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

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• 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
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
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