# Difference equations, calculus, prime numbers

Discussion in 'General Math' started by James Harris, Dec 3, 2003.

1. ### James HarrisGuest

I've been talking a lot about a partial difference equation I found
that sums over a certain range to give the count of prime numbers, but
it occurs to me that your calculus may be rusty, so you might be hazy
on some of what I mean.

Remember the following?

As dx approaches 0, f'(x) = (f(x+dx) - f(x))/dx

Now the idea is that you start with some dx and shrink it towards 0,

df(x) = f(x+1) - f(x)

which is a difference equation.

It's kind of obvious why it's called a *difference* equation as
well!!!

Now then, let's say you were given

df(x) = f(x+1) - f(x)

then it's not complicated how you get back to the differential
equation as you assume that the 1 is from the delta. So you have

df(x) = f(x+dx) - f(x), and dividing by dx, you have

df(x)/dx = (f(x+dx) - f(x))/dx,

and as dx approaches 0 in the limit you have

f'(x) = (f(x+dx) - f(x))/dx,

which completes the circle.

Now then, what I found is

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

Taking the first steps in approaching the differential, I have

dS(x,y)/dy = [p(x/y, y-dy) - p(y-dy, sqrt(y-dy))][ p(y, sqrt(y)) -
p(y-dy, sqrt(y-dy))]/dy,

where p(x, y) = x - S(x, y) + C.

In the limit as dy approaches 0, I have

S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))

and looking at

p(x, y) = x - S(x, y) + C,

I can differentiate with respect to y to get

p'_y(x,y) = - S'_y(x,y),

and making the substitution gives

p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),

which is the partial differential equation.

There's only one step in that process which might be confusing, which
is how I went from

[ p(y, sqrt(y)) - p(y-dy, sqrt(y-dy))]/dy to p'_x(y, sqrt(y)),

as dy approached 0, and the answer has to do with the directional
derivative as dy approaches 0, which has a vector that approaches that
of p'_x, and equals it in the limit (I think).

So what I just showed you is something that no one in recorded human
history has managed to do with any discovery related to prime numbers
before me.

It gives a *why* for the connection between the distribution of prime
numbers and continuous functions and opens up an unimaginably huge
area for future research.

James Harris

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

James Harris, Dec 3, 2003

2. ### Mu-PiGuest

Mu-Pi, Dec 3, 2003

3. ### C. BondGuest

Yes, and you are approaching the status of a newsgroup vandal. Your
continued posting of repetitive material and diatribes has elevated your
status from that of a simple troll or crank to that of a spammer, to put
it kindly. If you continue your tirades and constant reposts, you are
certainly using up bandwidth and degrading the SNR.
Your calculus is not only rusty and hazy, but completely wrong. (See
below.)
You are simply replaced unity with a differential. This time, however, in
a flash of brilliance you realized you must divide the right side by 'dy',
which you didn't do in previous posts. But your equation is still wrong!
You have taken a difference equation which solves a given problem, and
converted it to a finite difference equation which may have no relevance
whatsover wrt the original problem. Your results will be different for
every value of 'dy' and will only be correct when 'dy = 1.
Hahahaha. What a joke! This is the most spectacular example of handwaving
humbug I've ever seen. Go ahead and try to produce meaningful results with
your formulation. You programmed the original difference equation, didn't
you? Now do this one and post your results.

Hahahaha. I dare you!
No, you didn't. You haven't shown that your bizarre formulation actually
has any relevance to the prime counting function. You have shown *no*
results whatsoever.
It does no such thing. Go back to your playpen.

C. Bond, Dec 3, 2003
4. ### David C. UllrichGuest

No, it does nothing of the sort. You've explained where your "pde"
came from. You have not given _any_ reason to think that the
solution has anything to do with pi(x).

And you've continually ignored an explanation for why it
seems most likely that the pde has nothing whatever to do
with pi(x). Consider the precisely analogous process of
converting a difference equation to a differential equation,
in a much simpler case where we can see what the answer
is:

f(x+1) - f(x) = f(x), f(0) = 1.

The solution is f(x) = 2^x.

Now consider the corresponding differential equation:

f'(x) = f(x), f(0) = 1.

The solution is f(x) = e^x.

