<i>The other Hardy-Littlewood conjecture is the seemingly innocuous statement that there are more primes in the first n numbers than in a string of n numbers starting anywhere else on the number line.</i><p>I feel like some important detail must have been lost in her presentation of the theorem. She uses the concrete examples "There are 25 primes less than 100 and 168 less than 1,000," and then says that it's difficult "to believe that there are places along the number line where the primes bunch up enough to make up for those very dense areas".<p>Is she saying that it's difficult to believe that there is span of 100 consecutive integers starting at some i > 1 that contains more than 25 primes? And that there is some span of 1000 starting at i > i that has more than 168? Or is she saying that it's difficult to believe that there exists any n for which this is true?<p>That is to say, what does it mean for Conjecture 2 (quoted above) to be false? Does it mean that for all [1..n] there exists some span [i..i+n] that contains more primes? Or does it's falseness mean only that there exists at least one n for which [i..i+n] contains at least as many primes as [1..n]?<p>Her emphasis on the concrete n=100 and n=1000 makes me think she's claiming the universal "for all n", but this seems almost certainly false for at least some small n.
> I have reluctantly come to accept the fact that somewhere up there, in the vast expanse of primes, a cluster sits there outweighs the first chunk of prime numbers.<p>Just because it is possible doesn't mean it exists. Of course the odds of finding one are negligible so we'll probably never know for sure.