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Help with Twin Primes

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  • math4life2000
    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.
    Message 1 of 4 , Apr 1, 2002
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      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.
    • 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 2 of 4 , Apr 1, 2002
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        --- 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 3 of 4 , Apr 1, 2002
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          --- 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 4 of 4 , Apr 1, 2002
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            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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