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Patterns in prime numbers (themselves, not their frequency)

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  • isbat1
    I m interested in the idea that a number can be ruled out as a prime number due to patterns occuring within the digits of that number. For example, you might
    Message 1 of 4 , Aug 4, 2004
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      I'm interested in the idea that a number can be ruled out as a prime
      number due to patterns occuring within the digits of that number.
      For example, you might say, "a number containing five nines in a row
      can't (or likely won't) be a prime number, for this reason: ...",
      or "any number containing the sequence '1-2-3-4' followed by that
      same sequence again won't be prime, because: ..."

      Are there any rules which describe prime numbers in such a way?
    • Edwin Clark
      ... No, let N be any number then there is a prime containing a consecutive subsequence of digits equal to N. One way to show this is as follows: Let N =
      Message 2 of 4 , Aug 4, 2004
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        On Wed, 4 Aug 2004, isbat1 wrote:

        > I'm interested in the idea that a number can be ruled out as a prime
        > number due to patterns occuring within the digits of that number. 
        > For example, you might say, "a number containing five nines in a row
        > can't (or likely won't) be a prime number, for this reason: ...",
        > or "any number containing the sequence '1-2-3-4' followed by that
        > same sequence again won't be prime, because:  ..."
        >
        > Are there any rules which describe prime numbers in such a way?
        >

        No, let N be any number then there is a prime containing a consecutive
        subsequence of digits equal to N. One way to show this is as follows:
        Let N = 921024..4. Now add the digit 1 to N to get M = 921024..41. Now
        gcd(M,10^n) = 1. Choose n so that 10^n > M. Then k10^n + M will "contain"
        M as least significant digits for all positive integers k. There is a
        theorem due to Dirichlet which implies that there are infinitely many
        primes of the form k10^n + M where n and M are fixed. See
        http://mathworld.wolfram.com/DirichletsTheorem.html
        for a statement of Dirichlet's Theorem.
      • Edwin Clark
        ... Here s an example constructed using Maple: Start with the number M. Just make sure its least significant digit is not 0, 2 or 5. ...
        Message 3 of 4 , Aug 4, 2004
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          On Wed, 4 Aug 2004, Edwin Clark wrote:

          > On Wed, 4 Aug 2004, isbat1 wrote:
          >
          > > I'm interested in the idea that a number can be ruled out as a prime
          > > number due to patterns occuring within the digits of that number. 
          > > For example, you might say, "a number containing five nines in a row
          > > can't (or likely won't) be a prime number, for this reason: ...",
          > > or "any number containing the sequence '1-2-3-4' followed by that
          > > same sequence again won't be prime, because:  ..."
          > >
          > > Are there any rules which describe prime numbers in such a way?
          > >
          >
          > No, let N be any number then there is a prime containing a consecutive
          > subsequence of digits equal to N. One way to show this is as follows:
          > Let N = 921024..4. Now add the digit 1 to N to get M = 921024..41. Now
          > gcd(M,10^n) = 1. Choose n so that 10^n > M. Then k10^n + M will "contain"
          > M as least significant digits for all positive integers k. There is a
          > theorem due to Dirichlet which implies that there are infinitely many
          > primes of the form k10^n + M where n and M are fixed. See
          >   http://mathworld.wolfram.com/DirichletsTheorem.html
          > for a statement of Dirichlet's Theorem.
          >

          Here's an example constructed using Maple:

          Start with the number M. Just make sure its least significant digit is not
          0, 2 or 5.

          > M:=1111122222333330000004444444441:
          > for i from 1 do if 10^i > M then n:=i; break; fi;od:
          > print(M,n);
          1111122222333330000004444444441, 31

          Now we generate lots of primes of the form k*10^n + M:

          > for k from 1 to 1000 do
          > p:=k*10^n+M;
          > if isprime(p) then print(k,p); fi;
          > od:

          24, 241111122222333330000004444444441


          46, 461111122222333330000004444444441


          48, 481111122222333330000004444444441


          144, 1441111122222333330000004444444441


          157, 1571111122222333330000004444444441


          168, 1681111122222333330000004444444441


          199, 1991111122222333330000004444444441


          228, 2281111122222333330000004444444441


          232, 2321111122222333330000004444444441


          238, 2381111122222333330000004444444441


          261, 2611111122222333330000004444444441


          277, 2771111122222333330000004444444441


          282, 2821111122222333330000004444444441


          301, 3011111122222333330000004444444441


          310, 3101111122222333330000004444444441


          312, 3121111122222333330000004444444441


          337, 3371111122222333330000004444444441


          370, 3701111122222333330000004444444441


          385, 3851111122222333330000004444444441


          403, 4031111122222333330000004444444441


          415, 4151111122222333330000004444444441


          421, 4211111122222333330000004444444441


          454, 4541111122222333330000004444444441


          493, 4931111122222333330000004444444441


          522, 5221111122222333330000004444444441


          532, 5321111122222333330000004444444441


          598, 5981111122222333330000004444444441


          622, 6221111122222333330000004444444441


          652, 6521111122222333330000004444444441


          693, 6931111122222333330000004444444441


          729, 7291111122222333330000004444444441


          772, 7721111122222333330000004444444441


          778, 7781111122222333330000004444444441


          781, 7811111122222333330000004444444441


          825, 8251111122222333330000004444444441


          826, 8261111122222333330000004444444441


          862, 8621111122222333330000004444444441


          888, 8881111122222333330000004444444441


          940, 9401111122222333330000004444444441


          946, 9461111122222333330000004444444441


          996, 9961111122222333330000004444444441

          >
        • Edwin Clark
          Correction to my previous example. The example is okay, I just should have
          Message 4 of 4 , Aug 4, 2004
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            Correction to my previous example. The example is okay, I just should have
            said:

            > Start with the number M. Just make sure its least significant digit is not
            > 0,2,4,5,6,or 8, so that gcd(M,10^n) = 1.
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