increases for larger and larger numbers. - Hardy and Littlewood, they refined this a bit more, but essentially what they find, you know, is the exact same thing. They just have like a correction factor in front, but then this term is still the same.
We've invited Elmer, from the YouTube channel, Fern, to tell you about it. - Hardy sits at his breakfast table, when a strange letter arrives. It's sent by a completely unknown mathematician from India, Srinivasa Ramanujan.
and very little explanation. Hardy is astounded. He has never seen anything like it.Some are already known, others are completely new, unknown to the world.
How did he come up with this? Hardy shows a letter to Littlewood.The two stay up until midnight going through the manuscript, debating whether the mysterious author is a genius or a fraud.
Around 1917, they invented what's known as the circle method, and used it to tackle different problems in number theory. Hardy and Littlewood later developed this idea further, and for the next a hundred years, this would be the main method to tackle the weak Goldbach conjecture.- But let's remind ourselves of the problem.
And for a while everything seems kind of chaotic, circling around the origin, but then, when alpha hits 1 over 6, the tail unwinds, Hardy and Littlewood realized they could use this.
That's the Poincare recurrence. Hardy is great.
It's churning out both good bombs with triggers and bad bombs without triggers. Hardy writes them as "good" and "bad," and quips about the quotation marks.Obviously, you have a different moral perspective on it, he says.
And both are really great this time of year, very seasonal in terms of their ingredients. Hardy , but super healthy and fresh.Well, let's do it.
One of them was Hardy , the guy who discovered ramanujen-- whom you may have seen a movie about. Hardy was famous for saying that number theory had no applications at all.And he was really glad of that, because it meant his hands would never get dirty with warfare or anything nasty like that.
In other words, we integrate this expression from the first prime two up to that number. Now Hardy and Littlewood refine this argument further to account for the fact that primes aren't really independent. But all that added is just this correction factor upfront.
Only a really phenomenal, off-the-chart, genius could have come up with them. - So Hardy writes back. The rigorous and strict mathematician he is, he asks for proof.
- "An equation for me has no meaning unless it expresses a thought of God." Ramanujan never really learned how to provide proof to his theorems. So Hardy urges him to come to Cambridge to explain them in more detail. But there's a problem.
So, using this approach in a simple way won't work. So Hardy and Littlewood came up with a smarter way. They fundamentally reframed the problem by taking the sum inside the integral.
So I spend a lot of my time thinking about our next-door neighbor planet, Mars, and whether Mars ever had the appropriate environment for hardy microbial or bacteria-like lifeforms to get started. And if they're there, if there are Martian bugs, what be the best way of trying to find them?
That's the Poincare recurrence. So Hardy was a number theorist who famously wrote a defense of mathematics for its own sake in his essay, "A Mathematician's Apology."
foliage, defying the approaching winter in this part of Ontario, Canada. Lucien Hardy puts his laptop on the table, pointing out that he does most of his work in cafes, and figures that all he needs in his office is a cafe-like small table to set down his laptop.
Please join me in welcoming, to the stage, Mr. Quentin Hardy . Well, let's get right to the real wattage here.
Christian? Audigier? No. A different one. Ed Hardy ? Ed Hardy . Christian is the-- yeah, sorry.
these are essentially derivatives of Hardy Portuguese dishes you know kind of made for to beef up Sailors right so you have like really lots of meat laid in dishes
We were walking on fours, all four. Tom Hardy is somebody who I could either applaud or kill.
And one reason why 1729 is special is that it's called a Harshad number. And Hardy was blown away.
And one reason why 1729 is special is that it's called a Harshad number. And Hardy went to visit him.
And one reason why 1729 is special is that it's called a Harshad number. And Hardy took a train from Cambridge to London, took a taxi from the station to the nursing home, went into the hospital, sat next to Ramanujan.
And one reason why 1729 is special is that it's called a Harshad number. And Hardy said it wasn't very interesting, it was just 1729.
Currently the easiest way to do that is to ask a white person for their thoughts on people who wear Ed Hardy . Ed Hardy is a clothing company that makes a wide range of expensive T-shirts, hoodies, and jeans. These clothes are notable for their use of elements from classic tattoo designs such as skulls, hearts, and dragons. On the surface, the use of the words classic, tattoo, and T-shirt would suggest
The two stay up until midnight going through the manuscript, debating whether the mysterious author is a genius or a fraud. - But it sounded like Hardy felt these couldn't be crank results because you couldn't imagine them if you were just a crank. They're too fantastical to be the the work of that kind of second rate mind.
It's such a beautiful sentiment. Trying to remember how Hardy put it. It's something like he thought,
in the same direction, to the right. But now Hardy and Littlewood multiply the angle by alpha, where alpha is just some number between 0 and 1. You can think of it as a slider.
They grow in the cracks of the sidewalks that we walk on. They are so hardy that they can grow anywhere and prosper anywhere. So we began to think of these kids as dandelion children.
And then I'm just whisking in-- I could use a whisk, but there's not that much liquid in here-- a few tablespoons of olive oil. And it's hardy , and it's tasty, and it's really yummy.
I'm Jeff Ayers. I run our brand new broadcasting division, and I write and edit and produce our weekly podcast, "Fueled by Death Cast." My name is Alyssa Hardy , and I'm the content manager for Death Wish. So the reason why we brought Death Wish here today-- and I think it's relevant to a lot of Googlers-- is there's this great sort of just Homecoming King-esque type
It's like Thomas Hardy .
Christian is the-- yeah, sorry. So Ed Hardy . He loved Ed Hardy . And he wore all Ed Hardy .
A guy named Warren Hardy , no relation to the president, hiked around the back.
So Warren Hardy didn't spend five days, didn't spend 10 days, he spent 33 days on this wall
When you get to the 20th century, there are all kinds of famous number theorists. One of them was Hardy , the guy who discovered ramanujen-- whom you may have seen a movie about. Hardy was famous for saying that number theory had no applications at all.
We were walking on fours, all four. Look at Tom Hardy .
Along with Janet Hardy , she has co-authored several books, including "The Ethical Slut," a guide to infinite sexual possibilities.
to Tom hardy and then Charlize Theron.
And Tom Hardy walked through the door, and he felt exactly-- I felt exactly the same feeling
And one reason why 1729 is special is that it's called a Harshad number. And they arrived on Hardy 's desk.
And one reason why 1729 is special is that it's called a Harshad number. And GH Hardy was a formidable mathematician.
speculations are always fo Hardy if he had been elected president
But how would you go about building this machine out of mathematics? Because that is exactly what Hardy and Littlewood did. Their technique is so ingenious, I want to take you through it step by step.
That's the Poincare recurrence. Your book ends with a reply to Hardy that a certain number, which you could tell us about, he had once worried was boring, was, in fact, actually--
That's the Poincare recurrence. So what do you think of Hardy ?
I spend my day job thinking about hardy bacteria and how they could survive on the Martian surface and, more importantly, how we could detect
There is the obligatory blackboard, taking up most of one wall of his office. It doesn't take long for Hardy to spring up and start chalking it up with diagrams and equations, something that most of the quantum physicists I meet seem inclined to do.
So Ed Hardy . He loved Ed Hardy . And he wore all Ed Hardy . And we talked about it on that drive that I remember.