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14506RE: Primes in the concatenation with the digits of Pi

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  • cino hilliard
    Feb 2, 2004
      I have been investigating some old buddies in the realm of prime numbers. If
      we take for example
      the first 999 digits of Pi and append that to the sequence of numbers below
      we will get a greater
      than 1000 digit pseudoprime. Pari code is at the end to generate these and
      the pprimes thenselves.


      Notice 2623,2635. What, a kind of twin prime? Are there more of these?

      Dirichlet proved in 1837 that every arithmetic progression kn + a where
      (k,a)=1, n=1,2,3,..
      contains an infinite number of primes. The Full proof can be found in
      Lectures on Elementary
      Number Theory by Hans Rademacher pp121 - 135.

      If we set k = 10^999 and a = 314159...4209. then the terms in the above
      sequence are infinite.

      If we set n=1,k=1 we have the progression 1+a. So for a=1,2,3,..
      Dirichlet's theorem proves
      the infinity of primes per se.

      My question is this. Why can't we set n = 0 in the Dirichlet theorem once
      proved? Dirichlet used
      zeta(s) = Sum(n=1..infinity,1/n^s) in his proof. This restricts n=0 unless
      of course we let s=0 and
      this would mean zeta(s) = infinity which dosen't tell us much or too much
      depending on how you
      look at it. Nevertheless, the finished product, kn + a will accept n=0
      without restriction so the
      progression of integers 0 + a = 1,2,3... contain an infinite number of
      primes. I don't see a catch 22
      here since "that which is true is true and cannot be changed." But...

      Pari Code
      f(n) =

      Have fun in the facinating world of numbers,

      "Behavior is not for the pursuit of survival but because of it."

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