Fibonacci Numbers with the Lehmer Property

Abstract

Summary. We show that if m > 1 is a Fibonacci number such that φ(m) | m − 1, where φ is the Euler function, then m is prime.Let φ(n) be the Euler function of the positive integer n. Clearly, φ(n) = n − 1 if n is a prime. Lehmer [9] (see also B37 in [7]) conjectured that if φ(n) | n−1, then n is prime. To this day, no counterexample to this conjecture (and no proof of it either) has been found. Let us say that n has the Lehmer property if n is composite and φ(n) | n − 1. Thus, Lehmer's conjecture is that there is no number with the Lehmer property.Pomerance (see [14], [15]) showed that if L(x) denotes the number of numbers n ≤ x with the Lehmer property then the estimateholds, where log x stands for the natural logarithm of x. The exponent 3/4 of log x in the above bound was successively lowered to 1/2 by Zhun [18] and to 0 (at the cost of some extra power of log log x) by Banks and Luca [2]. In the recent paper [6], Diaconescu studied numbers with the Lehmer property and some extra structure and concluded that there should be only finitely many of them. For example, he showed that if k ≥ 1 is a fixed positive integer then there are only finitely many positive integers n with the Lehmer property which also satisfy the congruence φ(n) k ≡ 1 (mod n).Here, we study the numbers with the Lehmer property which belong to a familiar subset of positive integers, namely the Fibonacci numbers. Recall We also recall that if we write α = (1 + √ 5)/2 and β = (1 − √ 5)/2, then F n = (α n − β n )/(α − β) for all n ≥ 0. This is sometimes called the Binet formula. Furthermore, if we write (L n ) n≥0 for the Lucas sequence given by L 0 = 2, L 1 = 1 and L n+2 = L n+1 + L n for all n ≥ 0, then both the Binet formula L n = α n + β n and (1) L 2 n − 5F 2 n = 4(−1) n hold for all n ≥ 0.

Pell numbers with the Lehmer property

“…Many results concerning this problem can be found in the litterature (see [1], [9]). Not succeeding in proving that there are no numbers with the Lehmer property, some researchers concentrated on proving that there are no numbers with the Lehmer property in certain interesting subsequences of positive integers like in the Fibonacci sequence {F n } n≥0 and its companion sequence {L n } n≥0 (see [5] and [8]). In [4] and [6], it was shown that there are no numbers with the Lehmer property in the sequence of Cullen numbers {C n } n≥1 of general term C n = n2 n + 1, and in some appropriate generalization of the sequence of Cullen numbers, respectively.…”

Generalized Cullen Numbers With the Lehmer Property

“…At present no Lehmer number is known. Recently it was proved that certain sequences of integers such as the Fibonacci sequence do not contain a Lehmer number (see, for example, [1], [6]). …”

Repunit Lehmer numbers

“…In [5], it was shown that there is no Lehmer number in the Fibonacci sequence. We will refer to such numbers simply as repunits without mentioning the dependence on g. It is not known whether, given g, there are infinitely many repunit primes.…”