Re: Explaining the prime distribution

From: James Harris (jstevh_at_msn.com)
Date: 08/16/04 

Date: 16 Aug 2004 03:58:01 -0700

feoren@yahoo.com (C. Leth) wrote in message news:<40b90bde.0408151220.50ad3697@posting.google.com>...
> Your first points where you define functions are all derived from
> dS(x,y), so here I'm only going to talk about your function for that,
> since that must be correct in order for the rest to work.
>
> jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0408150646.61e270a@posting.google.com>...
>
> >p(x,sqrt(x)) = floor(x) - S(x,sqrt(x)) - 1
> Just a side note, this doesn't really matter, but since sqrt(x) is a
> function of x and not a seperate independent varaible, you only need
> one input. Also, floor(x) is unnecessary because floor(x) = x when x
> is natural, which it will always be.

Actually it won't as can be seen further down, where I end up with

p(x/y,y-1)

and the floor can conveniently be put in that one place and handle
everything.

Also, the function is p(x,y) where p(x,sqrt(x)) just happens to count
primes!

So it's a multi-variable function that can be evaluated over a certain
path to count prime numbers, but it makes sense, as evaluating it
requires both variables to write it as a I have.
 
> p(x) = x - S(x,sqrt(x)) - 1
>
> Anyway, on to the meat.
>
> > So I've finally outlined all the necessary functions, and it's now
> > time to get the full dS(x,y) function, which surprisingly enough is
> > easy to the point of trivial.
> >
> > Remember, dS(x,y) is the count of composites up to and including x
> > that have y as a factor which do not have any other primes less than y
> > as a factor.
> >
> > As a first step to calculating it, I use the fact that the total
> > number of naturals with y as a factor up to and including x is given
> > by
> >
> > floor(x/y)
> >
> > like with 10 again, the naturals with 3 as a factor are
> >
> > 3, 6, 9
> >
> > but wait, 3 gets included, but it is prime and not a composite, so
> > with primes I need to use
> >
> > floor(x/y) - 1
> >
> > to get a count of all composites up to and including x with y as a
> > factor, and here y is prime.
> >
> > But, what about primes less than y? Well I can count those out with
> >
> > p(y-1,sqrt(y-1))
>
> Right here, in your definition for dS(x,y), you use the function
> p(x,y) = floor(x) - S(x,y) - 1. S(x,y) is the sum of dS(x,i) where i
> = every prime from 2 to y. In your definition of dS(x,y) you have a
> function that uses dS(x,y) in its definition. Unless you can take
> p(x,y) out of your definition of dS(x,y), it will be a circular
> definition and therefore be impossible to evaluate.
>

It's a recurrence relationship. The recursion does not prevent
evaluation.

Evaluating dS(x,y) from y =2 to the sqrt(x), will naturally force

S(x,1) = 0 and p(x,1) = x-1

so you will not infinitely recurse.

Notice that S(x,1) can be talked out as saying that there are no
composites with primes less than 1 as a factor as in fact there are no
primes less than 1.

> > which you'll notice works up to 10, as
> >
> > p(3-1,sqrt(3-1)) = p(2,sqrt(2)) = 1
> > as, of course, 2 is the only prime less than 3, so I now have
> >
> > floor(x/y) - 1 - p(y-1,sqrt(y-1)), with y prime,
> >
> > and notice that gives me the correct count up to 10 for 3, as that
> > just leaves
> >
> > 9
> >
> > as 3 got subtracted off by 1, and 6 got subtracted off by the count of
> > primes less than 3.
> >
> > But I'm not done in general, as there may be composites now that
> > multiply times y that are less than x, which have primes less than y
> > as factors.
> >
> > For instance, up to 12 I have 2(2)(3), and what I've done so far would
> > still leave that in the count for dS(12,3).
>
> Ok, just trying for clarification. So far, if we took dS(12,3) we'd
> get floor(12/3) = 4 (Count(3, 6, 9, 12)). Subtract one for the prime,
> 3 (6, 9, 12). Subtract another for p(2,sqrt(2)), and we get 2 (9 and
> 12 or 6 and 9).
>

Yup, and 2(2)(3) is still left after the first corrections, though it
has 2 as a factor but 2 is, of course, a prime less than 3, so more
work is needed.

