Prime After Prime (2016) | Hacker News Reader

Prime After Prime (2016)

bit-player.org

Actually, even the first two tables comparing the frequency of 1,2,3,4,5,6 when obtained using primes vs. a fair die suggest that consecutive primes do not give a truly random (uncorrelated) way of choosing congruence classes mod 7.

If I throw a fair die 10^6 times, the probability of getting any given single outcome should behave according to Poisson statistics. On average, if I repeat a trial of 10^6 die-throwings many times, the number of outcomes of "4" (let's say) should be on average 10^6/6 = 166,667 , as mentioned in the article.

However, the exact number of times "4" comes up in a given trial itself follows a distribution around that average whose spread is about sqrt(166,667), or about 400. So the typical "error" in the frequencies given in the table should be ~few hundred.

By this reasoning, the deviations in the top table, the one given by the primes, are surprisingly small -- of order tens rather than hundreds. In other words, primes are more equitably distributed among congruence classes than we would expect independent die roll outcomes to be.

Yes, at the bottom of the article is an Addendum that covers this:

Addendum 2016-06-14. I noted above that the distribution of primes mod 7 seems flatter, or more nearly uniform, than the result of rolling a fair die. John D. Cook has taken a chi-squared test to the data and shows that the fit to uniform distribution is way too good to be the plausible outcome of a random process. His first post deals with the specific case of primes modulo 7; his second post considers other moduli.

6x±1: 5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,61,65,67,71,73,77,79,83,85,89,91,95,97,

101,103,107,109,113,115,119,121,125,127,131,133,137,139,143,145,149,151,155,157,161,163,167,169,173,175,179,181,185,187,191,193,197,199,203,205,209,211

Is this just a poor sieve for odd-number pairs or is there something more going on within the factors of 6x±1?

I think it's just a sieve that removes multiples of 2 and 3, leaving false positives that are multiples of 5, 7, 11, etc.

Right, which is why as the numbers get larger and the primes get more sparse, 30 eventually takes over from 6 as the sieve (2x3x5).

I think I watched a good video from Numberphile and/or Matt Parker on this but I can't seem to find it now. IIRC it was used as an alternative proof for Fermat's last theorem.

This explains the 6n situation pretty concisely though:

https://reflectivemaths.wordpress.com/2011/07/22/proof-prime...

It just eliminates the multiples of 2 and 3

6x ± 0: divisible by 2,3

6x ± 1: not divisible by 2,3

6x ± 2: divisible by 2

6x ± 3: divisible by 3

6x ± 4: divisible by 2

6x ± 5: modulus equivalent to 6x ± 1

https://web.archive.org/web/20191202152434/http://bit-player...

Now I've got Cyndi Lauper in my head.

Somehow Vic Damone claimed that spot for me a few years back, so I've got him in my head instead

Strange thing about that song - I can never remember what the verse melody goes like. I can remember the bridge and chorus always. Usually I would just sort of play through the song in my head until I get back to the verse but never seems to work. If I hear it I will have forgotten it again the next day.

Interesting. It seems to me that the two consecutive primes, modulo 7, are more likely to be an odd and even pair (i.e. the total of the 2 modulo is more likely to be odd) than an odd odd or even even pair.

Discussed at the time: https://news.ycombinator.com/item?id=11837511

Reply on news.ycombinator.com