```Message 4042 from Yahoo.Groups.Primeform
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Date: Fri, 12 Dec 2003 15:43:38 -0000
To: primeform@yahoogroups.com
Subject: Palindromic prime with 36401 prime digits
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N = 34*R(36400)-42000040044444004000024*10^2264*R(36400)/R(4550)-1
is a palindromic prime all of whose 36401 decimal digits are prime.

The previous record for a prime-digit palindromic prime was set in
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0308&L=nmbrthry&P=R61
at 30931 digits.

To prove that N is prime, I proceeded as follows.

1) Primo was used to prove that
p4546 = (34*(10^4550-1)/9-42000040044444004000024*10^2264)/38834
is prime.

2) 10^36400-1 is divisible by a pair of titanic primes, namely
p1914 = Phi(5200,10)/5990401
and
p1440 = gcd(Phi(9100,10),\
10^(4*455)+5*10^(3*455)+7*10^(2*455)+5*10^455+1+\
10^228*(10^(3*455)+2*10^(2*455)+2*10^455+1))
which were also proven by Primo.

3) Combining these 3 prime factors of N+1 with 114 smaller
proven primes, to form an OpenPfgw helper file, one obtains

> [N+1, Brillhart-Lehmer-Selfridge]
> Reading factors from helper file kp36400.fac
> Running N+1 test using discriminant 5, base 5+sqrt(5)
> Calling Brillhart-Lehmer-Selfridge with factored part 30.03%
> 34*R(36400)-42000040044444004000024*10^2264*R(36400)/R(4550)-1
> is Lucas PRP! (6681.1182s+0.2175s)

4) With those BLS tests validating a factorization percentage
in excess of 30%, the proof was completed by 11 square tests
and 2 cubic tests, using a Konyagin-Pomerance method.

The proof of the primality of p4546 may be found in