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3114Base 10 database, R(13860), R(15600) and etc., etc.

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  • Harvey Dubner
    Oct 5, 2001
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      During the last several weeks there have been many messages about a Dubner
      factor database, finding factors of R(13860) (a repunit), primality proving
      of palindromes, and other related topics. It is now time to give a
      detailed
      explanation about what has been going on. It is hard to brag about good
      results when almost no one knows what you are talking about.

      I got my first home computer in 1980. My first project was to look for
      repunit primes, and this was before I knew what a repunit was. This was
      the
      time when the Cunningham project was picking up steam, and new factoring
      methods were appearing regularly. I assisted in proving that the repunit
      R(1031) (1031 1's) was truly prime, and I still think this is the single
      most interesting Titanic prime. Searching for Titanic primes also became
      fashionable and I joined in.

      R(n) = (10^n-1)/9 . It is easy to show that the prime factors of R(n)-1
      are
      factors of R(n-1) and are all of the form, (10^s +/- 1), that is they are
      base 10
      factors as described in the Cunningham project. It seemed that I kept
      running into base 10 factors, so without any master plan I started
      collecting base 10 factors. I then found that I could make
      interestingTitanic primes that could be proved prime by using the base 10
      factors that I had been collecting. For example,

      5*R(12600)*10^68+1 12668 digits 1997 Most leading 5's
      1802616208*10*R(12600)/R(10) + 1 12601 digits 1997 Palindrome
      110101*10*R(10080)/R(6)+1 10081 digits 1997 4-way Palindrome

      These could be proved prime since R(12600) and R(10080) were over 33.33%
      factored. This was in 1997. The next largest repunit that was reasonably
      close to being 33.33% factored was 13860 which was about 29.4% factored.
      Virtually no improvement was made until a few months ago. Then several
      things happened.

      First, the 33.33% factoring requirement was lowered by theory to 30%.

      Second, I began to appreciate the resource that the Primenumber group
      represented. They had outstanding capability for primality proving and
      factoring, and individually they were actively ready to help each other. I
      asked for help from David Broadhurst (I had previously gotten help on other
      jobs from Hans and Bouk and I didn't want to overload them, probably an
      unnecessary precaution). David asked if I would make my base 10 database
      available (of course I did) which he set up as a Primenumber file and he
      distributed the 17 composites of R13860 that needed factoring.

      David then examined the base 10 files and found other 30% candidates. In
      particular, 15600 was about 16% factored but included a 1914-digit prp that
      was "easily" proved prime using Primo making 15600 over 29% factored.
      However I had never paid attention to 15600 so that there were many
      relatively untouched composites and only 214 more digits were needed.
      David
      and I rapidly found enough factors and then found six palindromic primes
      based on 15600. This all happened so fast that we never had time to tell
      the group about this and we didn't really want to interrupt the factoring
      efforts on 13860. Incidently, 13440 also became more than 33.33%
      factored.
      Thus 13440, 13800 and 15600 all became resources for creating large,
      interesting primes because of the efforts of many individuals.

      In a matter of weeks more progress was made than in the last five years.
      This was only possible because of the dedicated efforts of the Primenumber
      group. In particular we would like to thank ----- unfortunately, no
      fortunately, the list became so large that it overwhelmed the message. Let
      me just send a big Thank You to everyone.

      Hopefully this effort will continue. I am preparing a list of candidates
      larger than 15600 that should be investigated but more about this later.

      I would be happy to answer any questions either on or off list.

      Harvey
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