We have found a Konyagin-Pomerance proof of primality of
the 15601-digit base-10 palindrome N = (1989191989)15601
1) Larger palindromic primes have been proven by applying
Brillhart-Lehmer-Selfridge (BLS) tests to numbers such as
but this method appears to be limited to base-10
strings consisting almost entirely of 9's or 0's.
2) It is considerably more demanding to prove gigantic
palindromic primes of a more varied form. The previous
record for a Konyagin-Pomerance (KP) proof was set by
(1579393975)13861, recently proven by collaboration
of members of the PrimeNumbers e-group.
3) For several years one of us (HD) has kept a database
of prime factors of primitive parts of 10^n +/- 1. Until
very recently this recorded that only 16.36% of the digits
of 10^15600-1 had been factorized into proven primes.
4) We noted that the 1914-digit probable prime
prp1914 = Phi5200(10)/5990401
was now easy to prove, thanks to Marcel Martin's Primo.
In the event, it took less than a day on a 1GHz machine.
That left us 214 digits short of the 30% threshold, required
by KP. This gap was made up by renewed P-1 and ECM efforts.
5) While not sufficient, the BLS tests are necessary.
There were done, with great efficiency, by OpenPfgw.
6) Pari/gp was used for the final cubic test.
All elements of the proof were checked by Greg Childers.
7) Now that one has 30% of the digits of 10^15600-1, it is
straightforward to generate and prove further palindromic
primes between 10^15600 and 2*10^15600. For example
and 10787026001 were proven by the same method. However,
to progress to significantly larger palindromes of such
a varied form, more factorization effort would be required.
We thank our colleagues in the PrimeNumbers, PrimeForm
and OpenPfgw e-groups, and in particular Greg Childers,
Jim Fougeron, Marcel Martin and Chris Nash.
David Broadhurst and Harvey Dubner