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Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$ , where n is a positive integer and $F_n$ denotes the nth Fibonacci number. Leonetti and Sanna proved that $\mathcal {A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that $$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$ for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.

Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosystems, such as HFE of Patarin and UOV of Kipnis et al., is based on the postulated hardness of solving quadratic polynomial systems over a finite field. Lokshtanov et al. (2017) were the first to introduce a probabilistic algorithm that, in the worst-case, solves a Boolean polynomial system in time O ∗ ( 2 δ n ) , for some δ ∈ ( 0 , 1 ) depending only on the degree of the system, thus beating the brute-force complexity O ∗ ( 2 n ) . Later, Bjorklund et al. (2019) and then Dinur (2021) improved this method and devised probabilistic algorithms with a smaller exponent coefficient δ . We survey the theory behind these probabilistic algorithms, and we illustrate the results that we obtained by implementing them in C. In particular, for random quadratic Boolean systems, we estimate the practical complexities of the algorithms and their probabilities of success as their parameters change.

Istanbul, con i suoi oltre 16 milioni di abitanti, è oggi la maggiore città della Turchia e una delle megalopoli più grandi del mondo. Storica capitale dell'Impero Ottomano, oggi non è solo il maggiore centro abitato della Turchia, ma anche il suo cuore pulsante culturale, sociale ed economico. Da sola, la città genera oltre il 40% del gettito fiscale e circa il 40% del PIL della Turchia, con più del 20% della produzione industriale del Paese. Eppure, appena sessant'anni fa Istanbul era una città portuale stagnante e dimenticata, tagliata fuori dalle rotte del commercio internazionale, con meno di un milione di abitanti e una crescita praticamente nulla. La situazione di Istanbul rispecchiava la condizione della Turchia del secondo dopoguerra: un Paese scarsamente industrializzato e prevalentemente rurale, con strutture economiche basate principalmente sul settore agricolo, infrastrutture scarse e arretrate, poco integrato nel mercato internazionale. In seguito la Turchia fu protagonista di una crescita vorticosa che, seppure caratterizzata da un andamento estremamente altalenante, la proiettò saldamente all'interno dei meccanismi del mercato globale negli ultimi due decenni del XX secolo. Questi processi scossero profondamente Istanbul e l'intero Paese, attraversati da enormi cambiamenti nelle dinamiche non solo economiche, ma anche politiche e sociali. La città fu protagonista di un pluridecennale processo di trasformazione che la portò da poco più di 950 mila abitanti nel 1950 a una metropoli con oltre 8 milioni di abitanti negli anni Novanta. L'articolo analizza come questi processi storici abbiano modificato la struttura economica, sociale, demografica di Istanbul, ponendo le basi per farla diventare la città che è oggi.

Let $$(F_n)$$ ( F n ) be the sequence of Fibonacci numbers and, for each positive integer k, let $${\mathcal {P}}_k$$ P k be the set of primes p such that $$\gcd (p - 1, F_{p - 1}) = k$$ gcd ( p - 1 , F p - 1 ) = k . We prove that the relative density $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) of $${\mathcal {P}}_k$$ P k exists, and we give a formula for $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given k satisfies $$\varvec{r}({\mathcal {P}}_k) > 0$$ r ( P k ) > 0 , and we provide upper and lower bounds for the counting function of the set of such k’s. As an application of our results, we give a new proof of a lower bound for the counting function of the set of integers of the form $$\gcd (n, F_n)$$ gcd ( n , F n ) , for some positive integer n. Our proof is more elementary than the previous one given by Leonetti and Sanna, which relies on a result of Cubre and Rouse.

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn ) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn ) for n = 2 kpℓ , where k, ℓ ≥ 0 are integers and p is an odd prime number.

A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We improve this result by reducing the number of exceptions to $\exp\!\big(C (\log x)^{4/5} \log \log x\big)$, where $C > 0$ is a constant.

