Transcription
For more than 160 years, mathematicians have been trying to solve a deceptively simple puzzle about numbers. It looks innocent. It fits on a page. And yet, no one has been able to prove whether or not it's true. There's even a juicy $1 million reward for the person who finally cracks it. This is the Reeman hypothesis. Often called the biggest unsolved mystery in mathematics.
At first glance, it's about prime numbers, the building blocks of all whole numbers. Every number you've ever used can be broken down into primes. They're the raw ingredients of arithmetic. But here's the strange part. Primes don't follow an obvious pattern. They seem to be scattered almost randomly and yet zoom out far enough and a hidden order seems to emerge.
In 1859, a German mathematician called Bernard Reman proposed that this hidden order is governed by a single rule, one that connects all prime numbers to a mysterious formula. And for more than a century and a half, mathematicians have tested his claim again and again. Using supercomputers, they've checked more than 10 trillion cases, and every single one behaves exactly as Reman predicted. But checking trillions of examples is not the same as proving something is true forever.
That missing proof matters because if the hypothesis is correct, it would sharpen our understanding of everything from encryption to randomness. And it may even reveal an unexpected link between pure mathematics and quantum physics. In this video, we'll explore why this problem is so hard, why it's so important, and why recent breakthroughs suggest we may finally be learning how to approach it. And we'll ask a deeper question. Why does a pattern hidden inside prime numbers seem to echo the mathematics of the physical universe?
Chapter one, whether pattern breaks. In 1900 at the International Congress of Mathematicians in Paris, David Hilbert presented a list of problems he believed would shape the future of mathematics. Number eight on that list concerns something surprisingly simple, prime numbers. Now, Hilbert's name may sound familiar. We recently explored one of his other famous ideas, the thought experiment known as Hilbert's infinite hotel. But this problem was no thought experiment. It was about something far more fundamental.
Now here's a primer on prime numbers. Prime numbers are those which can be evenly divided by one and themselves. So that includes two, three, five, seven and so on.
>> So the reason that primes are important and that mathematicians really love prime numbers is because every whole number is built out of prime numbers being multiplied together. And so prime numbers are somehow the atoms of arithmetic.
12 for example is equal to 2 * 2 * 3 all prime numbers. Even enormous numbers can be broken down into a unique combination of primes. This is known as the fundamental theorem of arithmetic. Every whole number has one and only one prime factorization.
>> So this means that uh in the same way that a chemist might try and understand uh chemical compounds by understanding the constituent atoms, pure mathematicians try and understand whole numbers by understanding the constituent prime numbers. The fascinating thing about prime numbers is that even though they're these fundamental basic building blocks of whole numbers, they remain very mysterious to pure mathematicians, they've been studied since the ancient Greeks, but uh lots of the uh simplest questions that you might ask about prime numbers have been unsolved. And that's the reason that I'm really drawn to prime numbers.
Infinite number of prime numbers exist. But as numbers grow larger, primes also occur less frequently. In the late 18th century, Carl Frederick Gaus, then only a teenager, notice a remarkable trend. Although primes are scattered half-hazardly, the overall frequency follows a simple rule. As numbers increase, the density of primes decreases roughly inversely proportional to the logarithm of the number itself. This insight would be eventually formalized as the prime number theorem, a landmark result in mathematics.
>> And Gaus's guess is amazingly good that uh it's sort of as good as you could possibly hope for.
And yet the rule is inexact. When you compare Gaus's smooth prediction to the actual counterp primes, the two don't exactly line up perfectly. Those tiny discrepancies, the subtle wiggles are where our mystery begins. Half a century later, Bernard Reeman approached the problem from a radically different direction. Instead of counting primes directly, he turned to a strange function, one that appeared to encode the tiny gaps between Gaus's prediction and reality. So what exactly did Reman see? And why has no one been able to prove he was right?
