Cullen numbers with the Lehmer property

Abstract

In this note, we correct an oversight from the paper [2] mentioned in the title.

Corrigendum to “Cullen numbers with the Lehmer property”

“…Also, n cannot be divisible by a prime q ≥ 5, for otherwise, since n 1 = ρ w for some w ≥ 3, we would get that the number of prime factors p of C n with m p > 1 is at most 3+log(200, 000/q 3 )/ log 3 < 9.8, so k ≤ 9+4 = 13, contradicting again the result of Cohen and Harris. Hence, n = 2 α · 3 β and the proof finishes as in the paper [2] after formula (7).…”

“…So, from now on, we shall treat only the case when n 1 = ρ w . Comparing estimate (3) in the paper [2] with (1) leads to n 1/2 9(log n) 1/2 < 2.4 log n,…”

“…Note that inequality (6) from the paper [2] still holds whenever A = 0 for all primes p dividing n, and in particular for all n except maybe when n 1 = ρ w for some ρ ≥ 3 and w ≥ 3. So, from now on, we shall treat only the case when n 1 = ρ w .…”

“…The remaining of the argument from the paper [2] shows that the expression A is nonzero for all other primes q of C n , so all prime factors q of C n satisfy inequality (5) in the paper [2] with at most one exception, say p, which satisfies the inequality p ≤ (n2 n ) 1/3 + 1. Hence, instead of the inequality from Line 2 of Page 132 in [2], we get that…”

“…There is an error on Page 131 of the paper [2] mentioned in the title in justifying that the expression A is nonzero. After the sentence "Also, since m p divides n 1 , it follows that u ≤ w" on Page 131 in [2], the argument continues in the following way.…”