Original version of This story Appears in Quanta Magazine.
The simplest idea in mathematics is also probably the most confusing.
Increase. Here is a simple operation: one of the first mathematical truths we learn is that 1 plus 1 equals 2. However, mathematicians still have many unsolved problems with the increased pattern. “This is one of the most basic things you can do,” said Benjamin Bedert, a graduate student at Oxford. “Somehow, this is still very mysterious in many ways.”
Mathematicians also want to understand the limitations of additive power when exploring this mystery. Since the early 20th century, they have been studying the nature of the “no generalization” set, a set of numbers where no two numbers in the set will increase by one third. For example, add any two odd numbers and you will get a uniform number. Therefore, sets of odd sets are not summarised.
In a 1965 paper, prolific mathematician Paul Erds asked a simple question that illustrates the set without summary. But over the decades, the progress of the problem has been negligible.
“It’s a very basic thing, and we have very little understanding of us,” said Julian Sahasrabudhe, a mathematician at the University of Cambridge.
Until February this year. Sixty years after Erd raised his question, Bedert solved it. He shows that in any set consisting of integers (positive and negative counted numbers), there are large numbers that must not be summarised. His evidence goes deep into mathematics, honing techniques from different fields, revealing hidden structures not only in non-expensive sets, but also in various other environments.
“This is an amazing achievement,” Sahasrabud said.
Stuck in the middle
Erds knows that any group of wholes must contain a smaller, non-expensive subset. Consider the set {1, 2, 3}, which is not unsummarized. It contains five different subsets of no sum, such as {1} and {2, 3}.
Erd wondered how far this phenomenon stretched. If you have a suit of one million integers, how big is its largest subset of no sum?
In many cases, this is huge. If you randomly select a million integers, about half of them are weird, giving you a ~500,000 element.
In a 1965 paper, Erd Horse showed in the evidence that it was several lines of evidence and was praised as brilliant by other mathematicians – n Integers have at least no summarization subsets n/3 elements.
Despite this, he was not satisfied. His proof involves the average: he found a series of uncompiled subsets and calculated that their average size is n/3. But in such collections, it is generally believed that the largest subset is much larger than the average.
ERDS wants to measure the size of those subsets that do not contain sums.
The mathematicians quickly assume that as your suit grows bigger, the largest subset of no sum will become more than n/3. In fact, the deviation will become infinitely large. This prediction-the size of the largest subset of no sum is n/3Add some deviations to make infinite n– Now called the conjecture of no summarization set.
