Riesel number

A Riesel number is a positive integer k for which the integers k.2n-1 are all composite (that is, for every positive integer n). In 1956 Riesel formed a set of congruences whose solutions k had this property (so there is an infinite number of Riesel numbers). In particular, Riesel showed that the multiplier k=509203 had this property (as did 509203 plus any positive integer multiple of 11184810).

It is conjectured that k=509203 is the smallest Riesel number. To show that it is the smallest, one needs to find a prime of the form k.2n-1 for each of the positive integers k less than 509203. Primes have already been found for most of these k's. Wilfrid Keller is currently organizing a search to find primes for the remaining values.

Notice that the Riesel numbers are very similar to the Sierpinski numbers. The latter numbers were explored by Sierpinski in 1960, several years after Riesel wrote his paper, and their infinitude was proven using very similar congruences to those Riesel used. Ironically, the Sierpinski number received much more attention than the Riesel numbers, perhaps because the Riesel paper was in Swedish.

Related pages (outside of this work)

References:

BY88
D. A. Buell and J. Young, "Some large primes and the Sierpinski problem," SRC techn. Rep. 88004, Super-Computing Res. Center, Lanham, MD, (1988)
Keller92
W. Keller, "Factors of Fermat numbers and large primes of the form k· 2n + 1 ii," Hamburg, (September 1992) Manuscript.
Ribenboim95 (p 356-358)
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, 1995.  New York, NY, pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Riesel56
H. Riesel, "Naagra stora primtal," Elementa, 39 (1956) 258-260.  Swedish: Some large primes. [See the glossary entries Riesel number and Sierpinski number.]