Re: Sieve distinction, prime counting

mm_at_nowhere.net
Date: 03/03/05 

Date: Thu, 03 Mar 2005 10:05:42 +0100

jstevh@msn.com wrote:

> A key point that needs to be understood to get at why my prime counting
> discovery is important is how it's a big deal that it is fully
> recursive without needing to be given primes.

And that's a big difference with the original Legendre formula?
That's clear, either you are a liar or you understand nothing,
maybe both.

> Basically what I have is
>
> p(x,y) = floor(x) - S(x,y) - 1
>
> where S(x,y) is the sum of the dS(x,k) function where
>
> dS(x,k) = (p(x/k, k-1) - p(k-1, sqrt(k-1)))(p(k,sqrt(k) - p(k-1,
> sqrt(k-1)))
>
> and you can just pick some natural number x, like 100, and follow that
> formula and out will come the answer 25.
>
> So the complete math form doesn't require you do much of anything but
> follow the mathematical formula which directs you.
>
> However, for sieves, you can't just follow a formula but you also have
> to give it more information, like for Legendre's Formula, you need to
> have the list of primes up to the square root of 100, or it won't work.
>
> So you have to tell the sieve function that 2, 3, 5 and 7 are primes,
> and then it can count out for you that there are 25 primes up to 100.
>
> The "intelligence" you might say of my discovery that allows it to
> figure out that those numbers are prime without you telling it is in
>
> (p(k,sqrt(k) - p(k-1, sqrt(k-1)))
>
> which equals 0 if k is composite and 1 if k is prime, so it's a logic
> circuit.

Once more, you just prove how your knowledge in math is
incredibly low. The function that is equal to 0 or to 1
depending on the fact that an element belongs or not to
a given set exists. But, of course, it has an other name
than the one you are giving it (try a Google search with
"Characteristic function").

> It gives 1, which is true, if k is prime, and 0, as in false, if k is
> composite.
>
> So built into my prime counting function is a logical switch that flips
> on or off automatically based on whether or not k is prime, which is
> not seen in anything else in mathematical history.
>
> It's just extraordinary, and no one can point to anything even like it
> in this area.

Extraordinary. Establishing the primality of X by computing the
difference pi(X) - pi(X-1) is absolutely extraordinary.

 
> Now that logical switch

You mean 'characteristic function'?

> does slow it down, so if you want to count
> fast, you can speed it up by yourself telling it whether or not k is
> prime versus letting the mathematical function find out by recursion.
>
> Then it becomes a sieve as you're giving it more information.
>
> Now in any other method that you will see mathematicians talking about
> for counting primes you'll see a tell-tale list of primes being needed,
> and it's not because it's a luxury, but because it must be so, or their
> formulas won't work.

Bull***. For instance, in "Prime Numbers and Computer Methods
for Factorization", page 14, Hans Riesel explicitely wrote :

  Using formula (1.9) repeatedly, we can break down any Phi(x,a)
  to the computation of Phi(x,1) which is the number of odd
  integers <= x. However, because the recursion has to be used
  many times, the evaluation is cumbersome. It is far better
  to find a way to compute Phi(x,k) for some reasonably large
  value k, and then break down the desired value Phi(x,a) just
  to Phi(x,k) and no further.

That's not because the formulas don't work but because IT
IS FAR BETTER to do so.

mm