Beyond counting primes, partial difference equation


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Topic: Science > Physics
User: "James Harris"
Date: 18 Nov 2003 11:06:02 AM
Object: Beyond counting primes, partial difference equation
I've emphasized the gee-whiz feature of my discovery, and you can see
how mathematicians have reacted, so on to the science!!!
For those who missed it (hard to imagine given all my posts) a while
back I discovered the following partial difference equation:
dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))],
where
S(x,1) = 0, and p(x, y) = floor(x) - S(x, y) - 1.
Which probably doesn't tell you a lot.
Well it turns out that you can integrate that partial difference
equation, that is, sum it, by taking values from dS(x,2) to dS(x,y),
and adding them together to get S(x,y).
Like if you do that with x=10 and y=3, you'll get S(10,3) = 5, and
putting that into the equation for p(x,y) gives you 4, which is the
count of primes up to 10.
It works that way out as x goes to positive infinity. You have some
x, and the closest positive integer to sqrt(x), and the result is the
count of prime numbers up to and including x. Like p(11,3) = 5, as
now you have 11 in that count.
So that's a lot of the math aspect, but I'd like to go beyond that to
considering the behavior of the partial difference equation itself.
Not surprisingly from the p(x/y, y-1), you get some kind of
exponential drop in values for dS, and intriguingly, it's an
oscillator.
The oscillation is such that if y is NOT prime then dS is 0, but for a
while, that is until y>sqrt(x), you get positive integer values for dS
of exponentially decreasing size.
What I have then, besides being a prime counter when used one way, is
a mathematial model of a damped oscillator.
Comments please.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/
.

User: "Krzysztof Olczyk olczykdotkrzysztofatxldotwpdotpl"

Title: Re: Beyond counting primes, partial difference equation 18 Nov 2003 12:00:23 PM
"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0311180906.595c6d29@posting.google.com...

I've emphasized the gee-whiz feature of my discovery, and you can see
how mathematicians have reacted, so on to the science!!!

(...)

What I have then, besides being a prime counter when used one way, is
a mathematial model of a damped oscillator.

Comments please.

What for, then?
You are NOT reading them anyway.
You are still flattering yourself with your great discovery despite the fact
that competent people are claiming that you are talking "utter flannel".
Although it has been said it's wrong,
you are still presenting your incredible way
of "integration of difference equation".
I'm waiting for you to present way
of integrating the equation of form:
x + b = a,
where a and b are given.
--
Regards,
Krzysiek
.


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