We show that if a > 1 is any fixed integer, then for a sufficiently large x>1, the nth Cullen number Cₙ = n2ⁿ +1 is a base a pseudoprime only for at most O(x log log x/log x) positive integers n ≤ x. This complements a result of E. Heppner which asserts that Cₙ is prime for at most O(x/log x) of positive integers n ≤ x. We also prove a similar result concerning the pseudoprimality to base a of the Woodall numbers given by Wₙ = n2ⁿ - 1 for all n ≥ 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-5,
author = {Florian Luca and Igor E. Shparlinski},
title = {Pseudoprime Cullen and Woodall numbers},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {35-43},
zbl = {1153.11048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-5}
}
Florian Luca; Igor E. Shparlinski. Pseudoprime Cullen and Woodall numbers. Colloquium Mathematicae, Tome 107 (2007) pp. 35-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-5/