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.