# Re: Prime Numbers

`On Mon, Sep 17, 2012 at 12:27 AM, Rex Allen <rexallen31...@gmail.com> wrote:`
```
>
> On Sun, Sep 16, 2012 at 6:10 PM, Stephen P. King <stephe...@charter.net>wrote:
>>
>> HI Rex,
>>
>>     Nice post! Could you riff a bit on what the number PHI tells us about
>> this characteristic. How is it that it seems that our perceptions of the
>> world find anything that is close to a PHI valued relationship to be
>> "beautiful"?
>>
>>
>
> Thanks Stephen!
>
> Actually my initial example of "numeracy" isn't quite right, but it's not
> important to the rest of the argument.
>
> My main point is that you can get to the concept of "prime numbers" just
> using relative magnitudes that we have an innate sense of.
>
>
I think an easier way to intuit prime numbers that can't be represented as
rectangles, only a 1-wide "lines".

While the concept of primes is straight forward, there is an unending set
of not-so-obvious facts that we continue to discover about the Primes.  For
example:

The average distance between primes of size N is approximately the natural
log of N, yet we know of no way to predict where the next prime will
exactly be. ( http://en.wikipedia.org/wiki/Prime_gap )

Between N and 2N, there will always be at least one prime. (
http://en.wikipedia.org/wiki/Bertrand's_postulate )

There is a one-to-one correspondence, and method to get one from the other,
between perfect numbers and primes of the form ((2^p) - 1) (
http://en.wikipedia.org/wiki/Perfect_number#Even_perfect_numbers )

For any prime p, and any integer i where 0 < i < p, i^p divided by p has a
remainder of i.  This almost never works for composite numbers.  (
http://en.wikipedia.org/wiki/Fermat's_little_theorem )  the exception for
composite numbers where this does hold are known as Carmichael numbers (
http://en.wikipedia.org/wiki/Carmichael_number ) but they are rare.

And there are an infinite number of other such patterns waiting to be
discovered.

Jason

As for the significance of PHI - well - I guess there's probably some
> plausible sounding evolutionary story that could be told about that.
>
> Though how satisfying or useful an explanation like that is just depends
> on what you're after and what your interests are.
>
> An explanation that might be useful in one context might be useless in
> some other context.
>
> Explanations are observer dependent.
>
> Probably.
>
> Rex
>
>
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