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Science Forum Index » Cognitive Science Forum » Partial difference equation, counting primes
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| James Harris |
 Posted: Sat Nov 15, 2003 12:23 pm |
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Guest
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I think that the last topic I mentioned was too advanced, so here's
something simpler to test the rationality, and cognitive ability of
this newsgroup.
It just so happens that I found a partial difference equation which
gives the count of prime numbers when integrated over a certain range.
It also turns out that my find is unique in human history as NO OTHER
person besides myself has *ever* given a partial difference equation
integration method for counting prime numbers.
And, it's simple to implement!!!
Here's the difference equation and the instructions for the
integration:
dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,
sqrt(y-1))],
S(x,1) = 0.
And p(x, y) = floor(x) - S(x, y) - 1, and you get S as the sum of dS
from dS(x,2) to dS(x,y).
Note that it's a *discrete* function, so for you programmers that
means you need to use int's or long's or some discrete variable type.
So you only want the integer square root value, like for sqrt(10), you
want 3.
For programmers "you get S as the sum of dS from dS(x,2) to dS(x,y)"
means you sum up to and *including* y, as I don't doubt some of you
may write something like
for (i=2; i<y; i++ {}
which is WRONG, and if you do that you will probably get a wrong
answer.
Note: p(x,sqrt(x)) here gives the same value as the traditional pi(x).
For faster calculations you need to use
dS(x,y) = [ p(x/y, sqrt(x/y)) - p(y-1, sqrt(y-1)][ p(y, sqrt(y)) -
p(y-1, sqrt(y-1))]
when sqrt(x/y) < y-1.
That's a BIG deal, as the pure math implementation is VERY SLOW, and
even that quick speed-up won't push you very far.
However, the point is that the difference equation integration does
work, which those of you who can program can verify for yourselves.
Then you should search to see if ANYONE has ever used a partial
difference equation integration to get a count of prime numbers
because then you can see that yes, I'm the only one in recorded
history to present this method.
The above instructions are easy enough to program into a computer and
if you follow them, you'll notice that you do get a correct count for
prime numbers, and you should also notice that unless y is prime
dS(x,y) = 0, so yes, it'll also tell you *which* numbers are prime.
The research is a shining example of the importance of *independent*
researchers willing to check in areas that more staid academics think
are fallow.
The independent researcher is important for the future.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/ |
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