Let $\mathcal{A}$ be a random set constructed by picking independently each element of $\{1, \dots, n\}$ with probability $\alpha \in (0, 1)$. We give a formula for the probability that a rational number $q$ belong to the random ratio set $\mathcal{A} /\! \mathcal{A} := \{a / b : a,b \in \mathcal{A}\}$. This generalizes a previous result of Cilleruelo and Guijarro-Ord\'o\~nez. Moreover, we make some considerations about formulas for the probability of the event $\bigvee_{i=1}^k\!\big(q_i \in \mathcal{A} /\! \mathcal{A}\big)$, where $q_1, \dots, q_k$ are rational numbers, showing that they are related to the study of the connected components of certain graphs. In particular, we give formulas for the probability that $q^e \in \mathcal{A} /\! \mathcal{A}$ for some $e \in \mathcal{E}$, where $\mathcal{E}$ is a finite or cofinite set of positive integers with $1 \in \mathcal{E}$.

Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ F n , for every positive integer n not divisible by 3, where $$F_n$$ F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo $$F_n$$ F n , for every fixed integer $$a \ge 3$$ a ≥ 3 and for all positive integers n with $$\gcd (a, F_n) = 1$$ gcd ( a , F n ) = 1 . Our proof makes use of the so-called base-$$\varphi $$ φ expansion of real numbers.

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $${\mathbb {Q}}(\zeta _n)$$ Q ( ζ n ) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial.

Given two sets of positive integers A and B, let $$AB := \{ab : {a \in A}$$ , $${b \in B}\}$$ be their product set and put $$A^k := A \cdots A$$ (k times A) for any positive integer k. Moreover, for every positive integer n and every $$\alpha = \alpha (n) \in [0,1]$$ , let $$\mathcal {B}(n, \alpha )$$ denote the probabilistic model in which a random set $$A \subseteq \{1, \ldots , n\}$$ is constructed by choosing independently every element of $$\{1, \ldots , n\}$$ with probability $$\alpha $$ . We prove that if $$A_1$$ , $$\ldots ,$$ $$ A_s$$ are random sets in $$\mathcal {B}(n_1, \alpha _1), \ldots , \mathcal {B}(n_s, \alpha _s)$$ , respectively, $$k_1, \ldots , k_s$$ are fixed positive integers, $$\alpha _i n_i \rightarrow +\infty $$ , and $$1/\alpha _i$$ does not grow too fast in terms of a product of $$\log n_j$$ ; then $$|A_1^{k_1} \cdots A_s^{k_s}| \sim \frac{|A_1|^{k_1}}{k_1!}\cdots \frac{|A_s|^{k_s}}{k_s!}$$ with probability $$1 - o(1)$$ . This is a generalization of a result of Cilleruelo, Ramana, and Ramare [3], who considered the case $$s = 1$$ and $$k_1 = 2$$ .

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1 + s_1, L_2 + s_2, \dots, L_n + s_n)$ as $n \to +\infty$, for $(s_n)_{n \geq 1}$ a periodic sequence in $\{-1, +1\}$. We also carry out the same analysis for $(s_n)_{n \geq 1}$ a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$. These results are Lucas numbers-analogs of previous results obtained by the author for the sequence of Fibonacci numbers.

Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively studied by several authors, and in the last 10 years there has been a burst of activity in this field of research. This concise survey attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs. Previous surveys on this topic were given by Lenstra (1979), Vaughan (1989), and Thangadurai (2000).