Chapter 2, the line that controls the primes. It would be very helpful if we had a single formula that could tell us exactly where the primes are. So, I want you to imagine a graph that stays flat, then flat, then flat, and then suddenly jumps at two, flat again, then jumps at three, then at five, seven, and 11. Kind of like a staircase that only steps upward when we hit a prime number. So, mathematicians actually have a name for such a staircase. They call it pi of x. And no, despite the familiar name, it has nothing to do with the irrational number that goes like 3.14159. Pi of x, otherwise known as a prime counting function, tells us exactly how many primes are less than or equal to x. If we had an exact formula for pi of x, we wouldn't just know that primes go on forever. We would know precisely how they are distributed. But the prime counting function is really jagged. It's like a staircase that jumps up quite abruptly. So there's no obvious smooth formula that can capture its behavior.
So mathematicians have tried something else. Now in physics when a signal looks really chaotic like a clap echoing in a room, we can break it into simple waves. Smooth sign curves when of different frequencies that when added together recreate the original sound. But what if we can do something similar for the formula that describes the prime counting function? What if primes work the same way? What if that jagged staircase could also be built from smooth oscillations, each contributing one tiny ripple to the overall pattern?
In 1859, Bernard Reman discovered something astonishing. He found that a strange function now called the Reman Zeta function encodes exactly this kind of wavelike structure. When Reman studied where this function equals zero, he discovered that those zeros actually govern the distribution of prime numbers. everything about all the zeros of the zeta function. You'd know everything about the distribution of prime numbers.
>> Okay, let's take a beat to understand what the heck is going on. What does it mean to have zeros? When we solve an equation, we're usually looking for the values that make it equal to zero. For example, we can take the simple equation x^2 - 4 = 0. The solutions to that would be x = 2 and x = -2. In this case, those two values are the only zeros of the function. But Reayman was working with a far more complicated function. Instead of two zeros, his function had infinitely many. And all of them control what our prime counting function actually ends up looking like. Now, put a pin in that. Explain more about that later.
Now, Reman also wasn't working with just any ordinary number. He extended the zeta function into a realm of complex numbers. numbers that have both a real part and an imaginary part. Now, imaginary numbers may sound really strange, but they're simply numbers that are defined as the square root of minus1 written as i. A complex number can look like this, where a is the real part and b is the imaginary part. Instead of sitting on a simple number line, complex numbers live on a plane, where real numbers lie along one axis and imaginary numbers along the other.
When Reeman examined this complex version of the zeta function, he uncovered something remarkable. He saw that we could recover the prime counting function, the reman zeta function. They might look completely dissimilar, but you can actually convert one into the other. See this critical spot where the reman zeta function suddenly gets a bit strange? Mathematicians call that a pole. And that's what actually tells us that our prime counting function follows that smooth logarithmic shape that Gaus first identified. It tells us that as numbers get bigger, primes get more and more infrequent. And those zeros I promised to explain more about, well, you can think of them as these wavelike corrections that we can add to the smooth plateauing shape to make it resemble more of a jagged staircase.
Some of these zeros fall at predictable locations. the so-called trivial zeros. Now, it's a bit complicated to explain why they're evenly spaced like this, but all you need to know is that because they're negative, they'll add negligible corrections to our prime counting function, the the jagged staircase. But the others, the non-trivial zeros, they lie within a narrow vertical band between one, where the pole is, and zero. This region is known as the critical strip. So the reman hypothesis is the claim that all the non-trivial zeros of this reman zeta function have real part equal to 1/2 and so they all lie on this magic line in the complex plane with real part equal to 1/2.
>> This is often known as the critical line. Now Reman's bold claim was this all non-trivial zeros lie exactly on that line and that means that our prime counting function can predict where prime numbers appear.
Now the reman zeta function has very special mirrorlike properties. This critical line is the only position where the zero is its own mirror image keeping the system in balance. Remember these swirls from earlier. The loops that you see are actually what happens when you move up the critical line. Each time the curve crosses the center the function equals zero. And you can see that the loops are perfect reflections of each other. But if even one zero drifted away from the critical line, it would ruin the symmetry and cause unpredictable corrections to be added to our beautiful smooth logarithmic function that Gaus first identified as a teen. The further it was from that line, the stronger its chaotic influence would be. In other words, saying that all zeros lie on that one line is the same as saying that the appearance of primes is not random. Their apparent chaos is constrained by an underlying harmony.
