Pell numbers with the Lehmer property

Abstract

Abstract. In this paper, we prove that there is no number with the Lehmer property in the sequence of Pell numbers.

Pell numbers whose Euler function is a Pell number

“…Thus, McDaniel's result applies for us showing that 4 | q − 1 | φ(P n ) | P m , so 4 | m by Lemma 2.2. Further, it follows from a the result of the second author [5], that φ(P n ) P φ(n) . Hence, m φ(n).…”

“…For example, in [11], it is shown that 1, 2, and 3 are the only Fibonacci numbers whose Euler function is also a Fibonacci number, while in [4] it is shown that the Diophantine equation φ(5 n − 1) = 5 m − 1 has no positive integer solutions (m, n). Furthermore, the divisibility relation φ(n) | n − 1 when n is a Fibonacci number, or a Lucas number, or a Cullen number (that is, a number of the form n2 n + 1 for some positive integer n), or a rep-digit (g m − 1)/(g − 1) in some integer base g ∈ [2,1000] have been investigated in [10,5,7,3], respectively. Here we look for a similar equation with members of the Pell sequence.…”

Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property

“…Many researchers concentrated on proving that there are no numbers with the Lehmer property in certain in sequences of positive integers like the sequences of Fibonacci and Lucas numbers [3,1]. In [5,6], it was shown that there are no numbers with the Lehmer property in the sequence of Cullen numbers and in generalized Cullen numbers.…”

“…In [5,6], it was shown that there are no numbers with the Lehmer property in the sequence of Cullen numbers and in generalized Cullen numbers. Also, in [7] it was proved that there is no number with the Lehmer property in the sequence of Pell numbers. Also, in [9] it was proved that there is no Lehmer number of the form g n −1 g−1 for any n > 1 and 2 ≤ g ≤ 1000.…”

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

“…The upper bound given by Pomerance was a crucial step in proofs of many results concerning the existence of Lehmer numbers in certain sequences, such as the Fibonacci sequence [6], Pell numbers [3] or Cullen numbers [4,5].…”