answered about the types of patterns that can lead to this. “This is one of the most important things you can do,” he said Benjamin PederApplicitist Student at Oxford University. “Somehow, it’s still very mysterious in several ways.”

Upon investigation of this mystery, mathematicians also hope to understand the limits of the addition of the addition. Since the early twentieth century, they have been studying the nature of “sheet-free” groups-where he did not add two numbers in the group to a third. For example, add any two individual number and you will get my husband’s number. Thus, the individual numbers set is free.

In the 1965 sheet, a prolific production scientist Paul Erdis asked a simple question about the prevalence of free group groups. But for decades, progress in the problem was small.

He said: “It is a very essential thing that we have not shocked.” Julian SahsarbaMathematics scientist at Cambridge University.

Until February. Sixty years later, Erdős raised his problem and solved by Bedert. It has shown that in any group consisting of correct numbers – positive and negative count numbers – there A large sub -set of numbers that should be free. Its proof of mathematics and sharpening techniques of varying fields to detect the hidden structure not only in sheet -free groups, but in all other settings.

“It is a great achievement,” said Sahasrabudhe.

Stuck

Erdős knew that any group of correct numbers should contain a smaller sub -group of the summary. Consider the group {1, 2, 3}, which is not free. It contains five different free sub -groups, such as {1} and {2, 3}.

Erdős wanted to know the extension of this phenomenon. If you have a set with a million number of correct numbers, how extent the size of a summary is the summary of the summary?

In many cases, it is huge. If you choose a million correct numbers randomly, it will be about half of them strange, allowing you to a summary of the summary with about 500,000 elements.

In his research in 1965, he showed Erdős – in evidence that he was a few lines, and was praised by other mathematicians – that is, a group of N The correct numbers contain at least a free subsidiary N/3 elements.

However, he was not satisfied. His evidence is dealing with averages: He found a group of free sub -groups and calculating that their average size was N/3. But in such a group, the largest sub -groups are believed to be larger than average.

Erdős wanted to measure the size of these blab -free sub -groups.

Mathematics soon assumed that with the increase in the size of your group, it will become the largest subsidiary of the group N/3. In fact, a large deviation will grow without limits. This prediction-that the size of the altogether is the size of the sub-group N3 /In addition to some deviation that grows indefinitely NIt is now known as free guessing.