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For every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

For every nonconstant monic polynomial $g \in \mathbb{Z}[X]$, let $\mathfrak{M}(g)$ be the set of positive integers $m$ for which there exist an integer linear recurrence $(s_n)_{n \geq 0}$ having characteristic polynomial $g$ and a positive integer $M$ such that $(s_n)_{n \geq 0}$ has exactly $m$ distinct residues modulo $M$. Dubickas and Novikas proved that $\mathfrak{M}(X^2 - X - 1) = \mathbb{N}$. We study $\mathfrak{M}(g)$ in the case in which $g$ is divisible by a monic quadratic polynomial $f \in \mathbb{Z}[X]$ with roots $\alpha,\beta$ such that $\alpha\beta = \pm 1$ and $\alpha / \beta$ is not a root of unity. We show that this problem is related to the existence of special primitive divisors of certain Lehmer sequences, and we deduce some consequences on $\mathfrak{M}(g)$. In particular, for $\alpha\beta = -1$, we prove that $m \in \mathfrak{M}(g)$ for every integer $m \geq 7$ with $m \neq 10$ and $4 \nmid m$.

Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

Let f(n) be the number of distinct exponents in the prime factorization of the natural number n. We prove some results about the distribution of f(n). In particular, for any positive integer k, we obtain that $$\begin{aligned} \#\{n \le x : f(n) = k\} \sim A_k x \end{aligned}$$ and $$\begin{aligned} \#\{n \le x : f(n) = \omega (n) - k\} \sim \frac{B x (\log \log x)^k}{k! \log x}, \end{aligned}$$ as $$x \rightarrow +\infty $$ , where $$\omega (n)$$ is the number of prime factors of n and $$A_k, B > 0$$ are some explicit constants. The latter asymptotic extends a result of Aktas and Ram Murty (Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430) about numbers having mutually distinct exponents in their prime factorization.

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $, Comment: 6 pages

Given a real number τ, we study the approximation of τ by signed harmonic sums σ N ( τ ) : = ∑ n ≤ N s n ( τ ) / n , where the sequence of signs ( s N ( τ ) ) N ∈ N is defined “greedily” by setting s N + 1 ( τ ) : = + 1 if σ N ( τ ) ≤ τ , and s N + 1 ( τ ) : = − 1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence ( σ N ( τ ) − τ ) N ∈ N . Moreover, we give an accurate description of the behavior of the sequence of signs ( s N ( τ ) ) N ∈ N , highlighting a surprising connection with the Thue–Morse sequence.

For every integer $k \geq 2$ and every $A \subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^k\}$ and $D^{\underline{k}}(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^{\underline{k}}\}$, where $\|\cdot\|$ is the Euclidean norm and $A^{\underline{k}} := \{{\bf a} \in A^k : a_i \neq a_j \text{ for all } i \neq j\}$. Via an appropriate homeomorphism, $D^k(A)$ is a generalization of the \emph{ratio set} $R(A) := \{a / b : a,b \in A\}$, which has been studied by many authors. We study $D^k(A)$ and $D^{\underline{k}}(A)$ as subspaces of $S^{k-1} := \{{\bf x} \in [0,1]^k : \|{\bf x}\| = 1\}$. In~particular, generalizing a result of Bukor and T\'oth, we provide a characterization of the sets $X \subseteq S^{k-1}$ such that there exists $A \subseteq \mathbb{N}$ satisfying $D^{\underline{k}}(A)^\prime = X$, where $Y^\prime$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^k(A)$ to be dense in $S^{k-1}$. We conclude leaving some questions for further research., Comment: 7 pp. Accepted in Bull. Austr. Math. Soc

For each integer $b \geq 3$ and every $x \geq 1$, let $\mathcal{N}_{b,0}(x)$ be the set of positive integers $n \leq x$ which are divisible by the product of their nonzero base $b$ digits. We prove bounds of the form $x^{��_{b,0} + o(1)} < \#\mathcal{N}_{b,0}(x) < x^{��_{b,0} + o(1)}$, as $x \to +\infty$, where $��_{b,0}$ and $��_{b,0}$ are constants in ${]0,1[}$ depending only on $b$. In particular, we show that $x^{0.526} < \#\mathcal{N}_{10,0}(x) < x^{0.787}$, for all sufficiently large $x$. This improves the bounds $x^{0.495} < \#\mathcal{N}_{10,0}(x) < x^{0.901}$, which were proved by De Koninck and Luca.

