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Wieferich primes

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  • John McNamara
    This post is about the Wieferich primes, 1093 and 3511. (1) Concerning the two in concert producing a prime P (2) Concerning what I have found to be an
    Message 1 of 5 , Sep 4, 2001
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      This post is about the Wieferich primes, 1093 and
      3511.
      (1) Concerning the two in concert producing a prime P
      (2) Concerning what I have found to be an interesting
      composite for P+1 and possibilities for primes close
      to it
      (3) Concerning 3511 itself, a Wieferich prime.

      (1) Let f(x,y) = x^2 - y^2 -2xy.
      f(3511,1093) = 7294949 and is prime P.

      (2) 7294950 = 2*3*3*5*5*13*29*43 which made it easy to
      factorise into its 8 factors, of which 5 are less than
      (P+1)^(1/8) and all less than (P+1)^(1/4). This fact
      seem to me to make 7294950 fairly interesting as only
      6435 previous composites have the property of having 8
      prime factors, one in more than a thousand.

      (3) f(79,26) = 3511 itself remarkable as 79 = 3*26+1
      which makes it a "near" Lucas series with its
      characteristic second term 79 = 1mod3 and 1mod(the
      first term viz. 26 in this case).
      (I have postulated that any prime 1mod10 or -1mod10 is
      always expressible with x > 3y in f(x,y), and that x
      is the second term of a Fibonacci series T(1) = y,
      T(2) = x.)

      I have more interesting material but unless this
      creates interest, I will spare you it.
      John McNamara


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    • d.broadhurst@open.ac.uk
      ... The problem is that given two numbers one can find any number of features for them. The important thing is, does any of your message help the search for
      Message 2 of 5 , Sep 4, 2001
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        John McNamara wrote:
        > This post is about the Wieferich primes, 1093 and 3511.
        The problem is that given two numbers
        one can find any number of features for them.
        The important thing is, does any of your message help
        the search for the next Wieferich prime, above 46 trillion:
        http://www.loria.fr/~zimmerma/records/Wieferich.status
      • Phil Carmody
        ... There s a distributed hunt for them ongoing. http://catalan.ensor.org/ The client source is available, so anyone with a compiler can join. My alpha got to
        Message 3 of 5 , Sep 5, 2001
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          On Tue, 04 September 2001, d.broadhurst@... wrote:
          > John McNamara wrote:
          > > This post is about the Wieferich primes, 1093 and 3511.
          > The problem is that given two numbers
          > one can find any number of features for them.
          > The important thing is, does any of your message help
          > the search for the next Wieferich prime, above 46 trillion:
          > http://www.loria.fr/~zimmerma/records/Wieferich.status

          There's a distributed hunt for them ongoing.
          http://catalan.ensor.org/
          The client source is available, so anyone with a compiler can join. My alpha got to ~#5 on their tables a short while back.

          Phil

          Mathematics should not have to involve martyrdom;
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        • torasso.flavio@enel.it
          ... Notice that (1) should be read f(x,y) = x^2 - y^2 -xy to get f(3511,1093) = 7294949. Assuming this, other prime pairs (x,y) exist that satisfy f(x,y) = P
          Message 4 of 5 , Sep 5, 2001
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            --- In primenumbers@y..., John McNamara <mistermac39@y...> wrote:
            > (1) Let f(x,y) = x^2 - y^2 -2xy.
            > f(3511,1093) = 7294949 and is prime P.
            >
            > (2) 7294950 = 2*3*3*5*5*13*29*43 which made it easy to
            > factorise into its 8 factors, of which 5 are less than
            > (P+1)^(1/8) and all less than (P+1)^(1/4). This fact
            > seem to me to make 7294950 fairly interesting as only
            > 6435 previous composites have the property of having 8
            > prime factors, one in more than a thousand.

            Notice that (1) should be read f(x,y) = x^2 - y^2 -xy
            to get f(3511,1093) = 7294949.

            Assuming this, other prime pairs (x,y) exist that
            satisfy f(x,y) = P (prime) and P+1 have the property
            of having 8 prime factors.
            As an example, the prime pair (47,2) gives
            P=f(47,2)=2111 prime and P+1=2*2*2*2*2*2*3*11.

            Regards
            Flavio Torasso
          • d.broadhurst@open.ac.uk
            ... That s recently gone up to 150 trillion, and climbing fast: http://www.spacefire.com/numbertheory/wieferich.htm
            Message 5 of 5 , Sep 5, 2001
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              I wrote:
              > the search for the next Wieferich prime, above 46 trillion
              That's recently gone up to 150 trillion, and climbing fast:
              http://www.spacefire.com/numbertheory/wieferich.htm
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