We show that there exist infinitely many consecutive square-free numbers of the form , . We also establish an asymptotic formula for the number of such square-free pairs when does not exceed given sufficiently large positive number.
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.