Math lovers, unite! It's a great day when modern-day mathematicians solve or prove math problems from the past, and earlier this month, such a day occurred.
Two mathematicians have worked together to prove the first part of Paul Erdős' conjecture surrounding the additive properties of whole numbers. It is one of the most famous ones.
The paper is currently being peer-reviewed and has been pre-published in arXiv.
What is the conjecture?
Erdős' conjecture asks when an infinite list of whole numbers will be sure to contain patterns of at least three evenly spaced numbers, such as 26, 29, and 32. The famous Hungarian mathematician posed the problem around 60 years ago, one of the thousands of problems he asked throughout his long-standing career.
This particular problem has been a top contender for mathematicians, though.
"Pretty well any additive combinatorialist who’s reasonably ambitious has tried their hand at it," further explained Gowers. The conjecture belongs to the branch of mathematics called additive combinatorics.
As per Quanta Magazine, Erdős posed his problem as follows "Just add up the reciprocals of the numbers on your list. If your numbers are plentiful enough to make this sum infinite, Erdős conjectured that your list should contain infinitely many arithmetic progressions of every finite length — triples, quadruples, and so forth."
Even though countless mathematicians have tried to solve this conjecture, Bloom and Sisask's method is different so far, and doesn't require a strong knowledge of prime numbers' unique structure in order to prove they contain an infinite amount of triples.
"Thomas and Olof’s result tells us that even if the primes had a completely different structure to the one they actually have, the mere fact that there are as many primes as there are would ensure an infinitude of arithmetic progressions," wrote Tom Sanders of the University of Oxford in an email to Quanta Magazine.
It's an exciting time for mathematicians, however, there's still a fair amount of work yet to be done before the full Erdős conjecture is proven, as this was only the first part of it.
As Bloom told Quanta Magazine "It’s not like we’ve solved it completely,” Bloom said. “We’ve only just shed a little more light on the subject."