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4 fun Mersenne-divisors theorems

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  • leavemsg1
    Hi, Group. ... (all of them can be proved similar to this one) ... 1. theorem: if k 1 and p= 4k +1 is prime , then 2*p +1 is also prime iff 2*p +1 divides
    Message 1 of 1 , Nov 15, 2011
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      Hi, Group.
      ...
      (all of them can be proved similar to this one)
      ...
      1. theorem: if k >1 and p= 4k +1 is 'prime', then 2*p +1 is
      also 'prime' iff 2*p +1 divides Mp +2; a 'complimentary' proof
      to that of Euler's result for mersenne divisors [shown by modi-
      fying Lagrange's proof of the original if p= 4k +3,...].
      ...
      proof(forward): suppose that q= 2*p +1 is prime; q= 3 (mod 8)
      and 2 is a quadratic residue modulo q, and there exists an
      integer 'n' such that n^2= 4 (mod q). Thus, 2^p = 2^((q-1)/2)
      = n^(q-1) = -1 (mod q); adding 2 to each portion of this equa-
      tion shows that (2*p +1) | (Mp +2).
      ...
      proof(backward): let 2*p +1 be a composite factor of Mp +2 &
      let 'q' be the least prime factor; 2^p= 1 (mod q) and the order
      of 2 divides both p and q-1; so, the conclusion is the same as
      the original Lagrange proof, (2*p +1) +1 > q^2 > p^2 contradicts
      that p> 2.
      *QED
      ...
      2. if p= 4k +3 is 'prime', then 2*p +1 is also 'prime' iff 2*p +1
      divides Mp, or 2^p== 1 mod (2p +1).
      ...
      3. if k >1 and p= 4k +1 is 'prime', then 2*p +3 is also 'prime'
      iff 2*p +3 divides Mp -p, or 2^p -p==1 mod (2p +3).
      ...
      4. if p= 4k +3 is 'prime', then 2*p +3 is also 'prime' iff 2*p +3
      divides Mp -p -1, or 2^p -p == 2 mod (2p +3).
      ...
      Rewards,
      ...
      Bill
      (they are posted at my website... oddperfectnumbersc.com)
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