Physicist Michael Barry offers an analogy. If the distribution of primes is like music, then the zeros of the zeta function are the harmonies that shape it. Just as complex sound can be built from pure tones, the irregular wiggles in the primes can be understood as arising from the individual notes of these zeros. An idea evocatively known as the music of the primes.
It turns out that prime numbers do occur in various aspects of nature and in various aspects of musical composition. Um that uh when you have numbers like two and four that divide each other things neatly fit together. Whereas when you have um prime numbers like five and seven uh often they remain out of sync for a very long period of time. And so it's certainly been suggested that certain cichers uh their uh hibernation cycle um lasts a prime number of years as a way of staying out of sync with predators. And certain musicians um when they want to have a uh dissonant musical effect, they deliberately use uh prime number uh different prime number beats so that things stay um out of sync with one another for a maximally long period of time. And so even these very abstract objects in pure mathematics we start seeing crop up in uh nature and in music and uh in art because they have uh fundamental links to basic ideas like things being in sync or out of sync with one another.
Okay, to recap, the pole creates the smooth curve while the zeros create the ripples and together they shape the entire landscape of the primes. The reman hypothesis is to claim that those ripples are as tightly constrained as possible. More than a century and a half later, this remains our clearest window into how order and irregularity coexist in the primes. Hold on. Before we go further, I just need to take a pause to acknowledge that this is absolutely insane. If the reman hypothesis is true, there exists a function that helps us know exactly when prime numbers appear. It shows that prime numbers, these things we first defined in ancient Greece, aren't just a thing we arbitrarily identified, but they're so woven into the fabric of mathematics that that distribution is predictable. And that is amazing. Now, the only thing we're still missing, a proof.
Chapter three, a proof at last. In 2000, the Clay Institute identified seven of the most difficult unsolved problems in mathematics and offered a $1 million prize for a correct solution to each. The Reman hypothesis sits prominently among them. In 2018 at the H Highidleberg Laurate Forum, Sir Michael Ayia made his bid to secure the prize and his legend. Even at 90, Ayia remained a towering figure. Over the course of his career, he helped shaped entire fields from geometry to theoretical physics and mentored generations of mathematicians. A Fields Medal and Enable Prize placed him among the discipline's most decorated thinkers. But rather than resting on his laurels, he boldly attempted to tame the primes and announced that proof had been found. Immediate response, high tension. Aia was not the first to make this audacious claim, and many brilliant minds had met their match in attempting to conquer the reman hypothesis. Multiple promising proofs had been offered before, and each eventually revealed subtle flaws.
Aia framed his attempt as a proof of contradiction, a classic and widely used method in mathematics. He began by assuming the hypothesis was false. He then used mathematical reasoning to demonstrate why that was logically impossible, which in turn theoretically would prove the hypothesis to be true. To do so, he combined insights from two major 20th century mathematicians, John Vonoyman and Frederick Heaiserbrook, claiming that between the two, all the hard work was done 70 years ago. The outline of the proof fit on just a few slides. Aia pointed to what he called a miraculous link between the Reman hypothesis and the fine structure constant, a physical parameter that physicists use to describe the interaction between light and matter. Piers expressed cautious skepticism. The mathematical community then dove into a rigorous review and within weeks the experts had their verdict. The proof didn't hold. The hypothesis had survived yet another challenge. But AIA is just one of the many great mathematicians who have thrown themselves into this problem. And there is a reason why it remains yet unsolved.
Chapter 4. The breakthrough. The reman hypothesis is one of the central unsolved questions in mathematics. Not only because it is so vexing, but because its truth would place tight limits on prime fluctuations and many related results in number theory. Using supercomputers, we have now checked over 10 trillion zeros and every single one lies on the line just as Reman predicted in a handwritten journal. But the Reman hypothesis is a statement about infinitely many zeros. And no matter how large the computation, checking trillions of cases can never rule out the possibility that the next one, one far beyond our reach, breaks the patent. Without a proof, many of the strongest statements we can make about primes remain conditional.