> > But I have a way to handle that possibility!
> >
> > I simply use S(x/y, y-1) as the y-1 tells it to only count primes less
> > than y, and it's counting the composites which can multiply times y
> > and still be less than x, and now I'm nearly done as I get
>
> Here you use another function of dS(x,y) to define dS(x,y). Also, I
> know it doesn't matter for the function, but it's just more clear if
> you say S(floor(x/y),y-1).
>

That could be an alternate way to go, I'll admit.

I just like

p(x,y) = floor(x) - S(x,y) - 1

as the one place where floor() shows up.

As for the recursion, as I explained above, it's not a problem.

In case the recursion makes you really nervous you can look at some of
the explicit forms of the dS(x,y) function.

For instance, if x=N, where N is an even natural, then

dS(N,2) = N/2 - 1

dS(N,3) = floor((N-4)/6)

(to see calculation go to my blog

http://mathforprofit.blogspot.com/

and look for "tidbits" which will be near bottom)

and explicit forms exist for every prime out there in idea space you
could say.

But the recursion can be handled anyway by noting that S(x,1) = 0.

> > So I finally have a working dS function:
> >
> > dS(x,y) = (p(x/y,y-1) - p(y-1,sqrt(y-1))(p(y,sqrt(y)) -
> > p(y-1,sqrt(y-1)))
> >
> > and to help readability I usually use brackets:
> >
> > dS(x,y) = [p(x/y,y-1) - p(y-1,sqrt(y-1)][p(y,sqrt(y)) -
> > p(y-1,sqrt(y-1))]
> >
> > and just like that I have my dS function rigorously defined using some
> > very basic algebra and a few good ideas.
> >
> > James Harris
>
> I checked the function with your examples and they all worked.
> However, it's hard to know whether they work for all natural numbers
> or not. With a function like this, programming the whole thing then
> doing it on the computer would be faster than running through it 3 or
> 4 times by hand, but I can't picture a way to program it because of
> its circular nature.

You can find posted programs implemented it, but I'd prefer you
understand in detail from the derivation.

The derivation is supposed to show that it works: why and how.

Quite simply I count composites for a particular prime p, where I keep
from double counting by subtracting off the count of composites that
have primes less than p as a factor in two ways:

1. Subtract off the number of primes less than p
2. Subtract off the number of composites with primes less than p as a
factor

That route leads to recursion, which can definitely be handled, and it
stops at S(x,1) = 0, as there are no composites with primes less than
1 as a factor as there are no primes less than 1.

If something is not clear then I'd be happy to expand out on the
derivation.

 
> If worst comes to worst, this will still be very strong, though. The
> ultimate goal is an independent function that will give you a list of
> primes without having to know any previous primes and without huge
> factoring. Although your circular definitions mean that you don't
> quite have that, if you reworked your equations you'd very likely have
> a way to find primes through iterations using previous primes, which
> would be still be an amazing accomplishment.

Thanks! But it does work.

What you call a circular definition is technically called a partial
difference equation.

And you don't need previous primes.

My prime counting function finds them on its own, which is one of
those obvious facts that sci.math posters dismiss and I wonder about
the mind-control.

Like, that feature is really neat, never been seen before, and clearly
proves that my work is important, but sci.math posters would just
claim it's useless and stupid!

They did that for YEARS.

So you have a function that recurses to automatically select out only
prime numbers, something never before seen in the math literature, but
not only do mathematicians dismiss it as unimportant, posters from the
sci.math newsgroup specifically attack that unique feature as useless
and stupid, but AT THE SAME TIME claim that my work is really over two
hundred years old!!!

I think my research is neat, fascinating, and clearly something worth
recording for posterity, but over the last two years when I've talked
about it on newsgroups I've been insulted and my work dismissed, and
when I talk to mathematicians, I've been told it's of no interest or
just ignored.

I think they're weird. And mathematicians aren't acting at all like
the Hollywood version.

They're not quite acting like curious human beings.

They're being strangely incurious. It's bizarre.

It's like something is missing inside their heads as I'd think that
they'd be excited about more math to play with, but instead, like on
sci.math, they get angry and abusive.

James Harris

Relevant Pages

  • Re: Explaining the prime distribution
    ... >> Remember, dSis the count of composites up to and including x ... >> that have y as a factor which do not have any other primes less than y ... As for the recursion, as I explained above, it's not a problem. ... those obvious facts that sci.math posters dismiss and I wonder about ...
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  • Re: About random, primes and statistics
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