In the field of number theory, mathematicians study different types of sequences of numbers. This basic observation has significant implications. The number of distinct products, however, will be much larger - more on the order of a constant value times N 2. They showed that whenever the numbers used to generate the grid are consecutive (such as 1 through 12), the number of distinct sums will be 2 N − 1, where N stands for the number of numbers used to generate the grid. At the same time they proposed their conjecture, they solved a special case of the problem. If you compare the grid of products with the grid of sums, you might notice that there are many more distinct products than distinct sums.Įrdős and Szemerédi certainly did. Write out the numbers 1 to 12 along the bottom and up the side and fill out the grid accordingly.
![list of prime numbers 1-12 list of prime numbers 1-12](https://www.smartick.com/blog/wp-content/uploads/prime.19.jpg)
The most familiar grid in arithmetic is the times table. In a paper posted last year, a graduate student at Urbana-Champaign named George Shakan came the closest yet. Over the last few decades, mathematicians have clawed their way toward proving Erdős and Szemerédi’s threshold. “It somehow delves into the deep structure of the integers, meaning the interplay between multiplication and addition,” said Kevin Ford, a mathematician at the University of Illinois, Urbana-Champaign, who has worked on the problem.
![list of prime numbers 1-12 list of prime numbers 1-12](https://d1avenlh0i1xmr.cloudfront.net/bb1fa880-3934-4708-93b6-f948885016c6/slide1.jpg)
It’s as if the sum and product sets are magnetically charged to repel each other.Īnd like magnetism, the phenomenon is easy to observe but much harder to explain. If we flip the problem around and look at the number of duplicate entries, it says the following: If there are a lot of duplicates in the sum grid, then there cannot be that many duplicates in the product grid (and vice versa). Put more generally, the conjecture says that the additive and multiplicative properties of a set of numbers somehow force each other to behave in a certain way.