5 ms·

Prime After Prime (2016)

simonhughes22 6y ago
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.
mCOLlSVIxp6c 6y ago
https://web.archive.org/web/20191202152434/http://bit-player.org/2016/prime-after-prime https://web.archive.org/web/20191202152434/http://bit-player...
Zmetta 6y ago
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?
- ladberg 6y ago
  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.
  - caf 6y ago
    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).
- hinkley 6y ago
  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-primes-are-6n-1/ https://reflectivemaths.wordpress.com/2011/07/22/proof-prime...
- Vvector 6y ago
  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
erickhill 6y ago
Now I've got Cyndi Lauper in my head.
- airstrike 6y ago
  Somehow Vic Damone claimed that spot for me a few years back, so I've got him in my head instead
- Tycho 6y ago
  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.
deleted 6y ago
[deleted]
dang 6y ago
Discussed at the time: https://news.ycombinator.com/item?id=11837511 https://news.ycombinator.com/item?id=11837511
aardvarks 6y ago
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.
- caf 6y ago
  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.

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.

mCOLlSVIxp6c 6y ago

https://web.archive.org/web/20191202152434/http://bit-player.org/2016/prime-after-prime https://web.archive.org/web/20191202152434/http://bit-player...

Zmetta 6y ago

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?

ladberg 6y ago
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.
- caf 6y ago
  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).
hinkley 6y ago
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-primes-are-6n-1/ https://reflectivemaths.wordpress.com/2011/07/22/proof-prime...
Vvector 6y ago
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

erickhill 6y ago

Now I've got Cyndi Lauper in my head.

airstrike 6y ago
Somehow Vic Damone claimed that spot for me a few years back, so I've got him in my head instead
Tycho 6y ago
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.

deleted 6y ago

[deleted]

dang 6y ago

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

aardvarks 6y ago

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.

caf 6y ago
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.