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Could someone explain this in layman's terms? What does it mean to have infinitely many pairs of primes separated by 600?

by dmak 13y ago

lutusp 13y ago
If the original claim is true, it means that the maximum numerical separation between primes is 600, meaning primes may sometimes be separated by a smaller gap, but no two primes will be separated by a larger gap, a numerical distance greater than 600. What this means is, if there are no gaps between primes a and b greater than 600 (i.e. |a-b| > 600), if all prime pairs have a smaller gap, then given the size of the set under consideration, it follows that there are an infinite number of pairings {a,b} in which their separation is 600 or less (|a-b| <= 600). Also, in mathematics, finding an example of a property isn't very difficult. The difficulty lies in proving a theorem that makes a universal statement about that property -- that's the real challenge. If that answer wasn't clear, don't hesitate to ask a clearer question.
- ipince 13y ago
  I don't think this is right. There are certainly prime gaps that exceed 600. In fact, the higher you up (in the number line), the higher these prime gaps tend to get. http://en.wikipedia.org/wiki/Prime_gap http://en.wikipedia.org/wiki/Prime_gap What's being said here is that despite of that, there will always be prime gaps smaller than 600. No matter how high you go, you can always find a pair of primes that are separated by less than 600. In other words, pick the biggest N you can ever dream of, and there will exist primes p1 and p2 such that they are both bigger than N and their difference is smaller than 600.
  - lutusp 13y ago
    Yes, you're right, and it's too late for me to either delete or edit my original post. :( In light of that, I think the discussion and work revolves around discovering the smallest gap as the numbers themselves become larger and tend toward infinity. Obviously the very smallest prime gap is that between 2 and 3, i.e. 1, and there are a great number of primes separated by 2, and the Twin Prime Conjecture asserts that there will always be occasional pairs of primes separated by 2 no matter how large the numbers themselves become. The present work is in part meant to put that conjecture on a more analytical footing. I would love to delete my original post.
mikeash 13y ago
You have a good reply, but I'd like to give another answer with a slightly different approach. Pick a number. Make it as large as you like. A billion to the power of a billion to the power of two billion? Just pick something. Now whatever you pick, I can then proceed to find two prime numbers which are at most 600 apart. Maybe they're just slightly larger than the number you picked. Maybe they're vastly larger. But the point is, they exist. We can repeat the procedure with you picking larger and larger numbers and I can always find a suitable pair that's bigger than what you pick. (In theory. In practice, we may get to a point where it takes more time to locate the actual pair than we have time before the universe burns out or collapses. But we can prove that they exist, at least.)

Could someone explain this in layman's terms? What does it mean to have infinitely many pairs of primes separated by 600?

by dmak 13y ago

Could someone explain this in layman's terms? What does it mean to have infinitely many pairs of primes separated by 600?

lutusp 13y ago

If the original claim is true, it means that the maximum numerical separation between primes is 600, meaning primes may sometimes be separated by a smaller gap, but no two primes will be separated by a larger gap, a numerical distance greater than 600. What this means is, if there are no gaps between primes a and b greater than 600 (i.e. |a-b| > 600), if all prime pairs have a smaller gap, then given the size of the set under consideration, it follows that there are an infinite number of pairings {a,b} in which their separation is 600 or less (|a-b| <= 600). Also, in mathematics, finding an example of a property isn't very difficult. The difficulty lies in proving a theorem that makes a universal statement about that property -- that's the real challenge. If that answer wasn't clear, don't hesitate to ask a clearer question.

ipince 13y ago
I don't think this is right. There are certainly prime gaps that exceed 600. In fact, the higher you up (in the number line), the higher these prime gaps tend to get. http://en.wikipedia.org/wiki/Prime_gap http://en.wikipedia.org/wiki/Prime_gap What's being said here is that despite of that, there will always be prime gaps smaller than 600. No matter how high you go, you can always find a pair of primes that are separated by less than 600. In other words, pick the biggest N you can ever dream of, and there will exist primes p1 and p2 such that they are both bigger than N and their difference is smaller than 600.
- lutusp 13y ago
  Yes, you're right, and it's too late for me to either delete or edit my original post. :( In light of that, I think the discussion and work revolves around discovering the smallest gap as the numbers themselves become larger and tend toward infinity. Obviously the very smallest prime gap is that between 2 and 3, i.e. 1, and there are a great number of primes separated by 2, and the Twin Prime Conjecture asserts that there will always be occasional pairs of primes separated by 2 no matter how large the numbers themselves become. The present work is in part meant to put that conjecture on a more analytical footing. I would love to delete my original post.

mikeash 13y ago

You have a good reply, but I'd like to give another answer with a slightly different approach. Pick a number. Make it as large as you like. A billion to the power of a billion to the power of two billion? Just pick something. Now whatever you pick, I can then proceed to find two prime numbers which are at most 600 apart. Maybe they're just slightly larger than the number you picked. Maybe they're vastly larger. But the point is, they exist. We can repeat the procedure with you picking larger and larger numbers and I can always find a suitable pair that's bigger than what you pick. (In theory. In practice, we may get to a point where it takes more time to locate the actual pair than we have time before the universe burns out or collapses. But we can prove that they exist, at least.)