- Oct 5, 2001During 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 - Next post in topic >>