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Re: [PrimeNumbers] Digits strings for prime numbers

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  • Jack Brennen
    Regarding the possibility of an infinite string of binary digits, which contains an infinite number of 1 digits, and which yields only a single two-bit prime
    Message 1 of 2 , Mar 29, 2003
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      Regarding the possibility of an infinite string of binary digits,
      which contains an infinite number of '1' digits, and which
      yields only a single two-bit prime among all leading substrings...
      This is easy to prove existence.

      Effectively, one needs only to prove that for any integer k, there is
      a composite number of the form k*2^n+1, for some n>=1.

      Such a proof is easy:

      Let A=k*2^n+1. If A is composite, we're done. So assume A is prime.

      Let x be the order of 2 mod A.

      Let B=k*2^(n+x)+1. B is divisible by A, and is thus composite.
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