Pseudoprime Cullen and Woodall numbers

Florian Luca¹ ; Igor E. Shparlinski²

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Department of Computing Macquarie University Sydney, NSW 2109, Australia

Colloquium Mathematicum, Tome 107 (2007) no. 1, pp. 35-43.

Résumé

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 $n\ge 1$.

Détail

DOI : 10.4064/cm107-1-5

Keywords: fixed integer sufficiently large nth cullen number base pseudoprime only log log log positive integers complements result heppner which asserts prime log positive integers prove similar result concerning pseudoprimality base woodall numbers given n