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Comment by jtchang | Hacker News Reader

So if I understand this right...it means that no matter how far along the number line I go (and I can go a long ways out) I will always be able to eventually find a pair of primes that are separated by no more than 600?

The crazy thing to me is that primes get less and less dense as you get further and further out. It would seem to me that once you reach a certain threshold you won't be able to find two primes "close" to each other anymore because primes will be so far apart from each other. But this says I will always be able to find a pair pretty close together? (I may be looking a long time though)

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> It would seem to me that once you reach a certain threshold you won't be able to find two primes "close" to each other anymore because primes will be so far apart from each other. ]

That is not quite correct. The average gap between primes gets larger as you move further along the number line. But, you will still get the occasional consecutive odd numbers that are close to one another. In fact, there is an old conjecture (Twin Prime Conjecture[1]) that claims that there are an infinite number of pairs of consecutive primes separated by 2.

PS. I'm not a mathematician (engineer with some decent math background), so I might very well be wrong.

[1] http://en.wikipedia.org/wiki/Twin_prime

Yeah, this result is effectively getting a little bit closer to proving the Twin Prime Conjecture.

It would be interesting to see the proof get progressively closer to 2, but more and more difficult each time.

You will always be able to find a pair of primes that are "close" to each other. However, the theorem makes no claim how far such pairs are from each other ;). To your point, primes get progressively less dense, which must also mean that these pairs get sparser still. It's probably quite vast distances between them, but infinity is a long ways, so it's not entirely surprisingly you find such improbably events with regularity.

> Maynard’s approach applies not just to pairs of primes, but to triples, quadruples and larger collections of primes.

Quadruples implies that it does, in fact, make a claim how far such pairs are from each other. In fact, it sounds like it could be used to make a claim about any finite grouping of primes. It would be interesting to see what kind of pattern those follow.

There's no such thing as a "quadruple pair of primes". Assuming x is prime, one of x+2, x+4, and x+6 has to be nonprime. More specifically, one of them has to be dividable by 3... If x mod 3 is 1, then x+2 is divisible by 3; otherwise, if x mod 3 is 2, then x+4 is divisible by 3.

Given that a "pair of primes" simply refers to two primes within a fixed maximum distance of each other, not necessary separated by exactly 2, I assume a "quadruple of primes" would refer to the same thing. Not four numbers each separated by two, but four numbers all within a given distance of one another on the number line. (Whether this distance would be described as the total span encompassing all four numbers, or the maximum distance between any two of the four, I don't know.)

And of course, you could generalize four to k, and quadruple to k-tuple.

>Whether this distance would be described as the total span encompassing all four numbers, or the maximum distance between any two of the four, I don't know.

Actually, either one would imply a bound on the other (although a slightly looser one). If you know the total span, the same number is a looser bound on the maximum between any two of the four, and if you know the maximum of any two, 3 times that gives you a loose bound on the total span.

True. What I meant was, I'm just not sure which convention is commonly used when speaking of prime k-tuples.

In understanding how small prime gaps can occur so far out, it helps to think of primes as gaps in the projection of all prior prime numbers (i.e. the shadow of all prior primes).

It is then fairly easy to intuit that every so often there'd be gaps in the shadows right next to each other.

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