I've looked a bit at Richards' 1974 paper (http://projecteuclid.org/download/pdf_1/euclid.bams/1183535510 http://projecteuclid.org/download/pdf_1/euclid.bams/11

I've looked a bit at Richards' 1974 paper (http://projecteuclid.org/download/pdf_1/euclid.bams/1183535510 http://projecteuclid.org/download/pdf_1/euclid.bams/11835355...) , and understand better although not fully. Saying that Conjecture 2 is false means only that there exists some n for which the number of primes in [1..n] is less than the number of primes in [i..i+n]. Richards (I think) puts a minimum bound of 3500 on n (the 500th prime number is 3571) and an upper bound of 250000 (the 20000th prime number is 224737). Richards claims that for some threshold n between 3500 and 250000 beyond which there will always exist a region [i..i+n] with a greater density of primes than [1..n]. There is no claim (at least by Richards) that there exists a region as dense as [1..100] or [1..1000] somewhere out there to be found. And for his threshold n, the region of dense primes probably starts at some very large number, suggested as perhaps 2262^2262 (where 2262 is number of primes in [1..20000]). It's a fun paper, and interesting as an example of math at the dawn of the computer age. The explain, for example, that although the computer they used supported division, they didn't know how to program it to do so, so they did all the long division by hand instead. As they conclude the paper, "We could have solved the problem without the computer, but we probably wouldn't have."