## Message 4042 from Yahoo.Groups.Primeform

Return-Path: <d.broadhurst@...> X-Sender: d.broadhurst@... X-Apparently-To: primeform@yahoogroups.com Received: (qmail 97133 invoked from network); 12 Dec 2003 15:43:47 -0000 Received: from unknown (66.218.66.217) by m15.grp.scd.yahoo.com with QMQP; 12 Dec 2003 15:43:47 -0000 Received: from unknown (HELO n16.grp.scd.yahoo.com) (66.218.66.71) by mta2.grp.scd.yahoo.com with SMTP; 12 Dec 2003 15:43:46 -0000 Received: from [66.218.67.171] by n16.grp.scd.yahoo.com with NNFMP; 12 Dec 2003 15:43:40 -0000 Date: Fri, 12 Dec 2003 15:43:38 -0000 To: primeform@yahoogroups.com Subject: Palindromic prime with 36401 prime digits Message-ID: <brcnna+uhre@...> User-Agent: eGroups-EW/0.82 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Length: 1531 X-Mailer: Yahoo Groups Message Poster X-eGroups-Remote-IP: 66.218.66.71 From: "David Broadhurst" <d.broadhurst@...> X-Originating-IP: 212.56.75.172 X-Yahoo-Group-Post: member; u=35890005 X-Yahoo-Profile: djbroadhurstN = 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 http://physics.open.ac.uk/~dbroadhu/cert/p4546.zip and the proof of the primality of N is completed by http://physics.open.ac.uk/~dbroadhu/cert/kp36400.zip David Broadhurst