“The Most Important Unsolved Problem In Pure Math”: Where Is Humanity At With Prime Numbers?

As physics has its atoms, so math has the primes: numbers that are indivisible into any smaller constituent pieces. When we first learn about them, some time in grade school, they’re usually presented as an interesting little aside – strange blips in the number line that might occasionally make long division a bit more difficult, but ultimately, nothing that important.

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The truth couldn’t be more different. “Prime numbers are the most fundamental, ‘basic’ numbers from the point of view of multiplication,” explains number theorist Adam Harper, a professor in the University of Warwick’s Mathematics Institute. “For example, every positive whole number is expressible in exactly one way as a product of some prime numbers.”

“Prime numbers […] have been studied for thousands of years,” he says, “so it is very provocative that there are still so many questions about them which we don't understand very well.”

Searching for primes

With their easy definition and at least superficially simple properties, the study of primes goes all the way back to the very earliest days of math itself. Eratosthenes, some 22 centuries ago, was already on the search for a way to locate primes in the number line; his “sieve of Eratosthenes” is still one of the most efficient ways to find smaller primes, and the underlying question – simply, “which numbers are prime?” – is yet to be solved.

“Sieves are a vital tool in modern analytic number theory. They are used all the time,” says James Maynard, Professor of Number Theory at the Mathematical Institute of the University of Oxford and, in 2022, winner of the Fields Medal for his work towards understanding the structure of prime numbers. 

“This is actually the key limitation of the sieve of Eratosthenes – it works too well!” he explains. “Because the sieve of Eratosthenes tells you exactly which numbers are primes, understanding what is going on theoretically is roughly as difficult as understanding the primes themselves.” 

And therein lies the other part of primes’ mystique – because, as important as they are to the whole fabric of mathematics, that’s really only half the reason for their popularity. 

Primes may be basic – but as we’ve already seen, the questions surrounding them are mind-bendingly complex. “I don't think we'll ever be able to understand the primes ‘perfectly’,” Maynard tells IFLScience. “I think they are inherently complicated numbers and without perfect structure.” 

The prime problem

“Many famous, fundamental (and easy sounding) questions about primes, dating back hundreds or thousands of years, remain unsolved in their original forms,” Harper points out. There’s Legendre’s conjecture, which asks whether there always exists a prime number between two squares; Goldbach’s conjecture, which suggests that every natural number greater than 2 is the sum of two primes; the twin prime problem – are there infinitely many pairs of primes separated by two, such as 11 and 13? – or its slightly weaker sibling, the Chowla conjecture, which asks whether an integer having an odd or even number of prime factors implies that its neighbors have the same.

When it comes to the biggest challenge yet to solve, however, it seems there’s not much debate: “The Riemann Hypothesis has a good claim to be the most important unsolved problem in pure maths,” says Harper. 

It’s a problem almost perfectly designed to captivate: like so many questions around prime numbers, it’s easy to describe, but hard to understand; it’s esoteric enough to intrigue mathematicians, but powerful enough to bring down world economies – indeed, it occasionally makes its way into Hollywood plotlines for its blockbuster-like potential; if all else fails, solving it could win you a million dollars, which should be enough to get anybody interested.

At its core, though, it’s quite a simple question: “It is really a question about the distribution of prime numbers,” Harper explains. “If we count how many primes there are up to some point, how close must the answer be to the natural guess?”

Consider the number of primes below 10, for example: there’s 2, 3, 5, and 7, totaling four. Below 100, there are 25 prime numbers; between 0 and 1,000, there are 168, and between 0 and 10,000 – don’t worry, we won’t make you check – there are 1,229.

So, each time we increase the size of our interval by a factor of 10, the amount of it that is given over to prime numbers goes from 40 percent to 25 percent, to 16.8 percent, to 12.29 percent. In other words: primes are getting “rarer” – but how, exactly? 

We know that, as x gets larger, the number of primes below x grows roughly like x/lnx – but not exactly like it. And precisely the error margins that fit that function back to the true answers? That’s what the Riemann hypothesis is concerned with.

“It points to a ‘structural’ hidden pattern in the distribution of prime numbers,” Maynard says – one “which I don't think anyone would guess at first.”

“The Riemann Hypothesis would have amazing consequences for our understanding of primes, with many further applications in mathematics,” he tells IFLScience.

Where we’re at

So, with thousands of years of study under our belts, where exactly are we at with these prime problems? Well, with few exceptions, most still evade proofs – but “there has been progress,” Harper tells IFLScience. 

The past decade has seen movements toward a resolution of the twin prime conjecture, for example – first with Yitang Zhang’s proof that there are infinitely many pairs of primes separated by at most 70 million, then by a frenzy of work by mathematicians aiming to lower this admittedly large bound. 

“At times, the bound was going down every 30 minutes,” Terence Tao, another Fields medalist active in the prime number sphere, told Quanta magazine back in 2013. Within a couple of months, it had shrunk to 4,680; not long after, Maynard used novel sieve techniques to get the maximum difference down to just 600. Now, thanks to Tao and his “Polymath Project” collaboration, it stands at 246.

“The key benefit of sieve methods is that they are very flexible,” Maynard tells IFLScience, “so they can say something about challenging problems like the Twin Prime Conjecture or Goldbach's Conjecture, which mix addition and multiplication.” 

“This mixture means that several other techniques don't work at all in this situation,” he explains.

Combinatorial and probabilistic approaches to the primes are similarly bearing fruit. Since the 1970s, the distribution of the primes has been linked to the same kinds of random measures that describe quantum systems – though most of what’s been discovered fall short of actual theorems. Very recently, however, this has started to change: taking only relatively small intervals in the number line, mathematicians have found a way to delineate between truly random behavior, and more probabilistic patterns.

“A lot of my research proceeds by trying to understand some quite subtle probabilistic problems […] and then use the insights gained to actually prove things on the number theory side,” explains Harper, who, in 2023, conjectured what’s become known as “beyond square root” cancellation for counting primes. Only this year, his hunch was vindicated, at least to an extent: in September, Victor Wang and Max Xu put out a paper using Harper’s methods to near-prove an analog of Legendre’s conjecture for the Möbius function.

“Wang and Xu […] assum[ed] two other very strong conjectures,” Harper explains, but “it would be fantastic if we could weaken or remove these assumptions.” 

“This will certainly be challenging,” he tells IFLScience, “but seems to me quite a promising direction to explore. The problem no longer looks unattackable.”

Where we’re going

What, though, of the Big Ones? The Riemann hypothesis, for example, has been an open problem for more than 160 years at this point, and it’s tempting to think it may never be decided. But the experts are more optimistic: “I'm convinced that […] the Riemann Hypothesis must be true,” Maynard says, and “for a good reason – we just don't know what that is, and probably don't have the right mathematical machinery to think about it.” 

“But I'm sure a proof would be revolutionary,” he adds, “less because of the statement […] but more because the proof would surely give a new toolkit which would understand primes in much deeper ways than we currently do. It is this new understanding which would likely completely change the field in ways it would be very difficult to predict.”

While a proof of this landmark conjecture may be some way off still – though of course, you never know; as Harper points out, Fermat’s Last Theorem was thought all-but impossible until Andrew Wiles announced his solution seemingly out of the blue in 1993 – this is one area where “the journey is more important than the destination” is more than an empty platitude. “The real interest in getting a proof of the Riemann Hypothesis is the new ideas it would require,” Harper tells IFLScience.

“And the new understanding it would encode,” he adds. “We would not only know for sure that primes behave very nicely, but also why they are forced to do that.”