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $\alpha := \big(1 + \sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\mathbf{s} = (s_n)_{n \geq 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}} > 0$ such that \begin{equation*} \log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot C_\mathbf{s} \cdot n^2 , \quad \text{as } n \to +\infty . \end{equation*} Moreover, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \mathbb{E}\big[\log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n)\big] \sim \frac{3 \log \alpha}{\pi^2} \cdot \frac{15 \operatorname{Li}_2(1 / 16)}{2} \cdot n^2 , \quad \text{as } n \to +\infty , \end{equation*} where $\operatorname{Li}_2$ is the dilogarithm function.

For every positive integer n and every δ ∈ [ 0 , 1 ] , let B ( n , δ ) denote the probabilistic model in which a random set A ⊆ { 1 , … , n } is constructed by choosing independently every element of { 1 , … , n } with probability δ. Moreover, let ( u k ) k ≥ 0 be an integer sequence satisfying u k = a 1 u k − 1 + a 2 u k − 2 , for every integer k ≥ 2 , where u 0 = 0 , u 1 ≠ 0 , and a 1 , a 2 are fixed nonzero integers; and let α and β, with | α | ≥ | β | , be the two roots of the polynomial X 2 − a 1 X − a 2 . Also, assume that α / β is not a root of unity. We prove that, as δ n / log ⁡ n → + ∞ , for every A in B ( n , δ ) we have log ⁡ lcm ( u a : a ∈ A ) ∼ δ Li 2 ( 1 − δ ) 1 − δ ⋅ 3 log ⁡ | α / ( a 1 2 , a 2 ) | π 2 ⋅ n 2 with probability 1 − o ( 1 ) , where lcm denotes the lowest common multiple, Li 2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ = 1 . This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Matyas, who studied the deterministic case δ = 1 , and is motivated by an asymptotic formula for lcm ( A ) due to Cilleruelo, Rue, Sarka, and Zumalacarregui.

Let [Formula: see text] be a Lucas sequence and, for every prime number [Formula: see text], let [Formula: see text] be the rank of appearance of [Formula: see text] in [Formula: see text], that is, the smallest positive integer [Formula: see text] such that [Formula: see text] divides [Formula: see text], whenever it exists. Furthermore, let [Formula: see text] be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes [Formula: see text] such that [Formula: see text] divides [Formula: see text], as [Formula: see text].

For every $\tau\in\mathbb{R}$ and every integer $N$, let $\mathfrak{m}_N(\tau)$ be the minimum of the distance of $\tau$ from the sums $\sum_{n=1}^N s_n/n$, where $s_1, \ldots, s_n \in \{-1, +1\}$. We prove that $\mathfrak{m}_N(\tau) < \exp\!\big(-C(\log N)^2\big)$, for all sufficiently large positive integers $N$ (depending on $C$ and $\tau$), where $C$ is any positive constant less than $1/\log 4$., Comment: minor corrections to the text, a small improvement to Proposition 2.7

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (un)n≥0 be the Lucas sequence satisfying u0 = 0, u1 = 1, and un+2 = aun+1 + bun for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a 2 + 4b > 0. Also, let A be the set of all positive integers n such that |un| is a practical number. Melfi proved that A is infinite. We improve this result by showing that #A(x) ≫ x/ log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding A.Key words: Fibonacci numbers, Lucas sequences, practical numbers.

For each positive integer k , let A k be the set of all positive integers n such that gcd ( n , F n ) = k , where F n denotes the n th Fibonacci number. We prove that the asymptotic density of A k exists and is equal to ∑ d = 1 ∞ μ ( d ) lcm ( d k , z ( d k ) ) where μ is the Mobius function and z ( m ) denotes the least positive integer n such that m divides F n . We also give an effective criterion to establish when the asymptotic density of A k is zero and we show that this is the case if and only if A k is empty.

Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can find infinitely many integers $n\geq 0$ with the property that none of $f(n+1),f(n+2),\dots,f(n+k)$ is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers.

