Pell numbers whose Euler function is a Pell number

Abstract

We show that the only Pell numbers whose Euler function is also a Pell number are 1 and 2. 2010 Mathematics Subject Classification: 11B39.

The Euler Totient Function on Lucas Sequences

“…We also have that in our range of n that q(n) < (log(4n)) 1.45 , so that, by (7), we have l log α < 0.59 + log 1.45 + log log log(4n) = 0.59 + log 1.45 + log log(log 4 + log n).…”

“…Remark 1. Theorem 1 combined with the results in [14] and [7] completely exhausts the problem of finding solutions to ϕ (u n ) = u m where (u n ) n is a Lucas sequence of the first kind with recurrence relation u n = P u n−1 + u n−2 and P > 0. If, instead, we have P < 0 and (u n ) n is the corresponding Lucas sequence, then we can see that (|u n |) n is the corresponding Lucas sequence of −P .…”

“…For instance, in 2002, Luca gave a general result on such an equation, proving that if a few technical properties of the sequences in question hold, then the number of solutions to such an equation is finite with the upper bound being computable [10]. Since then, there have been a number of other similar results, in particular, when terms of a Lucas sequence get mapped to terms in the same Lucas sequence, see [1], [3], [5], [6], [7], [12], and [14]. There are also such results and study on the occurrence of the Euler totient function of terms in a Lucas sequence giving a factorial, a power of 2, or a product of a power of 2 and a power of 3, see [4], [9], [15], and [20].…”

“…To prove Theorem 1, we begin by following the proofs in [14] and [7], but we also borrow proof techniques from [3] and [7] too. Throughout this paper, we let (u n ) n denote the Lucas sequence of the first kind in question and (v n ) n denote the corresponding Lucas sequence of the second kind in question as well.…”

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In 2009, Luca and Nicolae [14] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca [7] proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here we generalise these two results and prove that for any fixed natural number P ≥ 3, if we define the sequence (un) n as u0 = 0, u1 = 1, and un = P un−1 + un−2 for all n ≥ 2, then the only solution to the Diophantine equation ϕ (un) = um is ϕ (u1) = ϕ(1) = 1 = u1.

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