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RE: [PrimeNumbers] Arithmetic Progression?

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  • Matt Insall
    [Andy Swallow] This shows that any arithmetic progression contains infinitely many prime numbers. [Matt Insall] This is a strange statement. When I first saw
    Message 1 of 5 , Jul 24, 2003
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      [Andy Swallow]
      This
      shows that any arithmetic progression contains infinitely many prime
      numbers.

      [Matt Insall]
      This is a strange statement. When I first saw it, I did not believe it,
      so I looked up ``arithmetic progression''. On MathWorld, I found

      http://mathworld.wolfram.com/PrimeArithmeticProgression.html

      in which the term being defined is not ``arithmetic progression'' but
      ``prime arithmetic progression''. In this definition, a ``prime arithmetic
      progression'' is an arithmetic sequence of primes. Thus I looked up
      ``arithmetic sequence'', to see if what I recalled was correct. The
      url for the MathWorld definition is

      http://mathworld.wolfram.com/ArithmeticSequence.html

      Now, correct me if I am wrong, but don't the even numbers form an arithmetic
      sequence? If one calls arithmetic sequences by the alternate name
      ``arithmetic
      progressions'', then your statement, Andy, seems incorrect, because the
      sequence
      of even numbers has only one prime in it. On the other hand, if the
      definition
      of ``arithmetic progression'' requires that the numbers in the
      ``progression''
      be primes, then your statement seems a bit odd also. For there are known
      finite
      prime arithmetic progressions (see again the entry on MathWorld). That is,
      these are prime arithmetic progressions with only finitely many primes in
      them.
      Thus, your statement seems to need revision, but I do not know what revision
      does the trick. Clearly some arithmetic progressions have very few primes
      in them
      (e.g. the sequence of even numbers), and the prime arithmetic progressions
      that are known are typically quite short.
    • Paul Jobling
      Ah, but there is another condition : {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1. The even numbers do not follow this pattern, as
      Message 2 of 5 , Jul 24, 2003
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        Ah, but there is another condition :

        {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.

        The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.

        Look up Dirichlet's theorem on primes in arithmetic progressions, say here:
        http://primes.utm.edu/glossary/page.php?prev=divides

        (Phew - did Google index your pages backwards, Chris??? prev=???)

        Regards,

        Paul.


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      • Paul Jobling
        Oh, and one more thing - there is a difference between arithmetic progressions containing ONLY primes, of which the longest known has 22 members; and
        Message 3 of 5 , Jul 24, 2003
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          Oh, and one more thing - there is a difference between arithmetic progressions
          containing ONLY primes, of which the longest known has 22 members; and
          *infinite* arithmetic progressions that contain an infinite number of primes
          (and an infinite number of composites).

          > -----Original Message-----
          > From:
          > Sent: 24 July 2003 14:47
          > To: 'Matt Insall'; 'primenumbers@yahoogroups.com'
          > Subject: RE: [PrimeNumbers] Arithmetic Progression?
          >
          >
          > Ah, but there is another condition :
          >
          > {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.
          >
          > The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.
          >
          > Look up Dirichlet's theorem on primes in arithmetic
          > progressions, say here:
          > http://primes.utm.edu/glossary/page.php?prev=divides
          >
          > (Phew - did Google index your pages backwards, Chris??? prev=???)
          >
          > Regards,
          >
          > Paul.
          >


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        • Zak Seidov
          Really, dear and highly respectful NP gurus, you may confuse each other as much as you wish, but think about poor laypeople... Please anybody explain in plain
          Message 4 of 5 , Jul 24, 2003
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            Really,
            dear and highly respectful NP gurus,
            you may confuse each other
            as much as you wish,
            but think about poor laypeople...

            Please anybody explain in plain language,
            what a great truth is in the assertion
            that in a*m+b with gcd(a,b)=1
            there are infinitely more primes;
            (I guess that squares -
            which are infinetly more rarer than primes -
            are still infinetly many in AP -
            or not?)
            is it OK, that still primes are infinitely less than
            composite terms of AP.
            I mean is there any AP with primes which are not
            infinitely small part of all terms?
            Sorry for silly Qs,
            Zak
            I'm in no case aiming to be sarcastic
            believe or not

            Are you sure you want to send this message?
            - sure not

            --- In primenumbers@yahoogroups.com, "Paul Jobling"
            <Paul.Jobling@W...> wrote:
            > Oh, and one more thing - there is a difference between arithmetic
            progressions
            > containing ONLY primes, of which the longest known has 22 members;
            and
            > *infinite* arithmetic progressions that contain an infinite number
            of primes
            > (and an infinite number of composites).
            >
            > > -----Original Message-----
            > > From:
            > > Sent: 24 July 2003 14:47
            > > To: 'Matt Insall'; 'primenumbers@yahoogroups.com'
            > > Subject: RE: [PrimeNumbers] Arithmetic Progression?
            > >
            > >
            > > Ah, but there is another condition :
            > >
            > > {n, n+m, n+2m, etc} contains infinitely many primes IF gcd(n,m)=1.
            > >
            > > The even numbers do not follow this pattern, as gcd(2,2)=2 <>1.
            > >
            > > Look up Dirichlet's theorem on primes in arithmetic
            > > progressions, say here:
            > > http://primes.utm.edu/glossary/page.php?prev=divides
            > >
            > > (Phew - did Google index your pages backwards, Chris??? prev=???)
            > >
            > > Regards,
            > >
            > > Paul.
            > >
            >
            >
            > __________________________________________________
            > Virus checked by MessageLabs Virus Control Centre.
          • Andrew Swallow
            Umm well sorry Matt, I missed out the important bit! But, prime arithmetic progression? Arithmetic sequence? The most commonly used term is arithmetic
            Message 5 of 5 , Jul 24, 2003
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              Umm well sorry Matt, I missed out the important bit! But, prime
              arithmetic progression? Arithmetic sequence? The most commonly used
              term is 'arithmetic progression', whatever MathWorld may say. And this
              usually refers to the infinite sequence, not a finite one.

              Anyway, let a and q be any two integers, which we may assume are
              positive. Then the set {qn+a} where n runs from 0 to infinity is
              called an 'arithmetic progression', and can also be called an
              arithmetic sequence if you so wish. Define pi(x,a,q) as the number of
              primes p such that p<=x and p=a mod q. This function is clearly only
              of interest when q and a have no common factor, so we'll assume that
              (q,a)=1. Then we have the prime number theorem for arithmetic
              progressions:

              pi(x,a,q) ~ x/(phi(q)log x)

              where $A$ may be taken as large as we like, for sufficiently large x.
              The ~ symbol denotes an asymptotic relationship, but it is possible to
              be more explicit about the accuracy of the approximation. This is a
              little stronger than Dirichlet's theorem, of course.

              The original question was concerned with how much variation there is
              in pi(x,a,q) when a varies, and I don't know much about this, other
              than asymptotically the number of primes is the same for all a such
              that (a,q)=1.

              It's of far more interest to vary q (as well as a), and indeed the
              consideration of such variations is extremely important when studying
              approximations to the Goldbach/Twin prime problems.

              As for 'prime arithmetic progressions', there exist (presumably) such
              things of arbitrary length. Finding them is another matter, but that
              they exist seems a reasonable assumption.

              Andy
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