The Euler Totient Function on Lucas Sequences

2021/10/08 by Saunders, J. C.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2110.04247

Abstract

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

Related

• Pell Numbers whose Euler Function is a Pell Number
• Fibonacci or Lucas numbers that are products of two Lucas numbers or two Fibonacci numbers
• Lucas Numbers with Lehmer Property
• Perfect numbers and Fibonacci primes (III)
• Diophantine Equation with Arithmetic functions and Binary recurrent sequences