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Math Help - What is pi(x)?

  1. #1
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    What is pi(x)?

    I was looking into prime numbers and i keep coming upon pi(x). Specifically How many primes are there?. It is obvious that they arent multiplying pi*x. So what are they doing? Im only in trig so please explain it in basic terms.

    Thanks!
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  2. #2
    o_O
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    If you scroll down a bit, it defines the function for you ...

    \pi (x) = The number of primes less than or equal to x

    So for example, \pi (10) = 4 because there are 4 primes less than or equal to 10 (2, 3, 5, and 7).
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  3. #3
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    Quote Originally Posted by o_O View Post
    If you scroll down a bit, it defines the function for you ...

    \pi (x) = The number of primes less than or equal to x

    So for example, \pi (10) = 4 because there are 4 primes less than or equal to 10 (2, 3, 5, and 7).
    Yes, however I want to be able to calculate how many prime numbers there are below a set number.

    Such as how would I go about finding how many primes are below 46,000?
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    Grand Panjandrum
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    Quote Originally Posted by Leelguy View Post
    Yes, however I want to be able to calculate how many prime numbers there are below a set number.

    Such as how would I go about finding how many primes are below 46,000?
    You could find them all and count them, other than that you will need one of the approximations for \pi(x) , look at the Wikipedia page for a start.

    (by the way \pi(46000)=4761)

    CB
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  5. #5
    Super Member redsoxfan325's Avatar
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    Li(x) is the best approximation and it has been proven that it asymptotically approaches pi(x), however Littlewood proved (in 1914?) that while for small n (n < 10^300) Li(x)-pi(x)>0, Li(x)-pi(x) changes signs infinitely many times as n --> infinity.


    Maple has the Li function built in.
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    Quote Originally Posted by CaptainBlack View Post
    You could find them all and count them, other than that you will need one of the approximations for \pi(x) , look at the Wikipedia page for a start.

    (by the way \pi(46000)=4761)

    CB
    Yes i have looked at that article and I know that it contains what I am looking for, however I havent taken calculus and I dont understand how they are displaying the formulas. Could you explain it in a way that someone in trig could understand it?

    You say that " \pi(46000)=4761", but what steps did you go through to get that? I know that it isnt pi*46000.
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  7. #7
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Leelguy View Post
    Yes i have looked at that article and I know that it contains what I am looking for, however I havent taken calculus and I dont understand how they are displaying the formulas. Could you explain it in a way that someone in trig could understand it?

    You say that " \pi(46000)=4761", but what steps did you go through to get that? I know that it isnt pi*46000.
    \pi(46000)=4761 isn't something you can calculate exactly with a formula. You can only approximate it using the log integral, Li(n) ( Li(46000)=4795, which is only a 0.7% error), and the larger the number, the more accurate (percentage-wise) the approximation will be. For example, Li(100)=30 and \pi(100)=25, for a 17% error. Li(1000)=178 and \pi(1000)=168, for a 5% error. Li(1000000)=78628 and \pi(1000000)=78498, for a 0.2% error. All of those calculations were done using Maple 12. The command for finding \pi(n) in Maple is "nops(select(isprime, [`$`(1..n)]))". Li(n) in Maple is just "Li(n)".

    If you don't understand calculus, you won't understand the meaning behind the log integral, Li(n). Just know it as the best approximation for \pi(n). There is no particular reason for using the notation \pi(n) either. There are only a limited number of letters, and they have to be recycled eventually.
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    Quote Originally Posted by redsoxfan325 View Post
    \pi(46000)=4761 isn't something you can calculate exactly with a formula. You can only approximate it using the log integral, Li(n) ( Li(46000)=4795, which is only a 0.7% error), and the larger the number, the more accurate (percentage-wise) the approximation will be. For example, Li(100)=30 and \pi(100)=25, for a 17% error. Li(1000)=178 and \pi(1000)=168, for a 5% error. Li(1000000)=78628 and \pi(1000000)=78498, for a 0.2% error. All of those calculations were done using Maple 12. The command for finding \pi(n) in Maple is "nops(select(isprime, [`$`(1..n)]))". Li(n) in Maple is just "Li(n)".

    If you don't understand calculus, you won't understand the meaning behind the log integral, Li(n). Just know it as the best approximation for \pi(n). There is no particular reason for using the notation \pi(n) either. There are only a limited number of letters, and they have to be recycled eventually.
    I actually have to recreate this method for finding how many primes are in a number for a programming class. Because of that, using Maple wont help me. Isnt there a mathematical formula of some sorts that I could use to find the amount of primes in a number?
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    Super Member redsoxfan325's Avatar
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    The following is NOT HTML code. It was the only way I could get the forum to preserve the indentations needed for Python, which is the language this is written in.

    [html]primes = [2]
    p=1
    i=3
    n=input("Up to what number? ")
    while i<=n:
    for j in primes:
    if i%j==0:
    break
    else:
    if j==primes[len(primes)-1]:
    primes.append(i)
    p=p+1
    i=i+1
    print ""
    print "Number of Primes: " + `p`[/html]

    This code is not very efficient (it takes almost 1 minute for n=100000, compared to about 3 seconds on Maple), but it should at least point you in the right direction.
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    Quote Originally Posted by redsoxfan325 View Post
    The following is NOT HTML code. It was the only way I could get the forum to preserve the indentations needed for Python, which is the language this is written in.

    [html]primes = [2]
    p=1
    i=3
    n=input("Up to what number? ")
    while i<=n:
    for j in primes:
    if i%j==0:
    break
    else:
    if j==primes[len(primes)-1]:
    primes.append(i)
    p=p+1
    i=i+1
    print ""
    print "Number of Primes: " + `p`[/html]This code is not very efficient (it takes almost 1 minute for n=100000, compared to about 3 seconds on Maple), but it should at least point you in the right direction.
    But it appears that you are just looping through counting each prime. This makes it very inefficient for counting large numbers, and multiple ones at a time (which is what im doing). I had thought that there was a formula used to find the number of prime numbers?
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  11. #11
    Super Member redsoxfan325's Avatar
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    I'm not too great at programming, but I would suggest that you check out the Sieve of Eratosthenes. It's supposedly efficient for n<10,000,000.
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  12. #12
    Grand Panjandrum
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    Quote Originally Posted by Leelguy View Post
    Yes i have looked at that article and I know that it contains what I am looking for, however I havent taken calculus and I dont understand how they are displaying the formulas. Could you explain it in a way that someone in trig could understand it?

    You say that " \pi(46000)=4761", but what steps did you go through to get that? I know that it isnt pi*46000.
    As I said before it is the number of primes less than or equal to 46000, and as I suggested I counted them.

    My Euler code to find the primes less than or equal to some limit is:

    Code:
    function primes(n)
    ##
    ##  function returns a row vector containing the primes
    ##  less than or equal n 
    ##
    ##  (c) Xxx Xxxxxx 1997 rewritten 2004 using MATLAB seive alg
    ##
    
    
      if size(n)!=[1,1]
        "n must be a scalar"
        rv=[];
        return rv;
      endif
    
      p=[1:2:n];
      q=length(p);
      p(1)=2;
    
      for k=3 to sqrt(n) step 2
        if p((k+1)/2)
          p(((k*k+1)/2):k:q)=0;
        endif
      end;
    
      rv=p(nonzeros(p>0));
      return rv;
    
    endfunction
    Then to find the number one just uses:

    Code:
    pp=primes(n);
    np=length(pp)
    CB
    Last edited by CaptainBlack; March 1st 2009 at 01:24 AM.
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