3 ms·

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 ;).

by lmkg 13y ago

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.

thedufer 13y ago
> 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.
- jychang 13y ago
  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.
  - tempestn 13y ago
    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.
    - thedufer 13y ago
      >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.
      - tempestn 13y ago
        True. What I meant was, I'm just not sure which convention is commonly used when speaking of prime k-tuples.

by lmkg 13y ago

thedufer 13y ago

> 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.

jychang 13y ago
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.
- tempestn 13y ago
  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.
  - thedufer 13y ago
    >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.
    - tempestn 13y ago
      True. What I meant was, I'm just not sure which convention is commonly used when speaking of prime k-tuples.