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The simplest ideas in mathematics can also be the most confusing.
Join more. It is a simple activity: One of the first mathematical facts we learn is 1 plus 1 equal to 2. But mathematicians still have many questions that have not been answered about the types of models that the addition can arise. This is one of the most basic things you can do Benjamin BedertA graduate at Oxford University. Somehow, it is still very mysterious in many ways.
When exploring this mystery, mathematicians also hope to understand the limit of additional power. Since the beginning of the 20th century, they have studied the nature of the free sets, the numbers of the numbers in which no two numbers in the set will add a third. For example, any two odd numbers and you will get an even number. Therefore, gathering odd numbers without sum.
In a 1965 article, the popular mathematician Paul Erdős asked a simple question about how unpopularized ministries. But for decades, the progress of the problem is negligible.
“It is a very basic thing that we have very little knowledge Julian SahasrabudheA mathematician at Cambridge University.
Until February. Sixty years after ERDőS raised his problem, Bedert solved it. He has shown that in any set includes integers, positive and negative counting numbers, A large number of numbers must be unexplained. His evidence reaches the depth of mathematics, honing techniques from different fields to explore the hidden structure not only in the sets without sum, but in all other types of settings.
This is a great achievement.
Trapped in the middle
ERDőS knows that any integer set must contain a smaller, no set of children. Consider gathering {1, 2, 3}, not free. It contains five different total outputs, such as {1} and {2, 3}.
ERDőS wants to know how far this phenomenon lasts. If you have a set with a million integers, how much does the child not have its largest amount?
In many cases, it is very large. If you choose a million integers randomly, about half of them will be strange, giving you a free set of children with about 500,000 factors.
In his 1965 article, ERDőS showed a few lines long, and was praised by other mathematicians as excellent. N The integers have a free set of children N/3 elements.
However, he was not satisfied. The evidence of him handled the average: he found a set of sets of unclean sets and calculated that their average size was N/3. But in such a collection, the biggest sub -sets are often thought to be much larger than the average.
ERDs want to measure the size of the sub -sets without largely.
Mathematicians soon hypothesized that when your set becomes bigger, the biggest free sets will become much bigger than N/3. In fact, the deviation will grow extremely large. This prediction-the size of the free set of children is the biggest N/3 plus a number of development deviations to infinity NIt is now called free guess.