Math query re infinite twin primes proof

http://blogs.smithsonianmag.com/smartnews/2013/05/an-aging-mathematician-made-a-major-dent-in-one-of-maths-oldest-problems/

My (non-mathematical) instincts are that you can't have an arbitrary finite limit for a property in an infinite series, so his "dent" is a de facto (reductive?) proof. Am I wrong? Math genii to thread, please.

Meaning his proof can't be right because he has established an upper bound for what the difference between to primes can be? ...or that his proof is obvious?

I think the article is misleading because it seems to imply that that his proof is that the difference between ANY two consecutive primes is bounded... whereas I think he just proved that there exist an infinite number of consecutive prime pairs whose difference is bounded. And he found that bound (which can, probably will be shortened eventually).

Re your original question, I don't see why you cannot have an arbitrary finite limit for a property in an infinite series. Can you put an example of what you mean?

oh! that makes sense! when i first read it, i had the same objection as MV.

i suppose the initial, gut-level objection to the "an arbitrary finite limit for a property in an infinite series" is that any such limit seems to strike against the essential unboundedness of infinitely nonrepeating series. this is erroneous, of course, because infinity is, in a sense, "bounded" by what it must contain

anyway, the dent, if it does operate in the manner you suggest, seems to proceed from (rather than subvert) the "anything and everything" implications of this sort of infinitude.

This is pretty poorly done.

"He was able to show that there is an infinite number of prime pairs separated by a measurable finite distance. In other words, there’s a limit to how far away primes can get from each other."

The first sentence is right, the second sentence is wrong.

This one is better:

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

Re original question: it is no different from saying "the set of numbers that end in either 0 or 2 is infinite, and among these numbers there are infinitely many pairs that differ by no more than 2" -- like 10 and 12, 30 and 32, 1000 and 1002.... This is another sequence that has "bounded gaps" in the relevant sense.

yeah, that was v helpful

but re: your last point: no, cuz the numerical "distance" between those numbers doesn't change in an unpredictable (and apparently unbounded?) manner.

change = increase, i mean

the simons foundation article is quite good. and yes this approach opens up a bunch of "obvious" next steps involving number crunching that can probably reduce the bound a fair bit, though its known that no amount of jiggling will reduce the bound to two with this particular approach (i.e. there are a few more huge breakthroughs left to go before we get to the genuine twin primes problem).

no, cuz the numerical "distance" between those numbers doesn't change in an unpredictable (and apparently unbounded?) manner.

OK, then, the sequence of numbers which are either a positive perfect square or 2 more than a positive perfect square.

1,3,4,6,9,11,16,18,25,27,....

There are larger and larger gaps, but there are still infinitely many gaps of size 2.

tricky, but i'm still not entirely convinced. you're setting up sequences that are limited in such a way as to guarantee the circumstance we're talking about. your sequences may be infinite in scale, but they're not infinite in variation. in fact, they're utterly constrained in variation.

the occurrence of primes within the sequence of integers doesn't seem to be constrained in this manner. these occurrences seem infinite not only in scale, but in variation. if the occurrences of a given thing are infinite in variation within an infinite sequence, my (completely untrustworthy) "gut feeling" says that there must be allowance for all but infinite distance between them.

that's the objection i thought MV was making - to the the misleading description of zhang's "dent".

"there must be allowance for all but infinite distance between them"

well we do have bounds on the growth of the maximal gap, of which this is the simplest:

https://en.wikipedia.org/wiki/Bertrand%27s_postulate

awesome! thanks for that. exactly why i called gut feelings "completely untrustworthy" as a guide to things i know nothing about.

...cuz the conjecture and proof aren't so obvious that a relative math novice might intuit them (or even work them out)

Here's a better (I think?) article about this:

http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.single.html

yeah ellenberg is another good source on this stuff -- talented mathematician, good populizer.

I think Guayaquil (eephus!) is at least as good as ellenberg.