This explains why a 2024 breakthrough by James Maynard and Larry Guth generated so much excitement. Because a direct proof remains so difficult, Maynard and Guth set a more modest goal.
So, um, as a workaround for the Reman hypothesis, the Reman hypothesis is like this big mountain that we don't know how to scale. And if we could scale, we could get to all this fertile land on the other side. Um, and my work with Larry is more a workaround that we can uh go a little bit around the side of the mountain. We don't have to go over the top to get to some of the nice fertile land.
Rather than trying to prove that every zero lies on the critical line, may not go asked if any zeros do drift off the line, just how many could there be? Remember how I said that complex sounds can often be broken up into pure frequencies? Well, similarly, Maynard and Goose created a related mathematical object made of thousands of oscillating pieces. If a zero existed off the critical line, they show how it caused this object to spike. And that meant that the waves that made up this object would have to combine in a way that was really unlikely and rare. This proved that offline zeros, if they do exist, must be vanishingly rare.
>> And so our work was saying that even though we can't prove the human hypothesis, uh we can show you that there aren't many counterexamples to the hypothesis with a better quantification of um how few the number of counter examples is. This was very satisfying to me because this area of number theory had been stuck for a very long time. Um there were lots of questions on the distribution of primes which had been stuck on the possibility that there were lots of zeros with real part equal to 3/4. So we thought that there should be no obviously we think that there should be no zeros with real part equal to 3/4 that would follow from the room hypothesis. But there was this bizarre potential conspiracy of there being lots of zeros with real part equal to three quarters that was limiting progress on lots of different questions on the uh distribution of primes. And my work with Larry managed to show that um there couldn't be that many uh possible counter examples with real part equal to three quarters which correspondingly allowed us to get improvements on lots of different questions about the distribution of primes. And that change in perspective is really important. It may reshape how this problem will someday be solved.
If the idea of mathematics pressing up against its own limits fascinates you, then head over to our podcast channel for an even deeper dive. One of our newest episodes explores a number so vast it stretches the limits of logic itself. Discovery that forces mathematicians to confront what can and can't be proven. Today's episode has you thinking about infinity, proof, and the boundaries of knowledge. This one goes even further. Powered by new scientists, decades of reporting, the world, the universe, and us is our award-winning weekly podcast hosted by Dr. Ran Hooper and Dr. Penny Sashe. It unpacks the most important developments in science each week, covering everything from consciousness and climate to dark matter and intelligent life. The podcast now has its own dedicated channel, which means every week we can respond in real time to the biggest breakthroughs in science, bringing in expert guests and unpacking what the headlines actually mean. We won't be publishing the podcast on this main channel anymore, so click the link in the description to subscribe to the new podcast channel and make sure you don't miss an episode. Now, back to the mystery that has challenged mathematicians for over a century.
Chapter 5. when numbers behaved like atoms. So while Maynard and Guth's breakthrough was not a rigorous proof of the reman hypothesis, it is a huge step forward. You know what? Sometimes maths is about the journey. The beautiful and cool approaches humans think up to tackle a seemingly impossible problem. But Maynard and Guth's work did not come out of nowhere. It is the latest chapter in a decadesl long shift in how mathematicians think about the reman hypothesis. Traditional number theoretic tools have struggled to crack the problem. But perhaps a better perspective has been hiding in physics all along.
