Difference equations, calculus, prime numbers

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

  1. James Harris

    James Harris Guest

    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,
    so let's say you start with dx=1, then you have

    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

    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,

    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"
    James Harris, Dec 3, 2003
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  2. James Harris

    Mu-Pi Guest

    Yes, you have. Shut up already and adjust your medications.
    Mu-Pi, Dec 3, 2003
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  3. James Harris

    C. Bond Guest

    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
    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. 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

    Start with the difference equation

    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.


    David C. Ullrich
    David C. Ullrich, Dec 3, 2003
  5. James Harris

    Mu-Pi Guest

    He already reached that status years ago.
    Mu-Pi, Dec 3, 2003
  6. James Harris

    Mu-Pi Guest

    LOL! Well said David!
    Mu-Pi, Dec 3, 2003
  7. James Harris

    Doug Norris Guest

    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 Norris, Dec 3, 2003
  8. James Harris

    Mu-Pi Guest


    The proof goes as follows:
    For every epsilon about the point.....
    Mu-Pi, Dec 3, 2003
  9. James Harris

    Uncle Al Guest


    Idiot. It has been variously shown that you couldn't pull the lint
    out of your own bellybutton without causing bleeding. You know near
    nothing about mathematics, and pervert it to suit your psychosis.

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

    It's not every braying jackass that gets a whole page at crank.net
    Uncle Al, Dec 3, 2003
  10. ;-))

    Dirk Vdm
    Dirk Van de moortel, Dec 3, 2003
  11. James Harris

    Sam Wormley Guest

    Prime Counting Function

    Prime Difference Function

    Crank Information
    Sam Wormley, Dec 4, 2003
  12. James Harris

    James Harris Guest

    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

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

    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.

    If he'd had a problem with the technique, why not reply about the
    difference equation example I used?

    James Harris

    "My math discoveries, found for profit"
    James Harris, Dec 4, 2003
  13. James Harris

    James Harris Guest

    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

    James Harris

    "My math discoveries, found for profit"
    James Harris, Dec 4, 2003
  14. 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. 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
    about everything, eh?

    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. James Harris

    C. Bond Guest

    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.

    My complaint is about that 'partial differential equation' and your claims about its relevance
    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
    challenge. Post your proof and your results. Otherwise -- SHUT UP!
    C. Bond, Dec 4, 2003
  17. Well...there might be an intermediate value theorem application in
    there somewhere. :)
    Thomas Bushnell, BSG, Dec 4, 2003
  18. James Harris

    James Harris Guest

    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"
    James Harris, Dec 4, 2003
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