04T-11-4

GÜNTER LÖH and WILFRID KELLER

Generalized Cullen primes

Preliminary report.

Numbers of the form n 2^n + 1 are called Cullen numbers. They became notable because of the rareness of primes in that sequence [see Math. Comp. 64 (1995), 1733--1741]. Cullen numbers may be generalized to the form n b^n + 1, where b \gt 2. These generalized Cullen numbers were introduced by H. Dubner in J. Recreational Math. 21 (1989), 190--194. He studied their possible primality status and remarked that for prime bases b \gt 3 there seemed to be almost an absence of primes. In fact, for b = 13, 17, 19, 23, 29, 31, 41, 47, 53, 71, 73, not a single prime was known at that time. It seemed unlikely, however, that for any of these bases non-existence of primes could be proved. By extensive computation it was subsequently shown that for a given base the ''smallest'' prime might appear for quite a large exponent n. With the help of an excellent program kindly supplied by Y. Gallot, the authors determined the first occurrence of a prime n b^n + 1 for b=17 (n=19650), b=19 (n=6460), b=23 (n=4330), b=31 (n=82960), and b=71 (n=13948). More generally, primes have been listed for all bases b in the interval 3 \le b \le 100, up to varying limits on n. The largest primes discovered during this investigation are 82960 * 31^{82960} + 1 (123729 digits), 89350 * 23^{89350} + 1 (121676 digits, the second prime for b=23), 117852 * 9^{117852} + 1 (112465 digits), and 105994 * 10^{105994} + 1 (106000 digits). These were all found by the first author. For more details, see http://www.rrz.uni-hamburg.de/RRZ/G.Loeh/gc/status.html.

(Received November 17, 2003)

Autor: Günter Löh, Stand: 20.04.2006 16:57 Uhr