So. If you've explained what you claim to have explained
then _I_ have just explained why e^x is an approximation
to 2^x.

Guffaw.
************************

David C. Ullrich

David C. Ullrich, Dec 3, 2003
5. ### Mu-PiGuest

He already reached that status years ago.

Mu-Pi, Dec 3, 2003
6. ### Mu-PiGuest

LOL! Well said David!

Mu-Pi, Dec 3, 2003
7. ### Doug NorrisGuest

Please remove "approaching the status of" in the sentence above. This
is not a limit problem; the function exists and *is* a vandal at that point.

Doug

Doug Norris, Dec 3, 2003
8. ### Mu-PiGuest

LOL!

The proof goes as follows:
For every epsilon about the point.....

Mu-Pi, Dec 3, 2003
9. ### Uncle AlGuest

[snip]

Idiot. It has been variously shown that you couldn't pull the lint
out of your own bellybutton without causing bleeding. You know near

Hey stooopid loud troll James Harris, put up or shut up,

http://www.rsasecurity.com/rsalabs/challenges/factoring/faq.html
http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html
http://www.crank.net/harris.html
It's not every braying jackass that gets a whole page at crank.net

Uncle Al, Dec 3, 2003
10. ### Dirk Van de moortelGuest

;-))

Dirk Vdm

Dirk Van de moortel, Dec 3, 2003
11. ### Sam WormleyGuest

Prime Counting Function
http://mathworld.wolfram.com/PrimeCountingFunction.html

Prime Difference Function
http://mathworld.wolfram.com/PrimeDifferenceFunction.html

Crank Information
http://www.crank.net/harris.html
http://www.crank.net/usenet.html

Sam Wormley, Dec 4, 2003
12. ### James HarrisGuest

Ok readers, notice that clear challenge from the poster, and now let's
see what he actually put down.

Notice no comments from the poster.
There's no rule in the calculus that says you can't take something
like the difference equation

df(x) = f(x+1) - f(x)

and move to the differential equation

f'(x) = [f(x+dx) - f(x)]/dx,

in the limit as dx approaches 0, and if the poster accepted that, why
attack my using the same principles with my partial difference
equation?

The poster is clearly irrational as he's so transparent. Why make a
post read by so many people that shows you fighting against
mathematics?

What kind of person would fight calculus?

Readers should note that the poster deliberately just went past the
part where I described going from a difference equation back to a
differential equation because the truth is something this person
doesn't like.

difference equation example I used?

James Harris

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

James Harris, Dec 4, 2003
13. ### James HarrisGuest

Since I know that "C. Bond" is a regular sci.math poster, I thought it
worth mentioning inconsistencies between posters, as David Ullrich is
one as well, and in fact, David Ullrich is an actual math professor at
Oklahoma State University, but notice, he doesn't offer the objection
raised by "C. Bond" in his post.

Well, I've demonstrated something that's possible with my discovery
that's impossible with any other so-called prime counting function.

That is, no one in recorded human history could even engage in the
exercise with a difference or partial difference equation that sums to
give a count of prime numbers before me.

Readers should remember that the fight here is with mathematicians
trying to dismiss my discovery, as I just want them to do their jobs
and record it somewhere. Then I'll be able to sell my story versus
being known as just some "crank".

So far I've given more than enough information to show that my
discovery is worth recording, while posters in reply have basically
just shown a stubborn desire to convince you that it's worthless.
I want readers to consider Ullrich's argument carefully, and remember,
he's an *actual* math professor.

Interesting assertion. But I'm showing details about my discovery
which show that it IS a unique first-find, and if Ullrich knew of a
partial difference equation or even a difference equation that counted
primes, then he could just give that in some attempted refutation.

Obviously, if I'm doing something that has been done before, or has
been possible before, he could just give an example, to show that what
I have isn't unique as I claim.

But he did not do that, and instead picked a difference equation that
has no relationship to prime numbers.

My point remains that *mathematicians* are here fighting against their
own claimed values by fighting to dismiss knowledge.

They did NOT have a partial difference equation like mine that sums to
give a count of primes that finds primes itself, and that leads to a
partial differential equation before my discovery.

If you let people like Ullrich get away with such clear lies then you
betray intellect itself, and the future of humanity.

