mdpl@allegra.UUCP (Mary Diane Palmer Leland) (11/15/84)
100656001 is a prime with 9 digits, so the conjecture that the number of digits in a palindromic prime is prime is false. However, it is easy to show that any palindromic number with an even number of digits is a multiple of 11. Thus, 11 is the only palindromic prime with an even number of digits. Mary Leland AT&T Bell Labs
bs@faron.UUCP (Robert D. Silverman) (11/16/84)
The number of palindromic primes is infinite. This follows from the fact the the class of automorphic binary quadratic forms is also infinite. See for example Dickson, Theory of Numbers pp. 111-115, Dover Press . It is easy to see that the number of digits must be odd because a palindrome of 2N digits will be a multiple of 10**N + 1. That the number of digits does not have to be prime is also easy to see by the example 131111131, which has 9 digits, yet is prime.
bs@faron.UUCP (Robert D. Silverman) (11/19/84)
Currently, the largest known palindromic prime is R1031, which is (10^1031-1)/9. It is a string of 1031 ones. It was just recently proved prime by Hugh Williams at the University of Manitoba. The proof utilised a recent factorization by Atkins and Rickert at the Univerisity of Illinois of 10^103+1. It has been known to be a probable prime for quite some time. R317 is also prime.
gjk@talcott.UUCP (Greg J Kuperberg) (11/20/84)
> > Currently, the largest known palindromic prime is R1031, which > is (10^1031-1)/9. It is a string of 1031 ones. It was just recently > proved prime by Hugh Williams at the University of Manitoba. The > proof utilised a recent factorization by Atkins and Rickert at > the Univerisity of Illinois of 10^103+1. It has been known to be > a probable prime for quite some time. R317 is also prime. Of course, if you're willing to be live in base 2, the 15 or so largest known primes happen to all be palindromic. They are also strings of ones. How big is the highest one, folks? I think it's somewhere around 2^130,000 (give or take a hundred orders of magnitude).
steven@mcvax.UUCP (Steven Pemberton) (11/21/84)
In article <118@talcott.UUCP> gjk@talcott.UUCP (Greg J Kuperberg) writes: > Of course, if you're willing to be live in base 2, the 15 or so largest > known primes happen to all be palindromic. They are also strings of ones. > How big is the highest one, folks? I think it's somewhere around 2^130,000 > (give or take a hundred orders of magnitude). A Dutch paper recently dedicated a whole page to printing the decimal representation of the highest (all 39,000+ digits) which had been calculated here at the CWI using the following B program: WRITE 2**132049-1 Steven Pemberton, CWI, Amsterdam; steven@mcvax
bs@faron.UUCP (Robert D. Silverman) (12/04/84)
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.