Their first hint came back in 1972 over an afternoon cup of tea. Mathematician Hugh Montgomery described a formula for the spacing between zeta zeros. Physicist Freeman Dyson immediately recognized it. It was identical to the statistics governing energy levels in certain quantum chaotic systems. Suddenly the zeros of the zeta function began to look less like abstract numbers and more like the spectrum of a physical system. Inspired by this parallel, Elaine Cons pursued a bold idea where the zeros are literally the energy levels of an as yet undiscovered quantum system. He constructed a vast mathematical framework built from structures called adelles in an attempt to give primes a physical state space. Though cons didn't prove the hypothesis, he showed that this spectral viewpoint could be made mathematically precise. In the 1990s, John Keiting and Nina Snith went further using random matrix theory originally developed in nuclear physics. They predicted the detailed statistical properties of the zeta function with striking accuracy. Patterns that had resisted pure number theory for decades emerged naturally from a physics inspired model.
Seen in this light, Maynard and Guth's work is not an isolated breakthrough. It is the latest expression of a deeper idea that the primes may ultimately yield to the methods borrowed from physics. And taken together, these developments suggest something radical that the reman hypothesis is not just a statement about numbers. It may reflect a deeper structure shared by mathematics and quantum physics. a bridge connecting prime numbers, the metaphorical atoms of arithmetic, and the very real atoms of the physical universe.
>> And so, although this seems like a very abstract problem, very much in the realm of pure mathematics, one of the amazing things about the Rema hypothesis is that it has a cascade of consequences if it's true. And so would have lots of different applications to um mathematics but also outside of mathematics to questions in computer science as well.
That possibility is exhilarating. But what would it actually mean? Suppose the reman hypothesis was proven tomorrow. What beyond a million dollar prize changing hands would truly change? The answer would reshape large parts of mathematics and ripple beyond it.
Chapter six. a hidden unity.
>> You told me uh that this morning someone had proven the reman hypothesis. I would obviously be super excited, but I wouldn't be so excited because the reman hypothesis is true. I know it's clearly going to be true. I believe the reman hypothesis very strongly. The reason I'm excited is that any proof of the reman hypothesis would develop all of these new tools that would be able to do much more than would surely be able to do much more than just prove the reman hypothesis. They would give us a whole new set of techniques and a whole new set of tools.
A proof of the reman hypothesis would be nothing short of transformative. It would give mathematicians the strongest possible control over the fluctuations in the primes, turning many long-standing conditional results into firm conclusions. At present, mathematicians can rule out large regions where zeros cannot exist, but not all of them. Proof would eliminate every remaining possibility except for the critical line itself. Instead of relying on delicate casebyase arguments, many results would follow from a single unifying principle. As mathematician Peter Sarnac put it, currently we have a screwdriver for these problems, but the hypothesis would be a bulldozer.
>> Thing it would open up all kinds of new avenues in mathematics, uh, but also much much wider than mathematics. And so maybe we could start having real mathematical proofs of lots of these open questions in computer science that boil down to questions about primes. For example, modern encryption relies on the difficulty of factoring very very large numbers into primes. While that difficulty does not hinge on the reman hypothesis, a proof would still sharpen our theoretical understanding of the distribution of primes, refining the mathematical bedrock beneath critical modern digital security. More broadly, because the SATA function mirrors patterns that also appear in quantum physics, a proof of clarify, a mathematical framework shared by number theory and certain physical systems. At its heart, the hypothesis is about beauty. It says that in apparent chaos and randomness, there's order. One subtle enough to elude us for now, but optimistically not forever.
Chapter 7. Beyond the Millennium Prize. Someday, maybe sooner than we expect, someone will finally resolve perhaps the most famous and vexing of the seven millennium prize problems. But the true significance of the Reman hypothesis transcends the race to claim its million-doll reward. For more than a century and a half, this search has reshaped our understanding of order and randomness. The primes appear scattered, yet they obey structure. The hypothesis suggests that this structure is fundamental and not accidental. It takes a rare problem to make mathematicians wax poetic. Reflecting on the reman hypothesis, Larry Guth involves Rainor Maria Rilka. Live the questions now. Perhaps then someday far in the future, you will gradually live your way into the answer. If you want to explore a mystery that pushes the balance between randomness and order to its absolute limit, watch our podcast episode, Scientists Found a Number That Defies Mathematics, where we dive into the astonishing busy beaver numbers. Figures so vast they challenge what mathematics itself can prove.