The Simple Math Problem That Took 244 Years to Crack

Añadido: 2026-05-08

[00:00:00]Listen to this number.

[00:00:01]1 + 1/8 + 1/27 + 1/64 The reciprocals of cubes added up forever. The sum converges. It settles slowly on a single value.

[00:00:16]1.2020569.

[00:00:20][music] We have computed this number to over a trillion digits. We can write any decimal place of it on demand. We see it in the radiation of distant stars. We see it in the magnetic moment of the electron. It is one of the most computed numbers in mathematics. And until 1978, we did not even know whether it was a fraction.

[00:00:41]The sum has a name now. It is called Apéry's constant. The function it comes from has a name, too. The Riemann zeta function evaluated at three. Zeta of three. Many of the deepest questions in number theory and in physics flow through this function.

[00:00:58]We know the answer at every even integer.

[00:01:02]At two, at four, at six. The formulas have been clean since Euler in 1734, but the odd integers refused. For three. For five. For seven. The cleanest version of the question is the sum a fraction? Sat open for a quarter of a millennium.

[00:01:20]Until a 61-year-old French mathematician walked into a conference in Marseille in June of 1978 and announced he had solved it. And nobody in the room believed him.

[00:01:30]To understand why this took two and a half centuries, you have to understand what happened in 1734.

[00:01:37]The Basel problem, find the sum of 1/n² where n runs through the integers, had been open for 90 years. Every great mathematician of the 17th century had tried. Newton missed it. Leibniz missed it. The Bernoulli brothers, the strongest analysts of their generation, circled it for decades. Then Leonhard Euler, 27 years old, [music] working in St. Petersburg, found a single trick. He factored the sine function as if it were a polynomial, an infinite product over its [music] zeros. He matched coefficients. The answer fell out of the page. π² / 6. That trick and the techniques that followed gave Euler a closed form for every even indexed zeta value. π² / 6. π to the fourth over 90.

[00:02:24]π to the sixth over 945. [music] They all landed cleanly. They all involved π to a power. Then Euler tried the odd values. [music] Zeta of three.

[00:02:35]Zeta of five. Zeta of seven. Nothing.

[00:02:39]The same trick that had worked so beautifully on the even side broke on the odd.

[00:02:44]The symmetry that made the sine product work, the fact that even powers of negatives are positive, failed for cubes, for fifth powers, >> [music] >> for any odd exponent. The whole machine collapsed. And no one for the next two and a half centuries found a replacement. By the 20th century, zeta of three was no longer just an open problem. It was a known impossibility. A wall. There is a deeper version of the question Euler couldn't answer.

[00:03:12]Even if you could not write zeta of three as a function of pi, could you at least say it was a fraction? That is a much weaker question. And it had no answer, either.

[00:03:23]Proving a number is not rational is not the same as computing it. You can know a billion digits of a number and have no idea whether the underlying value is some hidden fraction p over q. It might be.

[00:03:36]The decimal expansion alone tells you nothing. Every fraction with a large enough denominator looks indistinguishable from random noise. To prove a number is irrational, you need to bound how well it can be approximated by fractions.

[00:03:49]If a number is rational, the closest fraction with denominator q can never get nearer than 1 over q squared. Beat that bound, find rational approximations that are too good to be possible, and you have proved the number is not a fraction at all. In principle, in practice, for zeta of three, nobody could find approximations that good.

[00:04:11]By 1978, the problem had a reputation.

[00:04:14]Mathematicians who attempted it lost months. The textbooks listed it as an open problem and moved on.

[00:04:21]So, when an unknown speaker walked up to the chalkboard at the Journées Arithmétiques de Marseille that June, no one in the room expected what came next.

[00:04:29][music] He did not introduce himself dramatically. The talk was scheduled at 2:00 in the afternoon. The title on the program was understated. Sur l'irrationalité de zeta de trois, on the irrationality of zeta of three.

[00:04:44]He wrote a recurrence on the board. n³ * a sub n = 34n³ - 51n² + 27n - 5 - * a sub n - 1 >> [music] >> - n - 1³ * a sub n - 2. The starting values were simple. a sub 0 = 1. a sub 1 = 5. The next few followed. 73 1,445 33,001 819,005 The integers exploded. He claimed this single recurrence and a partner sequence built the same way were enough to prove zeta of three was irrational.

[00:05:27]The audience laughed. The recurrence looked pulled from [music] nowhere.

[00:05:31]There was no derivation. There was no clear motivation. It simply appeared on the board, fully formed, like a magician producing a rabbit.

[00:05:40]Mathematicians began calling out across the room. Someone threw paper airplanes.