After all, people like Ullrich who fight against the truth for their
own selfish reasons have been around for a long time, while the needs
of humanity for the truth to solve its problems have been around just
as long.

Who will you stand with, Ullrich or those before you who fought for
truth?

James Harris

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

James Harris, Dec 4, 2003
14. ### Dik T. WinterGuest

Indeed. And there is nothing in calculus that states that the solution
to the difference equation is in any way related to the solution of the
differential equation.

Dik T. Winter, Dec 4, 2003
15. ### David C. UllrichGuest

Fascinating. The fact that two people raise two different objections
is somehow in your favor. If a hundred people pointed out a hundred
different objections that would be conclusive proof you were right

Truly fascinating.
Um, first, you're changing the topic - the question was whether
you'd given an explanation for the connection between distribution
of primes and pi(x).

Second, no you're not the first person in history to do that.
The fact that you're able to type the words "you're lying"
infinitely many times does not change the fact that Legendre
beat you by about 200 years.
Again, changing the topic. Again, I and many people _have_ given
the refutation you ask for. And you're simply ignoring the point I
just made, which is that you have _no_ reason to think that the
solution to the thing that you insist on calling a pde has anything
to do with pi(x).
Exactly what is the significance of the words "leads to a partial
differential equation", _if_ the solution to the pde has nothing
whatever to do with counting primes?
Who _are_ these people you address these grand questions to?
************************

David C. Ullrich

David C. Ullrich, Dec 4, 2003
16. ### C. BondGuest

And?... If there are no comments here, then move on to portion *with* comments.
I stand by my remark that replacing unity in the original difference equation with a finite
difference treated as an infinitesimal will change the values you calculate and will not
produce correct values for the prime count unless you put the unity back in its place.

The attack was *specifically* aimed at your conclusion that you have produced a continuous
function which relates to prime counting. Comparing limiting processes using differentials
with finite differences is often done in a course on difference equations -- but there it is
done to show that the results *not* the same. Your claim is that you can replace a finite
difference with an infinitesimal and, in the limit, create a continuous function which solves
the same problem as the original difference equation. Not true.

Observe that James Harris, if that really is his name (I still think it might be an anagram
for "M. Harie Ass, Jr."), can only attack his critics by misrepresenting their position. His
repeated insinuation that not attacking the original finite difference equation props up the
faulty and misguided attempts to create an extension to continuous functions is bogus.
Your math is dead wrong. I am defending math and calculus against *your* pernicious attempts
to bulldoze over them. Instead of condemning me, you should be proving your case. After all,
my criticism was of the mathematics and your response should be to address the criticism.
You tell us. You're the one making and defending the errors.
Well, I certainly don't like the following part of your post:
But I don't like it because it *isn't* true. It is dead wrong. If your equation is, as you
claim, a partial differential equation, then it is an equation involving two or more
independent variables. Those variables appear to be 'x' and 'y'. The partial derivative of 'y'
wrt 'x' is then zero. Hence, p'_x(y,sqrt(y)), using your notation, equals zero. There is no
vector or directional derivative involved. Hence, p'_x(y,sqrt(y)) = 0.
The technique in question involves your so-called partial differential equation. You did not
use this technique in the original difference equation.

to prime counting. That is what I responded to. Your dissembling.

James, you would do much better in countering your critics if you addressed the same arguments
they criticized. Erecting a straw man, in this case your original difference equation, to wave
around when the criticism is elsewhere merely marks you as a charlatan or an individual with
severe Attention Deficit Disorder.

Produce results with your partial differential equation which prove that it is a solution of
the prime counting function. You have been asked repeatedly to support your claim with
evidence. You have produced no results and no evidence. I claim you will *not* do so, and that
you *cannot* do so, because you are *wrong*! If you want to refute my challenge, stick to the

C. Bond, Dec 4, 2003
17. ### Thomas Bushnell, BSGGuest

Well...there might be an intermediate value theorem application in
there somewhere. Thomas Bushnell, BSG, Dec 4, 2003
18. ### James HarrisGuest

OOPS! That's not correct as it sums to give a count of *composites*
which you then subtract to get a count of primes. Kind of a technical
point but certainly correctness and precision requires it be said a
different way.

James Harris

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

James Harris, Dec 4, 2003