Check out this sequence of numbers: 5, 7, 9. Can you identify the pattern? Here’s another sequence with the same pattern: 15, 19, 23. And one more: 232, 235, 238.
For nearly a century, mathematicians in the field of combinatorics have been trying to determine if an endless list of numbers contains such a sequence, known as an arithmetic progression. In other words, is there a way to be mathematically certain that a set contains a sequence of three or more evenly spaced numbers, even if you don’t know much about how the numbers in the set were selected or what the progression might be?
Progress on the question has been slow, but last year, Meka and Zander Kelley, a Ph.D. computer science student at the University of Illinois Urbana-Champaign, made a significant breakthrough. The researchers are outsiders in combinatorics, which is concerned with counting configurations of numbers, points or other mathematical objects. And the duo didn’t set out to tackle the mystery of arithmetic progressions.
Kelley and Meka were instead investigating abstract games in computer science. The pair sought a mathematical tool that might help them understand the best way to win a particular type of game over and over again. “I’m super-interested in a collection of techniques that fall under this umbrella called structure versus randomness,” Kelley says. Some of the earliest progress on arithmetic progressions relied on such techniques, which is what led Kelley and Meka to dive into the topic.
2024-02-26 08:00:00
Originally from www.sciencenews.org