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Re: Help with Twin Primes

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  • jbrennen
    ... a ... I assume that you mean all twin prime pairs (p, p+2) such that p+1 is a perfect number. If that is your intent, you found the only such pair
    Message 1 of 4 , Apr 1, 2002
      --- In primenumbers@y..., "math4life2000" <Math4life@a...> wrote:
      > I am looking for all twin primes such that their arithmetic mean is
      a
      > perfect number.
      >
      > For example:
      >
      > If 5 and 7 are our primes then 6 is our arithmetic mean.
      >
      > Thank you for your help.

      I assume that you mean all twin prime pairs (p, p+2) such that
      p+1 is a perfect number. If that is your intent, you found the
      only such pair already. All perfect numbers greater than 6 are
      congruent to 1 (modulo 3), meaning that the integer immediately
      before the perfect number is divisible by 3 (making it non-prime).

      On the other hand, if the two primes don't have to be part of
      a single twin prime pair, there could be many solutions. The next
      solution after (5,7) would be (13,43).
    • jbrennen
      ... Before somebody corrects me on this, let me rephrase: All EVEN perfect numbers greater than 6 are congruent to 1 (modulo 3). Obviously, an odd perfect
      Message 2 of 4 , Apr 1, 2002
        --- In primenumbers@y..., "jbrennen" <jack@b...> wrote:

        > All perfect numbers greater than 6 are congruent to 1 (modulo 3)

        Before somebody corrects me on this, let me rephrase:

        All EVEN perfect numbers greater than 6 are congruent
        to 1 (modulo 3).

        Obviously, an odd perfect number (if such exists) cannot be the
        number in the middle of a twin prime pair.

        As a side note, there are probably no odd perfect numbers, so
        the original statement was probably true, but until somebody
        proves the nonexistence of odd perfect numbers, I must watch
        myself :-)
      • Steven Whitaker
        ... There aren t any others. [Apologies to the real mathematicians for the standard of the proof. I hope the logic is clear] Given that these 3 numbers must be
        Message 3 of 4 , Apr 1, 2002
          On Mon, 01 Apr 2002 21:38:27 -0000, you wrote:

          >I am looking for all twin primes such that their arithmetic mean is a
          >perfect number.
          >
          >For example:
          >
          >If 5 and 7 are our primes then 6 is our arithmetic mean.
          >
          >Thank you for your help.


          There aren't any others.

          [Apologies to the real mathematicians for the standard of the proof. I
          hope the logic is clear]

          Given that these 3 numbers must be adjacent integers, then one of them
          must be divisible by 3. Neither of the primes is 3 (because no number
          adjacent to this is perfect), so it must be the perfect number that is
          divisible by 3. It must be an even perfect number (because otherwise the
          primes would both be even) and is therefore the product of a power of
          two and a Mersenne prime. Obviously a power of two is not divisible by
          three and so the Mersenne prime must be 3. And the only perfect number
          of this form is 6, which gives you the solution you already have.

          Regards
          Steve nice but dim
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