We prove that if [Formula: see text] is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers [Formula: see text] does not exceed [Formula: see text] and such that [Formula: see text] divides [Formula: see text] is less than [Formula: see text] as [Formula: see text]. This generalizes a result of Luca and Tron about the positive integers [Formula: see text] dividing the [Formula: see text]th Fibonacci number, and improve a previous upper bound due to Alba González, Luca, Pomerance and Shparlinski.

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting $f(n)$ denote the number of binomial coefficients $\binom{n}{k}$, with $0 \leq k \leq n$, that are not practical numbers, we show that \begin{equation*} f(n) < n^{1 - (\log 2 - \delta)/\log \log n} \end{equation*} for all integers $n \in [3, x]$, but at most $O_\gamma(x^{1 - (\delta - \gamma) / \log \log x})$ exceptions, for all $x \geq 3$ and $0 < \gamma < \delta < \log 2$. Furthermore, we prove that the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We also pose some questions on this topic., Comment: 10 pages, no figures

A natural number is a binary k’ th power if its binary representation consists of k consecutive identical blocks. We prove, using tools from combinatorics, linear algebra, and number theory, an analogue of Waring’s theorem for sums of binary k’th powers. More precisely, we show that for each integer k> 2, there exists an effectively computable natural number n such that every sufficiently large multiple of Ek:=gcd(2k - 1,k) is the sum of at most n binary k’th powers. (The hypothesis of being a multiple of Ek cannot be omitted, since we show that the gcd of the binary k’th powers is Ek.) Furthermore, we show that n = 2O(k3). Analogous results hold for arbitrary integer bases b>2.

The quotient set, or ratio set, of a set of integers A is defined as R ( A ) : = { a / b : a , b ∈ A , b ≠ 0 } . We consider the case in which A is the image of Z + under a polynomial f ∈ Z [ X ] , and we give some conditions under which R ( A ) is dense in Q p . Then, we apply these results to determine when R ( S m n ) is dense in Q p , where S m n is the set of numbers of the form x 1 n + ⋯ + x m n , with x 1 , … , x m ≥ 0 integers. This allows us to answer a question posed in Garcia et al. (2017) [5] . We end leaving an open question.

Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

Let ( u n ) n ≥ 0 be a nondegenerate Lucas sequence satisfying u n = a 1 u n − 1 + a 2 u n − 2 for all integers n ≥ 2 , where a 1 and a 2 are some fixed relatively prime integers; and let g u be the arithmetic function defined by g u ( n ) : = gcd ⁡ ( n , u n ) , for all positive integers n. Distributional properties of g u have been studied by several authors, also in the more general context where ( u n ) n ≥ 0 is a linear recurrence. We prove that for each positive integer λ it holds ∑ n ≤ x ( log ⁡ g u ( n ) ) λ ∼ M u , λ x as x → + ∞ , where M u , λ > 0 is a constant depending only on a 1 , a 2 , and λ. More precisely, we provide an error term for the previous asymptotic formula and we show that M u , λ can be written as an infinite series.

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

Let F and G be linear recurrences over a number field K , and let R be a finitely generated subring of K . Furthermore, let N be the set of positive integers n such that G ( n ) ≠ 0 and F ( n ) / G ( n ) ∈ R . Under mild hypothesis, Corvaja and Zannier proved that N has zero asymptotic density. We prove that # ( N ∩ [ 1 , x ] ) ≪ x ⋅ ( log ⁡ log ⁡ x / log ⁡ x ) h for all x ≥ 3 , where h is a positive integer that can be computed in terms of F and G. Assuming the Hardy–Littlewood k-tuple conjecture, our result is optimal except for the term log ⁡ log ⁡ x .

For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erd��s-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $��_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ representations start with the base $p$ representation of $k - 1$, but at most $3x^{0.835}$ exceptions. We also generalize a result of Lengyel by giving a description of $��_2(H(n,2))$ in terms of an infinite binary sequence., 12 pages, 3 figures