A Riesel number is a positive integer k
for which the integers
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.
See Also: SierpinskiNumber
Related pages (outside of this work)