Pseudoprime Cullen and Woodall numbers
Volume 107 / 2007
Colloquium Mathematicum 107 (2007), 35-43
MSC: 11A07, 11B83, 11N25.
DOI: 10.4064/cm107-1-5
Abstract
We show that if $a>1$ is any fixed integer, then for a sufficiently large $x>1$, the $n$th Cullen number $C_n = n2^n +1$ is a base $a$ pseudoprime only for at most $O(x\log \log x/\! \log x)$ positive integers $n\le x$. This complements a result of E. Heppner which asserts that $C_n$ is prime for at most $O(x/\! \log x)$ of positive integers $n\le x$. We also prove a similar result concerning the pseudoprimality to base $a$ of the Woodall numbers given by $W_n=n2^n-1$ for all $n\ge 1$.
