Claude failed to solve the Riemann hypothesis. On the way it moved a 167-year-old number from 41.6% to 67.2%, in a day and a half.

This is it, on Anthropic's own site. It went up yesterday.

Everything I just told you is in here. So is the part nobody explains, and that is the part I want to do now.

This is where most people saw it. Six million views in a day.

But look at the actual sentence. It increased the lower bound for the fraction of zeros of the Riemann zeta function from 41.6 percent to 67.2 percent.

That is the real claim. In a few minutes you are going to understand every word of it.

So where did this come from? A guy who works at Anthropic went out for a run. Halfway through he pulled out his phone and told Claude to have a proper go at it.

He is not a mathematician. He dropped out of school at sixteen. He says he did none of the maths himself.

So what is the problem? It starts with something small.

Six splits into two threes. Seven does not split at all. Two, three, five, seven, eleven. Those are the prime numbers.

And every other number is just primes multiplied together. Twelve is two times two times three. That is it.

So primes are the bricks. Every number is a building made out of them.

So where do the primes sit? You would think there is a pattern. Nobody has ever found one.

Two, three, five, seven. Then a gap. Eleven, thirteen. Then a bigger gap. Sometimes two of them sit right next to each other. Sometimes you count a hundred numbers and find none.

People have looked for that rule for two thousand years. Every number is built out of these, and nobody knows where they land.

Then a German mathematician called Bernhard Riemann found something strange.

Count the primes as you walk up the numbers. That count looks random. Riemann worked out it is really a pile of waves, added on top of each other.

One wave gives you a rough guess. Add another and it gets sharper. Add them all and it stops being a guess. It tells you exactly which numbers are prime. Every one, forever.

So where do the waves come from? This is the one bit you need.

Riemann had a machine. You give it a spot on a grid, like a square on a chessboard, and it gives you back a number. At almost every spot you get an ordinary number.

But at a few special spots you get exactly zero. Every one of those spots makes one wave.

So find the zero spots, and you know where every prime is.

Here is the strange part. Riemann started finding those spots, and they were not spread all over the grid.

They sat on one straight line going up. Not near it. On it.

So he wrote down a guess. Maybe all of the wave-making spots, all the way up, forever, sit on that line.

That guess is the Riemann hypothesis. He wrote it in 1859 and moved on. Nobody has proved it since.

Why not just check them? People did. Computers have checked the first ten trillion spots, one by one. Every single one is on the line.

And it still proves nothing. There are infinitely many of them, and infinity does not care about your first ten trillion.

If one spot further up is off the line, the whole thing falls apart.

So why does anyone care about a line? Because mathematicians got tired of waiting.

There are thousands of results that start like this: if Riemann was right, then this is true. Whole careers. Whole parts of maths. All sitting on a guess nobody has proved.

It is a building where nobody ever checked the foundation, and everyone kept adding floors anyway.

That is why there is a million dollars on it.

So if you can never check them all, how does anyone get anywhere?

They stopped trying to prove all of them are on the line. They started proving that some percentage of them is.

Think of a crowd so big you can never count it. You cannot check every person. But you can still prove at least half of them are wearing blue.

That is the game. How big a percentage can you prove?

There is a scoreboard for this, and almost nobody outside maths has ever seen it. I had not.

In the nineteen forties someone proved a small slice. By the seventies it was a third. By the late eighties, forty percent. Each of those took years and a brand new idea.

Until last week the record was forty one point six percent. Decades of work. Still not even half.

So what did Claude do? It got that number to sixty seven point two percent. Two out of every three. For the first time ever, most of them.

Two groups of mathematicians had each found a new trick for pushing the number up. One of those tricks is from the year two thousand. Nobody ever tried using both at the same time.

Claude tried both at the same time. That connection had been sitting there for twenty six years.

So how did it go? Badly, at first.

The first time round it came up with six hundred and fifty ideas and tried them all. Every single one was wrong.

That is where a normal person stops. The guy told it to try again.

The second time it did something different. It made about sixty copies of itself and gave every copy a job.

Two of them found the ideas that mattered. Thirty came back with nothing at all. The rest fed those two, checked them for mistakes, and wrote it up.

Together they ran two thousand four hundred commands and wrote hundreds of little programs to check each other. It took a day and a half.

And the guy watching all this did no maths at all. He said so himself.

Almost everything he typed was some version of: keep going. Or: believe in yourself.

Anthropic wrote that this seems to have helped, because Claude started out not believing it could get anywhere. They think it picked that up from us: that these problems are too hard.

It had to be talked into trying. I cannot stop thinking about that.

Here is the part I like most.

It sent copies of itself to attack its own answer and look for holes. It downloaded fifty four research papers to check nobody had already found this. Then it proved the whole thing again from scratch, to see if it landed in the same place.

Then it offered to write the paper, and said a human should check it.

And someone did. Two mathematicians at Anthropic went through it. Two more from outside read it, including Brian Conrey, the man who set that forty percent record in the eighties.

Then the main part was written out again in a language a computer can check line by line, and the computer agreed.

It has not been in a journal. But four named mathematicians read it, and a machine checked it.

Now the bit your feed will skip.

It is not close either. Proving most of the spots are on the line is not the same as proving all of them are, and Anthropic say straight out that they do not think this road leads there.

It added one brick to a wall people spent eighty years building.

And that version of Claude is not out. You cannot go and do this today.

So what is left when you take the hype away?

An AI made a real piece of knowledge. Not a summary of something a person already knew. Something new that nobody had written down, and experts checked it, and it held up.

Anthropic call it the latest example of how fast this is moving. Latest, not first.

And it came out of a failure. From a guy who is not a mathematician. On a run. By telling a machine to try one more time.

Anthropic finished their post with one line.

Claude did not think it could do anything here. They reckon it learned from us how hard these problems are, and that AI has limits.

And they wrote: perhaps Claude, like many of us, underestimates the rate of AI progress.