On Fibonacci powers

Andrejić, Vladica

doi:10.2298/petf0617038a

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…Since r < 10 14 , a calculation of McIntosh and Roettger (see [1] and [10]) shows that rkF z.r/ in this range confirming thus a conjecture of Wall [14]. Put r WD r i and WD i .…”

Section: Bounding`even Bettermentioning

confidence: 61%

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φ(Fn ) = Fm

Luca¹,

Nicolae²

2009

Integers

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“…Since r < 10 14 , a calculation of McIntosh and Roettger (see [1] and [10]) shows that rkF z.r/ in this range confirming thus a conjecture of Wall [14]. Put r WD r i and WD i .…”

Section: Bounding`even Bettermentioning

confidence: 61%

“…1/ n ; (1) or F 2n D F n L n , as well as several others which we will mention when they will be needed. 1/ n ; (1) or F 2n D F n L n , as well as several others which we will mention when they will be needed.…”

Section: Introductionmentioning

confidence: 99%

φ(Fn ) = Fm

Luca¹,

Nicolae²

2009

Integers

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“…We checked with Mathematica that Wall's conjecture is true for all p < 1400. In fact, in [1] it is mentioned that recently McIntosh and Roettger [12] verified Wall's conjecture for all p < 10 14 and found it to be true. In particular, it is true for p 1 .…”

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confidence: 92%

Fibonacci Numbers with the Lehmer Property

Luca

2007

Bull. Polish Acad. Sci. Math.

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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.

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On Fibonacci powers

Cited by 2 publications

References 14 publications

φ(Fn ) = Fm

φ(Fn ) = Fm

Fibonacci Numbers with the Lehmer Property

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