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The curious number 9839389.

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  • Max B
    9839389 = 7 * 43 * 97 * 337 Sum the primes between the smallest and largest prime factors: 7+11+13+...+331+337 = 10181, a prime. Sum the composites between the
    Message 1 of 3 , Nov 3, 2002
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      9839389 = 7 * 43 * 97 * 337

      Sum the primes between the smallest
      and largest prime factors:
      7+11+13+...+331+337 = 10181, a prime.

      Sum the composites between the
      smallest and largest prime factors:
      8+9+10+...+335+336 = 46751, a prime.

      Can anyone find larger palindromes
      like 9839389?

      "In mathematics you don't understand things.
      You just get used to them."
      --Johann von Neumann
    • richard_heylen
      ... Yes. They don t seem to be rare. The smallest is 15251=101*151 The sum of the primes 101+103+107+109+113+127+131+137+139+149+151=1367, a prime the sum of
      Message 2 of 3 , Nov 3, 2002
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        --- In primenumbers@y..., "Max B" <zen_ghost_floating@h...> wrote:

        > Sum the primes between the smallest
        > and largest prime factors:
        > 7+11+13+...+331+337 = 10181, a prime.
        >
        > Sum the composites between the
        > smallest and largest prime factors:
        > 8+9+10+...+335+336 = 46751, a prime.
        >
        > Can anyone find larger palindromes
        > like 9839389?

        Yes. They don't seem to be rare.
        The smallest is 15251=101*151
        The sum of the primes
        101+103+107+109+113+127+131+137+139+149+151=1367, a prime
        the sum of the composites
        (151^2-101^2+151+101)/2-1367=5059 a prime
        There are more than 300 such palindromes below 2^32

        The largest I can conveniently find is
        4273223724=2^2*3^2*7*11*467*3301
        where the sum of the primes up to and including 3301 is 700897 and
        the sum of the composites 4749053 are both primes.

        For large solutions to this problem I'd suggest sticking to even
        palindromes.
        You can divide the problem into two parts. Firstly find pairs of
        primes such that the sums of the primes and composites between them
        are both primes. Secondly, using the larger prime of the pair and
        quite a lot of thought you should be able to solve a degree 1 modular
        equation in quite a few variables to generate some candicates you can
        then check for smoothness. This could be tricky. One neat way of
        generating smooth palindromes is to look for palindromes with low
        digit values which are divisible by other, shorter low digit value
        palindromes. You can then add them together (with appropriate shifts)
        to generate more palindromes of which a large part of the
        factorisation is already known. For instance some of the common small
        largest factors of palindromes are 101, 9091 (divides 10^5+1), 9901
        (divides 10^6+1), 333667 (divides 1001001). Also 3637 but I don't
        know why.
        If you can find a large factor p of two low weight palindromes of
        different sizes such that the sum of the primes and composites from 2
        to p are prime then you're probably set for some really big solutions
        to this problem.

        Richard
      • Max B
        Using some of Richard s lucid tips below, and my usual brute-force blatant bricoleur approach (too clumsy to divulge here), I managed to find: 819870212078918
        Message 3 of 3 , Nov 4, 2002
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          Using some of Richard's lucid tips below, and my
          usual brute-force blatant bricoleur approach
          (too clumsy to divulge here), I managed to find:

          819870212078918 =
          2 * 7 * 11 * 13 * 17 * 17 * 10771 * 131561

          where the sum of primes from 2 to 131561 is
          766715749 and the sum of composites is
          7887498391, both primes.

          But I'm sure this solution could be easily eclipsed
          by the 'heavyweights' on this list.

          Thanks Richard!


          ----- Original Message -----
          From: richard_heylen
          To: primenumbers@yahoogroups.com
          Sent: Sunday, November 03, 2002 9:02 PM
          Subject: [PrimeNumbers] Re: The curious number 9839389.

          [...]

          The largest I can conveniently find is
          4273223724=2^2*3^2*7*11*467*3301
          where the sum of the primes up to and including 3301 is 700897 and
          the sum of the composites 4749053 are both primes.

          For large solutions to this problem I'd suggest sticking to even
          palindromes.
          You can divide the problem into two parts. Firstly find pairs of
          primes such that the sums of the primes and composites between them
          are both primes. Secondly, using the larger prime of the pair and
          quite a lot of thought you should be able to solve a degree 1 modular
          equation in quite a few variables to generate some candicates you can
          then check for smoothness. This could be tricky. One neat way of
          generating smooth palindromes is to look for palindromes with low
          digit values which are divisible by other, shorter low digit value
          palindromes. You can then add them together (with appropriate shifts)
          to generate more palindromes of which a large part of the
          factorisation is already known. For instance some of the common small
          largest factors of palindromes are 101, 9091 (divides 10^5+1), 9901
          (divides 10^6+1), 333667 (divides 1001001). Also 3637 but I don't
          know why.
          If you can find a large factor p of two low weight palindromes of
          different sizes such that the sum of the primes and composites from 2
          to p are prime then you're probably set for some really big solutions
          to this problem.

          Richard
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