In the third century BCE, a clever Greek chap by the name of Eratosthenes came up with a novel new piece of mathematics: a “sieve”, by which one could painstakingly trawl through every integer in order and discard any with more than two factors, leaving only the primes behind.
It was smart for the time – but objectively, you’ve gotta admit it’s pretty elementary. No offence to Eratosthenes, but that’s probably exactly the method a schoolchild would use to find and list the primes, too. But here’s the thing: more than two millennia later, it’s still one of the best methods we have for the task.
It’s a testament to just how peculiar and pesky the primes are. Finding any kind of sense or pattern to these numbers has, for centuries, been mathematics’ white whale: cropping up seemingly randomly in the number line, they evade prediction and categorization, forming a natural blockade against progress in myriad open problems.
At least, until recently. Last year, a trio of mathematicians found what seemed to be a kind of order in the primes – and it came from somewhere completely unexpected.
“This paper connects two fundamental areas of number theory: prime numbers and partitions,” said Ken Ono, Marvin Rosenblum Professor of Mathematics at the University of Virginia and one of the authors behind the new discovery, in a recent statement.
“Although prime numbers have been studied for centuries, many of their most basic properties remain elusive,” he said. “What we proved gives infinitely many new ways to detect prime numbers without having to check divisibility, which is one of the reasons primes are so difficult to detect.”
It’s big news – so much so that Ono was named as a runner up for the 2025 Cozzarelli Prize in the physical sciences, which recognizes teams “whose PNAS articles have made outstanding contributions to their field.” So it behooves us to ask, really: what’s the big deal?
The genetics of math
Prime numbers – that is, numbers whose only divisors are one and themselves – are one of those things that are way more important than they appear.
When you first meet them, they’re sort of a curiosity – an “oh, also this” mention when you’re learning about division and factorization. They’re the outcasts of the number line; the ones that have no real factors, and seemingly only exist to make long division more difficult.
In truth, though, the primes are like the atoms of math. They’re the fundamental building blocks of all other numbers, despite being so unpredictable – and that, in turn, makes them extremely valuable in the modern world. “One of the most widely used applications of prime numbers in computing is the RSA encryption system,” wrote Ittay Weiss, then a teaching fellow in the University of Portsmouth’s Department of Mathematics, in a 2018 article for The Conversation. “The system […] allows for the secure transmission of information – such as credit card numbers – online.”
“Large prime numbers are used prominently in other cryptosystems too,” Weiss, who was not involved in the new research, added.
The basic idea behind all these systems is the same, though: it relies on the fact that finding primes is a very difficult task. After all these years, finding some kind of new insight into them would require a really innovative perspective – something nobody had tried before.
Luckily, that’s exactly what Ono and his colleagues had.
Partitioning the problem
Let’s leave the prime numbers for a little bit, and move somewhere a little closer to combinatorics than the heart of number theory. Here, there’s a different – and visually at least, more literal – kind of “building blocks” of numbers, and it’s these that make up the second part of the breakthrough.
Integer partitions, perhaps even more so than primes, look deceptively simple. They are, basically, just a way of splitting up integers additively – for example, the integer partitions of 4 will be 3+1, 2+2, 2+1+1, and 1+1+1+1 (by convention, partitions are usually written largest-to-smallest like this).
You might be wondering what applications such an elementary object could have – and the answer is, quite a lot, especially if you’re interested in number theory or geometry or integrability in general. That’s because partitions have a natural connection to a type of equation known as Diophantine equations – equations like Pythagoras’s or Markov’s, for which there are multiple if not infinitely many sets of rational solutions.
That connection has been known for a long time. It’s only now, however, that Ono and his teammates have noticed something incredible: that “the prime numbers […] are the solutions of infinitely many special ‘Diophantine equations’ in well-studied partition functions,” they explain in their new paper.
“In other words, integer partitions detect the primes in infinitely many natural ways.”
A whole new world
The link between these two longstanding areas of math is nothing short of astonishing. “It is remarkable that such a classical combinatorial object – the partition function – can be used to detect primes in this novel way,” Kathrin Bringmann, a mathematician at the University of Cologne who was not involved with the new research, told Scientific American this week.
It’s not only the result that’s unexpected. Every part of this breakthrough seems to beggar belief: it was, Ono points out, inspired by a question from a student; it connects two areas that didn’t seem to be related; it doesn’t even rely on any new math – “as excited as I am about it, [this] represents theoretical math that could’ve been done decades ago,” Ono said. “If there was a time machine, I could go back to 1950, explain what we’ve done, and it would generate the same level of excitement […] and the experts at that time would understand what we did.”
But most incredible of all – it works. “We’re actually nailing all the prime numbers on the nose,” Ono told Scientific American. “It’s almost like our work gives you infinitely many new definitions for prime […] That’s kind of mind-blowing.”
With a new inroad into prime numbers, there’s no telling which problem might fall next. While solving longstanding puzzles like Goldbach’s conjecture is probably an overly tall order, some mathematicians see this new breakthrough as a potential signpost towards cracking other areas of math: “These kinds of results often stimulate fresh thinking across subfields,” Bringmann said. For example, “are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?”
As for our credit card transactions online – there’s no need to worry about an immediate breakdown in security. For all that prime numbers are important for cryptosecurity – and they are – it’ll take more than a newfound connection to bring down the world as we know it, Ono says.
“The good news is that the world will still be safe,” he confirmed. “But understanding primes remains a crucial area of research, especially in the era of quantum computing.”
Of course, “if someone successfully builds an efficient quantum computer, it would upend the way prime numbers are used in cryptography,” he added. “That’s something the mathematical community is preparing for.”
The paper is published in PNAS.
Source Link: "Remarkable" Pattern Discovered Behind Prime Numbers, Math's Most Unpredictable Objects
