Patterns in Primes

Y'all have probably been over this, but I must have missed it. I read an
interesting book on mathematics (once) and was informed that no clear
pattern had emerged pertaining to the way prime numbers reveal
themselves in ascending order. I'm not sure of the term for our system
of counting (decimal, base 10?) but have any other "bases", if that's
the right word, revealed a pattern. In otherwords would octopuses with
their 'octo' system understand the pattern better? Keep it simple folks.

Best.

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> Y'all have probably been over this, but I must have missed it. I
read an
> interesting book on mathematics (once) and was informed that no
clear
> pattern had emerged pertaining to the way prime numbers reveal
> themselves in ascending order. I'm not sure of the term for our
system
> of counting (decimal, base 10?) but have any other "bases", if
that's
> the right word, revealed a pattern. In otherwords would octopuses
with
> their 'octo' system understand the pattern better? Keep it simple
folks.
>
> Best.

Hi Alison!

Prime numbers from 5 upward are of form 6n+1 or 6n-1. So this pattern
is more obvious in base six.

Kalle

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> Y'all have probably been over this, but I must have missed it. I read an
> interesting book on mathematics (once) and was informed that no clear
> pattern had emerged pertaining to the way prime numbers reveal
> themselves in ascending order.

The basic "pattern", if you can call it that, is given by the prime number theorem: that the density of primes is 1/ln(x). That can be extended so that the zeros of the Riemann Zeta funcion show the oscillations, but that doesn't help much in practice.

> From: kalleaho <kalleaho@mappi.helsinki.fi>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, March 07, 2002 3:04 PM
> Subject: [tuning] Re: Patterns in Primes
>
>
> Prime numbers from 5 upward are of form 6n+1 or 6n-1. So this pattern
> is more obvious in base six.
>
> Kalle

wow, that's really interesting!

i checked for n = 0, 1, 2, ... 22 and the result gives the
first 31 primes in sequence, plus multiples of 5, 7, 11, etc.

how long has this been known?

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> wow, that's really interesting!
>
> i checked for n = 0, 1, 2, ... 22 and the result gives the
> first 31 primes in sequence, plus multiples of 5, 7, 11, etc.
>
> how long has this been known?

Dirichlet proved that primes in linear progression, a*n+b, occur infinitely often and at the expected density if a and b are relatively prime; this was over 150 years ago. Hence there are infinitely many primes of the form 6*n+1 and 6*n-1. It's been conjectured that this extends to sets of higher degree polynomials (Bateman-Horn conjecture) and if you check n^2 + n + 41 you find its ability to produce primes is remarkable. This is, believe it or not, related to the fact that exp(pi sqrt(163)) is very nearly an integer.

By plotting the primes on spirals in 2D you can get lines which look a little like the lines on the temperament graphs.