The Euler totient function on Lucas sequences

Abstract

In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are [Formula: see text], and [Formula: see text]. In 2015, Faye and Luca [Pell numbers whose Euler function is a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] proved that the only Pell numbers whose Euler totient function is another Pell number are [Formula: see text] and [Formula: see text]. Here, we add to these two results and prove that for any fixed natural number [Formula: see text], if we define the sequence [Formula: see text] as [Formula: see text], [Formula: see text], and [Formula: see text] for all [Formula: see text], then the only solution to the Diophantine equation [Formula: see text] is [Formula: see text].