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1
Content available remote

Erratum to the Paper "Fibonacci Numbers with the Lehmer Property" (Bull. Polish Acad. Sci. Math. 55 (2007), 7-15)

100%
Luca F.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 3
289-290
2
Content available remote

Fibonacci Numbers with the Lehmer Property

100%
Luca F.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 1
7-15
EN
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
3
Content available remote

On the system of Diophantine equations $a^2 + b^2 = (m^2+1)^r$ and $a^x + b^y = (m^2 + 1)^z$

100%
Luca F.
Acta Arithmetica
|
2012
|
tom 153
|
nr 4
373-392
4
Content available remote

Perfect powers in q-binomial coefficients

100%
Luca F.
Acta Arithmetica
|
2012
|
tom 151
|
nr 3
279-292
5
Content available remote

Arithmetic properties of members of a binary recurrent sequence

100%
Luca F.
Acta Arithmetica
|
2003
|
tom 109
|
nr 1
81-107
6
Content available remote

The diophantine equation $x^2 = p^a ± p^b + 1$

100%
Luca F.
Acta Arithmetica
|
2004
|
tom 112
|
nr 1
87-101
7
Content available remote

Acknowledgment of priority: "On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ" (Colloq. Math. 92 (2002), 111-130)

64%
Luca F., Pomerance C.
Colloquium Mathematicum
|
2012
|
tom 126
|
nr 1
8
Content available remote

The range of the sum-of-proper-divisors function

64%
Luca F., Pomerance C.
Acta Arithmetica
|
2015
|
tom 168
|
nr 2
187-199
EN
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.
9
Content available remote

Roughly squarefree values of the Euler and Carmichael functions

64%
Banks W., Luca F.
Acta Arithmetica
|
2005
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tom 120
|
nr 3
211-230
10
Content available remote

On the composition of the Euler function and the sum of divisors function

64%
De Koninck J.-M., Luca F.
Colloquium Mathematicum
|
2007
|
tom 108
|
nr 1
31-51
EN
Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.
11
Content available remote

Pseudoprime Cullen and Woodall numbers

64%
Luca F., Shparlinski I.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 1
35-43
EN
We show that if a > 1 is any fixed integer, then for a sufficiently large x>1, the nth Cullen number Cₙ = n2ⁿ +1 is a base a pseudoprime only for at most O(x log log x/log x) positive integers n ≤ x. This complements a result of E. Heppner which asserts that Cₙ is prime for at most O(x/log x) of positive integers n ≤ x. We also prove a similar result concerning the pseudoprimality to base a of the Woodall numbers given by Wₙ = n2ⁿ - 1 for all n ≥ 1.
12
Content available remote

On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ

64%
Luca F., Pomerance C.
Colloquium Mathematicum
|
2002
|
tom 92
|
nr 1
111-130
EN
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.
13
Content available remote

Multiplicatively dependent triples of Tribonacci numbers

64%
Gómez C., Luca F.
Acta Arithmetica
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2015
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tom 171
|
nr 4
327-353
EN
We consider the Tribonacci sequence $T:= {T_n}_{n≥0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.
14
Content available remote

On the moments of the Carmichael λ function

64%
Luca F., Sankaranarayanan A.
Acta Arithmetica
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2006
|
tom 123
|
nr 4
389-398
15
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Squares in Lehmer sequences and some Diophantine applications

64%
Luca F., Walsh P.
Acta Arithmetica
|
2001
|
tom 100
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nr 1
47-62
16
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Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

64%
Luca F., Munagi A.
Colloquium Mathematicum
|
2014
|
tom 134
|
nr 2
193-209
EN
We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n≥1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u = {uₙ}_{n≥1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.
17
Content available remote

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

64%
Gómez Ruiz C., Luca F.
Colloquium Mathematicum
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2014
|
tom 137
|
nr 2
171-188
EN
A generalization of the well-known Fibonacci sequence ${Fₙ}_{n≥0}$ given by F₀ = 0, F₁ = 1 and $F_{n+2} = F_{n+1} + Fₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${Fₙ^{(k)}}_{n≥-(k-2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $Fₙ² + F²_{n+1}²= F_{2n+1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes a recent result of Chaves and Marques.
18
Content available remote

Composite positive integers with an average prime factor

64%
Luca F., Pappalardi F.
Acta Arithmetica
|
2007
|
tom 129
|
nr 2
197-201
19
Content available remote

Triangular numbers whose sum of divisors is also triangular

64%
Iannucci D., Luca F.
Acta Arithmetica
|
2007
|
tom 129
|
nr 1
23-40
20
Content available remote

On the equation x² + dy² = Fₙ

64%
Ballot C., Luca F.
Acta Arithmetica
|
2007
|
tom 127
|
nr 2
145-155
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