Prime Spirals - Numberphile

JAMES GRIME: One of the reasons we're fascinated by

prime

numbers is that their behavior is quite strange. On the one hand, they seem random. They are appearing everywhere. Sometimes you have these long spaces between the

prime

numbers. And then suddenly, like on buses, a couple of prime numbers appear at once. On the other hand, there are things we can predict about prime numbers and when they will appear, which is a little unexpected that you can do that. They are not completely random. So one of the first things I want to show you is something nice and simple.

So everyone can do this at home. Let's write the numbers in a square spiral. Start with 1 in the middle. Then you write 2. But you circle it: 4, 5, 6, 7, 8. See the pattern, then? It is a square spiral. 12, 13, 14, 15... it is called the Ulam spiral... Stanislaw Ulam, was a Polish mathematician. And he left Poland just before World War II and went to the United States. And he worked on the Manhattan Project. After World War II, he entered academia. The story of this spiral is that he was sitting at a very boring conference in academia. It was in 1963. He's obviously a fan of Vi Hart or someone like that.

He sat there scribbling during this boring lecture. And he's writing down the numbers. Let's see, 30, 31, 32. The next thing he did was start circling the prime numbers. So let's do that. 2 is prime, and then 3, and then 5, and 7, and 11, 13, not 40, 41, 43, is prime, and so on. And he noticed, and maybe you can see, these stripes, the prime numbers seem to be lined up in diagonal lines. And if you make it bigger, if you make more and more numbers and write them in a spiral, that tends to be the case. I have one here. This is a great spiral of Ulam.

I think this is something huge. I think this is like 200 times 200. So there are 40,000 numbers or something here. Can you see, though, can you see the stripes? There are definitely some streaks here, these diagonal lines. Thus, the prime numbers appear to be located on diagonal lines. Or to put it another way, some diagonal lines have many prime numbers and other diagonal lines do not have many prime numbers. Then you can see the stripes starting to form. BRADY HARAN: Are they continuous stripes? They seem a little destroyed to me. JAMES GRIME: Yeah, they're not continuous stripes. But they have a higher than average number of prime numbers.

So these stripes could be a good place to look for more prime numbers, larger prime numbers, and new prime numbers. One thing people might say is, oh, we're just seeing random patterns. Actually, those aren't stripes at all. It's just the human brain. Look, if you compare it to randomness... this one, the same size, are random numbers. And as you can see, it's practically white noise. I can't really see any pattern in this. You can see that's random. And you can see that that's more than just being random. BRADY HARAN: These huge prime numbers that are found, are they found on diagonals?

Like this larger known cousin, was it diagonal? JAMES GRIME: The largest known prime was a Mersenne prime, which is type 2 to the power of n minus 1. It's one less than a power of 2, which is one way of looking for large primes. It's computationally somewhat easier to do. Maybe not the most fruitful way because they are quite rare, the Mersenne cousins. This could be another way to do it because this strip here, this diagonal, has an equation. This equation is for this one here, this half line, which means it starts at three and goes to infinity. The equation for this is 4x squared minus 2x plus 1.

Let me try it. Let's do the first one here. So if x is equal to 1, yeah, that's 3. If we try the next one here, coming. Then, 56 plus 57. Is that a prime number, Brady? BRADY HARAN: 57 is not a prime number. JAMES GRIME: It's not a prime number. So the next one is not a prime number, but 57 would be the next number on that line. BRADY HARAN: So that's one of the breaks on our dotted line? JAMES GRIME: Yeah, all of these lines, in fact, the horizontal lines, the vertical lines, and the diagonal lines, they're all like that.

All quadratic equations are like this. So what we are saying is that some quadratic equations have more prime numbers than others. And that's the conjecture, really. That has not been proven. But that's the guess. It seems to be the case. So here are lines that have seven times as many prime numbers as other lines. And the best we have found is a diagonal line that has 12 times more prime numbers than the average. BRADY HARAN: Great, does that line have a name? JAMES GRIME: I can write it for you. I think I had it somewhere. BRADY HARAN: Yeah, I'd love to know what that line is.

The golden line. JAMES GRIME: This golden line that Brady has now decided to call it is a quadratic equation. Everything starts again quite simply. But the number you add is not plus one. It is also something huge. This square spiral is called the Ulam spiral. But there is one that I like even more. It is called Sack's spiral. And it works like this. You write the square number on a line. The square numbers are 1, 4, yes, that is, 2 squared, 3 squared is 9, 16, 25, etc. Then you write the square numbers on a line. I then connect them with what is called an Archimedean spiral.

And then I would place the other numbers in that spiral and space them evenly. So it says 1, 2, 3, 4, 5, 6, 7, 8, 9. And if you mark the prime numbers for that, I already have it figured out, this is the image you get. And you can see the relationships, you can see the pattern, I think even more strikingly. Look at these curves. These are the cousins. BRADY HARAN: And obviously, you're never going to get a prime number because those are the squares. JAMES GRIME: Those are your squares, that big space there is the squares. So it seems that we have formulas, equations... some formulas, anyway, that have more prime numbers than others.

So if we can understand these formulas that contain this rich amount of prime numbers, then it would help us solve important conjectures in mathematics, such as the Goldbach conjecture and the twin prime conjecture. So prime numbers are not as random as you might think. There are equations that help us find prime numbers. And now I want to show you some equations that will help you find prime numbers. BRADY HARAN: We'll have more information on ways to look up prime numbers very soon from this interview with James Grime, unless you're watching this in the future, in which case this material may already be on YouTube.

But you get the idea. But I have a little confession to make. In fact, I've recorded some stuff about

spirals

and prime numbers before, not with James Grime, but with James Clewett. And I almost forgot about it and never got around to editing it. This was about a year and a half ago. I went back and took a look, and it was really interesting. So I turned it into a video too. Now you can wait for it to appear in your subscriptions in the next few days, or if you can't wait, you can go check it out now.

I have made the links available. The video is now live, so go ahead and check it out. Thanks for watching. Lots more videos, things I recorded, some of them quite a while ago, and things we have yet to record. There's some really cool stuff coming up on "Numberphile," so be sure to subscribe.