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2526Wieferich primes

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  • John McNamara
    Sep 4, 2001
      This post is about the Wieferich primes, 1093 and
      (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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