[00:05:45]Non-French speakers in the audience reported hearing only, quote, "a sequence of unlikely assertions." When a heckler demanded to know where the identities had come from, the speaker shrugged. He answered in French. Elles poussent dans mon jardin. They grow in my garden.

[00:06:03]Most of the audience walked out convinced they had witnessed either a hoax or a breakdown. Henri Cohen, sitting in the room with Hendrik Lenstra and Alfred van der Poorten, was not so sure.

[00:06:14]It would take Cohen six weeks, locked in with the manuscript, to realize the man at the chalkboard had been right the entire time. Here is what the recurrence does.

[00:06:24]Take the two integer sequences it generates. Call them a and b. They both grow extremely [music] fast. a sub n is already in the hundreds of thousands by n = 5. But their ratio, b sub n over a sub n, approaches a single fixed value as n increases. [music] That fixed value is zeta of three.

[00:06:44]The speed at which the ratio approaches zeta of three is the trick. There is a classical result in number theory. If a number is rational, written as p over q, then no fraction can approximate it better than within 1 over q squared.

[00:06:59]That is the limit. Any fraction with denominator q can miss the target by no less than 1 [music] over q squared.

[00:07:06]Apéry's sequences blew through that bound. His ratios approached zeta of three faster than the bound allowed. The denominators a sub n grew, yes, but the error shrank even faster. If zeta of three had been rational, these approximations would have been mathematically impossible. They existed.

[00:07:24][music] Therefore, zeta of three was not rational. That was the proof. Three lines when [music] you wrote it cleanly.

[00:07:31]The whole argument used nothing Euler did not have in 1734.

[00:07:35]The recurrence relations were elementary. The Diophantine approximation argument was 18th century.

[00:07:42]There was no modern machinery, no deep new theory, only the courage to believe an integer sequence pulled out of nowhere. Six months later, Frits Beukers in the Netherlands rewrote the proof using shifted Legendre polynomials, and it became the version taught in graduate courses.

[00:07:58]But the original raised a darker question. How had everyone missed this for 244 years? His name was Roger Apéry.

[00:08:06]He was 61 years old. He had not published a major theorem in 20 years.

[00:08:12]His work in the 1950s had been in algebraic geometry. Solid, technical, unfashionable. Through the '60s and '70s, he had drifted away from research mathematics and into university politics. He was a leftist firebrand. He had survived the Second World War as a member [music] of the French Resistance, forging identity papers from his bedroom at the École Normale. He had been captured by the Germans in 1940 and repatriated for poor health. The French mathematical establishment did not warm to him. He was difficult, combative, prone to long silences in seminars, and to sudden outbursts in others. He published rarely. He delivered lectures in a blasé, sketched style. He had two divorces. He wrote almost nothing down.

[00:08:58]This was the man who walked into Marseille and announced he had solved a 244-year-old open problem. The proof, when it appeared in print, was four pages long in a single 1979 issue of the journal Astérisque. Don Zagier in Bonn ran the numerics on a hand calculator and watched the integers obey the recurrence past every bound he could test. When Henri Cohen presented the cleaned-up version at Helsinki two months later, he stood at the podium and confirmed the result was correct. Apéry then took the podium and explained where the ideas had come from.

[00:09:33]The Marseille audience had been right about one thing. The proof was, in the words of one witness who later wrote it up, a mixture of miracles and mysteries.

[00:09:42]But it was right.

[00:09:44]And then, almost as an afterthought, the world realized something else. Apéry had proved the one number, the sum of the reciprocal cubes, was not a fraction.

[00:09:54]His proof said nothing about zeta of five. Or zeta of seven, or zeta of nine, or any other odd value of the zeta function above three. In 2000, Tanguy Rivoal in France proved that infinitely many of the odd zeta values are irrational.

[00:10:11]But he could not point to a single one and say which. The next year, Wadim Zudilin proved that at least one of zeta of five, zeta of seven, zeta of nine, and zeta of 11 is irrational. We still do not know which. That is where the question stands today.

[00:10:27]48 years after Apéry's recurrence, we know one number on the list is irrational.

[00:10:32]We know infinitely many on the list are irrational. We cannot identify a second one.

[00:10:38]Zeta of three itself, meanwhile, is everywhere. The cosmic microwave background, the photons left over from the Big Bang, has a number density set by zeta of three, the most precisely measured prediction in all of physics.

[00:10:52]The magnetic moment of the electron, agreeing with experiment to one part in a trillion, has Apéry's constant baked into the calculation.

[00:11:01]The number is etched into the structure of the physical world. We have computed it to over a trillion digits. We can extract any binary digit on demand. We can find it inside the radiation of every black body object in the universe, and we still know almost nothing about the next one.

[00:11:18]Apéry proved one number was irrational.

[00:11:21]48 years later, every other one on the list is still waiting for its 61-year-old.