Sierpiński Problem

Question
What is the smallest Sierpiński number?

It was proved by John Selfridge in 1962 that $78\ 557$ is a Sierpiński number.

In 1967, Wacław Sierpiński and John Selfridge conjectured that $78\ 557$ is in fact the smallest Sierpiński number.

To prove that this is the case, all odd positive integers smaller than $78\ 557$ are not Sierpiński numbers.

That is, it must be shown that, for all $1 \le k \le 78\ 557$, where $k$ is odd, there exists some $n \in \Z$ such that:
 * $k 2^n + 1$

is prime.

By March 2002, there were seventeen such $k$ whose status was still unknown.

That was when the distributed computing project Seventeen or Bust was established.

Its aim is to test all these remaining seventeen numbers by exhaustively checking all values of $n$ until finding a value of $n$ for which $k 2^n + 1$ is prime.

As of 14 July 2011, eleven of those remaining seventeen numbers have been found to be non-Sierpiński, by establishing a value of $n$ for which $k 2^n + 1$ is prime, as follows:

Also see
The Sierpinski Problem: Definition and Status