Problem 1113

Erdős Problem #1113: declared status 'open'. Formalized: yes. A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by some $p\in P$. Are there Sierpinski numbers with no finite covering set of primes? Current best: Erd\H{o}s and Graham [ErGr80] asked whether there are Sierpinski numbers for which a covering system is not 'responsible', for which the best interpretation seems to be the above question. They also prove that for every $l\geq 1$ there is an $m$ such that $2^km^i+1$ is composite for all $1\leq i\leq l$ and $k\geq 0$. Prize: no. OEIS: A076336. Tags: covering systems, number theory.