Prime numbers, my find, and discovery

I should be a rather happy guy.  After all, over 18 months ago I found
this partial difference equation I call dS(x,y), and the sum of dS
from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and
including x.

Afer talking with mathematicians all over the world by email and
Usenet, and searching math references, both bought and on the
Internet, I know that I have a first-find.

Somehow, I am the first human being in recorded human history to find
a partial difference equation that sums to give the count of prime
numbers.  This post is about some of the significance of that beyond
it being a first-find.

Prime numbers have fascinated people for some time, and mathematicians
especially.  The great mathematician Karl Gauss is credited with
making an important hypothesis in the field of prime numbers, as he'd
noticed something.

Gauss noticed that the count of primes numbers could be approximated
by x/ln x, for instance, the count of primes up to 1000 is 168, and
1000/ln 1000 approximately is 144.76.  The count of primes up to 10000
is 1229, and 10000/ln 10000 is approximately 1085.73, which is a
closeness that continues as you go higher.

Gauss wondered what the discrete count of prime numbers could have to
do with continuous functions like x/ln x, and while mathematicians
made progress in finding relations that gave limits, like Chebyshev's
use of the zeta function discovered by Euler, they never found a
reason why.

I may have found that reason.

Not surprisingly, a first-find in the area of prime numbers *should*
be a big deal, but despite the ease with which I link my discovery to
some of the biggest names and biggest issues in mathematics, there is
the value to society of the discoverer.

Since when has modern society decided that discoverers should be
attacked instead of cheered?

Now you may have seen a LOT of postings from people trying to attack
the worth of my find, which can be a healthy process--if they stick
with the facts.

Unfortunately posters have shown a dismaying tendency to lie, but
that's minor to the problem I've faced where mainstream mathematicians
have tried to ignore or downplay my result.

I have a first-find in the area of prime numbers, and my not being a
mathematician does not mean that mathematicians can just deny the
reality if it suits them.  While they may feel they have many reasons
to attack the value of an important find from a non-mathematician,
those reasons cannot be in the best interests of society.

If Gauss were alive today, would he cheer me?

I like to think he would, as he was someone interested in asking
questions *and* in getting answers.  First and foremost I think he
would have been driven to find out just where my discovery led, and if
it was the answer to the question that intrigued him.

As I've found a partial difference equation, it leads to a partial
differential equation.  That partial differential equation may answer
many questions.

Or more importantly, it should raise many more.

You should not allow mathematicians to continue to pervert a process
that has helped humanity for so many thousands of years.  You must not
show a loss of faith in the future of humanity, as if discoverers are
no longer needed.

Academic institutions can no more constrain who can make a major
discovery, than they could limit who will be a great painter,
composer, or architect.

Maybe that's part of the problem as we know that architects require a
lot of schooling beyond just art, as they need to know physics, like
materials science, and engineering, among many other things.

So it's easy to assume that a great building can only come from
someone heavily trained in academia who can manage a huge structure.

However, sometimes something a little smaller in terms of physical
size can be huge in terms of social value, and the person who built
it, might be someone from just around the corner, outside of academia.

Maybe I'm pushing the analogy, but I hope that you'll agree that at
the end of the day, what's important is the *information* and petty
squabbles and personal attacks are irrelevant, and often forgotten
over history anyway.

It's the knowledge that remains--pure.

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

and S(x,y) is the sum of dS from dS(x,2) to dS(x,y).

And p(x,sqrt(x)) is the count of prime numbers up to and including x.

That's pure knowledge.  Information discovered by me, and hey, it
wasn't like it just jumped in my lap you know.  There's a value to
cheering on discovery, and not attacking it.

The value is hope for the future.  Hope that there may be answers out
there from unlikely sources.  Hope that every person can be valuable.

Maybe mathematicians want a reality that has them ordained as the only
route for new mathematical knowledge.  Possibly they wish control over
the creative process, and total dominion over mathematical discovery.

But hey, they're only human.


James Harris

"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/