Sierpinski number

An integer $k$ is a Sierpinski number if for every positive integer $n$, the number $k{2}^n+1$ is composite.

That such numbers exist is amazing, and even more surprising is that there are infinitely many of them (in fact, infinitely many odd ones). The smallest known Sierpinski number is 78557, but it is not known whether or not this is the smallest one. The smallest number $m$ for which itis unknown whether or not $m$ is a Sierpinski number is 10223.

A process for generating Sierpinski numbers using covering sets of primes can be found at

http://www.glasgowg43.freeserve.co.uk/siercvr.htm

Visit

http://www.seventeenorbust.com/

for the distributed computing effort to show that 78557 is indeed the smallest Sierpinski number (or find a smaller one).

Similarly, a Riesel number is a number $k$ such that for every positive integer $n$, the number $k{2}^n-1$ is composite. The smallest known Riesel number is 509203, but again, it is not known for sure that this is the smallest.