φ(Fn ) = Fm

Abstract

We show that 1, 2 and 3 are the only Fibonacci numbers whose Euler functions are also Fibonacci numbers.

Pell numbers whose Euler function is a Pell number

“…where T 2 and T 3 denote the sums of the reciprocals of the primes in P n satisfying (ii) and (iii), respectively. The sum T 2 was estimated in [10] using the large sieve inequality of Montgomery and Vaughan [13] (see also page 397 in [11]), and the bound on it is (2.11)…”

“…therefore, q 1 < k(P n /P n−1 ) < 3k. Using the method of the proof of inequality (13) in [11], one proves by induction on the index i ∈ {1, . .…”

“…There are many papers in the literature dealing with diophantine equations involving the Euler function in members of a binary recurrent sequence. 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.…”

“…. We have the following result. For the proof, we begin by following the method from [11], but we add to it some ingredients from [10].…”

Some arithmetic functions of factorials in Lucas sequences

“…Some of the most familiar arithmetic functions in number theory are: The study of arithmetic functions of a linearly recurrent sequence has been a popular area of research. For example, in [10] it was shown that 1, 2, and 3 are the only Fibonacci numbers whose Euler function is also a Fibonacci number, while in [3] it was proved that if {u n } n≥0 is a Lucas sequence with b ∈ {±1}, then there are only finitely many effectively computable n such that φ(|u n |) is a power of 2, extending the previous works [8,12] which dealt with the above problem for the particular Lucas sequences of the Fibonacci and Pell numbers. In [9] it was shown that the largest Fibonacci number whose Euler function is a repdigit (i.e., numbers with only one distinct digit in its decimal expansion) is F 11 for which φ(F 11 ) = 88, while in [11] it was proved that there are no repdigits with two or more digits which are multiply perfect, namely numbers who divide the sum of their divisors.…”