In mathematics, an Eisenstein prime is an Eisenstein integer
that is irreducible (or equivalently prime) in the ringtheoretic sense: its only Eisenstein divisors are the units {±1, ±ω, ±ω^{2}}, a + bω itself and its associates.
The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.
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✪ Prime Spirals  Numberphile

✪ Number Theory: Lecture 12

✪ Eisenstein's Criterion
Transcription
JAMES GRIME: One of the reasons we're fascinated by primes is that they are quite weird in the way they behave. On one hand, they kind of feel random. They are turning up all over the place. Sometimes you have these long gaps between primes. And then suddenly like buses, you get a couple of primes turn up at once. On the other hand, there are things that we can predict about primes and when they're going to turn up, which is slightly unexpected that you can do that. They're not completely random. One of the first things I want to show you, then, is a nice easy thing. So everyone can do this at home. We're going to write the numbers in a square spiral. Start with 1 in the middle. Then you write 2. But you go around it 4, 5, 6, 7, 8 do you see the pattern, then? It's a square spiral. 12, 13, 14, 15 it's called an Ulam spiral Stanislaw Ulam, he was a Polish mathematician. And he left Poland just before World War II, and he went to America. And he worked on the Manhattan Project. After World War II, he went into academia. The story of this spiral is, he sat in a very boring lecture in academia. It was in 1963. And so he's obviously a fan of Vi Hart or someone. He sat there doodling during this boring lecture. And he's writing out the numbers. Let's see, 30, 31, 32. The next thing he did was start to circle the prime numbers. So let's do that. 2 is a 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 lining up on diagonal lines. And if you do this larger, if you do more and more numbers, and you write them out in a spiral, that tends to be the case. I've got one here. This is a big Ulam spiral. I think this is something huge. I think this is like 200 by 200. And so there's 40,000 numbers or something here. Can you see, though, can you see the stripes? There's definitely some stripes here, these diagonal lines. So prime numbers seem to be lying on diagonal lines. Or to put it another way, some diagonal lines have lots of primes, and some diagonal lines don't have lots of primes. So you can see the stripes start to form. BRADY HARAN: Are they continuous stripes? They look a bit broken up to me. JAMES GRIME: Yeah, they are not continuous stripes. But they have more than average number of primes. So these stripes might be a good place to look for more primes, bigger primes, new primes. One thing people might say is, oh, we're just seeing patterns in randomness. Those aren't really stripes at all. It's just the human brain. See, if you compare it with randomness this, the same size, these are random numbers. And you can see, it's pretty much white noise. I can't really see any pattern in this. You can see that that is random. And you can see that that is something more than just being random. BRADY HARAN: These ginormous primes that get found, are they found on diagonals? Like this largest prime known, was that on diagonal? JAMES GRIME: The largest prime known was a Mersenne prime, which is of the type 2 to the power n minus 1. It's one less than a power of 2, which is a way to look for large primes. It's computationally kind of easier to do. Perhaps it's not the most fruitful way because they are quite rare, Mersenne primes. This might be another way to do it because this stripe here, this diagonal, has an equation. This equation is for this one here, this half line, which means it starts at three and goes off to infinity. The equation for that is 4x squared minus 2x plus 1. Let me just try it. Let's do the first one here. So if x is equal to 1, yeah, that's 3. If we tried the next one here, x equals 2, it's 13. And this one here, that's 31. And well, best do one more, just to show you what comes next, 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 isn't a prime number, but 57 would be the next number on that line. BRADY HARAN: So that's one of the breaks in our dotted line? JAMES GRIME: Yeah, so all these lines, the, in fact, horizontal lines, vertical lines, and diagonal lines, they are all like this. All the quadratic equations are like that. So what we're saying is, some quadratic equations have more primes on them than others. And that's the conjecture, actually. That hasn't been proved. But that is the conjecture. It seems to be the case. So there are lines here that have seven times as many primes as other lines. And the best we've found is a diagonal line that has 12 times as many primes as the average. BRADY HARAN: Cool, has that line got a name? JAMES GRIME: I can write it out 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, it's a quadratic equation. It starts off quite simply again. But the number you add on is not plus one. It's plus something huge. This square spiral is called Ulam's spiral. But there's one that I like even more. It's called a Sack's spiral. And it works like this. You write the square number in a line. The square numbers are 1, 4, yeah, that is 2 squared, 3 squared is 9, 16, 25, and so on. So you write the square numbers in a line. Then I connect them with what is called an Archimedean spiral like that. And then I would the other numbers on that spiral and evenly space it. So it goes 1, 2, 3, 4, 5, 6, 7, 8, 9. And if you mark off the primes for that, I've got this already sorted out for you, this is the picture you get. And you can see the relations, you can see the pattern, even more strikingly, I think. Look at these curves. These are the primes. BRADY HARAN: And obviously, you'll never get a prime along there because those are the squares. JAMES GRIME: Those are your squares, that big gap there is the squares. So it looks like we have formulas, equations some formulas, anyway, that have more primes than others. So if we can understand these formulas that contain these rich number of primes, 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 of. There are equations to help us find prime numbers. And now I want to show you some equations that help you find prime numbers. BRADY HARAN: So we'll have more about ways to search for prime numbers coming really soon from this interview with James Grime unless you're watching this in the future, in which case this stuff might already be on YouTube. But you get the idea. But, I have a bit of a confession to make. I've actually recorded some stuff about the spirals and prime numbers before not with James Grime, but with James Clewett. And I kind of half forgot about it and never got around to editing it. This was, like, a year and a half ago. I went back and had a look, and it was actually really interesting. So I've turned that into a video as well. Now you can wait for that turn up in your subscriptions, in the next few days, or if you can't wait, you can go and have a look at it now. I've made the links available. The video's already up, so go ahead and have a look. Thanks for watching. Plenty more videos, both of stuff I've recorded, some of it quite a while ago, it turns out, and stuff we've still got to record. Really exciting stuff coming soon on "Numberphile," so make sure you've subscribed.
Contents
Characterization
An Eisenstein integer z = a + bω is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:
 z is equal to the product of a unit and a natural prime of the form 3n − 1,
 z^{2} = a^{2} − ab + b^{2} is a natural prime (necessarily congruent to 0 or 1 mod 3).
It follows that the square of the absolute value of every Eisenstein prime is a natural prime or the square of a natural prime.
In base 12, the natural Eisenstein primes are exactly the natural primes end with 5 or 3 (i.e. the natural primes congruent to 2 mod 3), the natural Gaussian primes are exactly the natural primes end with 7 or 3 (i.e. the natural primes congruent to 3 mod 4).
Examples
The first few Eisenstein primes that equal a natural prime 3n − 1 are:
Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example:
 3 = −(1 + 2ω)^{2}
 7 = (3 + ω)(2 − ω).
Some nonreal Eisenstein primes are
 2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω.
Up to conjugacy and unit multiples, the primes listed above, together with 2 and 5, are all the Eisenstein primes of absolute value not exceeding 7.
Large primes
As of March 2017^{[update]}, the largest known (real) Eisenstein prime is the seventh largest known prime 10223 × 2^{31172165} + 1, discovered by Péter Szabolcs and PrimeGrid.^{[1]} All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and all Mersenne primes are congruent to 0 or 1 mod 3; thus no Mersenne prime is an Eisenstein prime.
See also
References
 ^ Chris Caldwell, "The Top Twenty: Largest Known Primes" from The Prime Pages. Retrieved 20170314.