Two Plausible Things That Cannot Both Be True | Hacker News Reader

Two Plausible Things That Cannot Both Be True

blogs.scientificamerican.com

17 pointsjaybosamiya8 comments

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 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".

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?

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]?

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've looked a bit at Richards' 1974 paper (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."

> 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.

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.

The author is making the following reasoning: Conjecture 1 and Conjecture 2 can not both be true. Conjecture 1 is believed to be true. Therefore Conjecture 2 is false, which implies that there must exist a heavy cluster of primes out there.

Can anyone summarize this for people (such as myself) who have a hard time understanding what's going on with all this prime number stuff?

There is no reasonable formula for generating prime numbers. So let's treat primes as if they were empirical data points instead of deterministic mathematical constants. From this lens, the statistical properties of primes are very simple to predict. That's what all these conjectures are about: finding out what sort of probability distribution would produce random numbers that look like the primes.

And people just now realized that you could look at prime numbers this way?

As stated in the article, those conjectures were made in 1923. The recent quanta article https://www.quantamagazine.org/20160313-mathematicians-disco... involves a new correlation discovered in the primes, which means that the statistical model is more interesting than we previously thought.

For a mathematician's perspective on the recent breakthrough, I recommend https://terrytao.wordpress.com/2016/03/14/biases-between-con....

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