Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site faron.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!harpo!decvax!linus!faron!bs From: bs@faron.UUCP (Robert D. Silverman) Newsgroups: net.math Subject: palindromic primes Message-ID: <174@faron.UUCP> Date: Tue, 4-Dec-84 10:05:24 EST Article-I.D.: faron.174 Posted: Tue Dec 4 10:05:24 1984 Date-Received: Fri, 14-Dec-84 07:31:45 EST Organization: The MITRE Corp., Bedford, Ma. Lines: 68 The numbers comprised of all 1's are called repunits. A repunit of x digits is abbreviated as Rx. The repunits are prime for x = 2, 19, 23, 317, and 1031 and for no others < 2000. Tring to prove R19 and R23 prime by factoring them is like trying to hit a tack with a sledge hammer. The new Cohen-Lenstra algorithm is excellent but the older methods due to Brillhart, Lehmer, Selfridge, and Williams are much easier to program an quite effective up to about 50 digit numbers. See for example: "New Primality Criteria and Factorizations of 2^n +- 1" Brillhart et.al. Mathematics of Computation #29 (1975) "Some Algorithms for Prime Testing using Generalized Lucas Functions", Williams and Judd, Math. Comp. #30 (1976) "Factorizations of b^n +- 1 b = 2,3,5,6,7,10,11,12 up to high powers", Brillhart, Lehmer, Selfridge, Tuckerman, and Wagstaff, Contemporary Mathematics #22, American Mathematical Society 1983. Now for the final time here are the repunit primes in various bases: Base 2: 2^n - 1 is prime for n = 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,4423,9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, and 132049 Base 3: (3^n - 1)/2 is prime for n = 3,7,13,71,103,521 Base 5: (5^n - 1)/4 is prime for n = 3,7,11,13,47,127,149,181,619, and 929 Base 6: (6^n - 1)/5 is prime for n = 3,7,29,71,127,271, and 509 Base 7: (7^n - 1)/6 is prime for n = 5,13,131, and 149 Base 10: see above Base 11: (11^n -1)/10 is prime for n = 3,17,19, and 73 Base 12: (12^n - 1)/11 is prime for n = 3,5,19,97, and 109 If anyone knows of any others I'd like to hear about it.