We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
For a wide class of integer linear recurrence sequences 
$(u(n))_{n=1}^\infty $, we give an upper bound on the number of s-tuples 
$\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements 
$u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.
Let s be a fixed positive integer constant and let 
$\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set 
${\mathcal M}\subseteq \mathbb {F}_p^*$ of size 
$|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer 
$H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer 
$\lambda $ can be represented in the form 
$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$
When 
$s=1$, we show that, for almost all primes p, if 
$|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and 
$H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer 
$\lambda $ can be represented in the form 
$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$
We obtain asymptotic formulas for the number of matrices in the congruence subgroup which are of naive height at most X. Our result is uniform in a very broad range of values Q and X.
$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$
Let 
$f(X) \in {\mathbb Z}[X]$ be a polynomial of degree 
$d \ge 2$ without multiple roots and let 
${\mathcal F}(N)$ be the set of Farey fractions of order N. We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields 
${\mathbb Q}(\sqrt {f(r)})$ for 
$r\in {\mathcal F}(N)$, with a given discriminant.
We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map 
$\Phi :X{\longrightarrow } X$ defined over K, a point 
$\alpha \in X(K)$ and a subvariety 
$V\subseteq X$, then the set of all nonnegative integers n such that 
$\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, 
$\Phi $, 
$\alpha $ and V) such that for each positive integer M,$$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$
We show that in a parametric family of linear recurrence sequences 
$a_1(\alpha ) f_1(\alpha )^n + \cdots + a_k(\alpha ) f_k(\alpha )^n$ with the coefficients 
$a_i$ and characteristic roots 
$f_i$, 
$i=1, \ldots ,k$, given by rational functions over some number field, for all but a set of elements 
$\alpha $ of bounded height in the algebraic closure of 
${\mathbb Q}$, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.
A Ducci sequence is a sequence of integer 
$n$-tuples obtained by iterating the map Such a sequence is eventually periodic and we denote by 
$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$
$P(n)$ the maximal period of such sequences for given odd 
$n$. We prove a lower bound for 
$P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo 
$n$.
We obtain a new lower bound on the size of the value set 
$\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial 
$f\in \mathbb{F}_{p}[X]$ over a finite field of 
$p$ elements when 
$p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of 
$f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in 
$\mathscr{V}(f)$.
We improve some previously known deterministic algorithms for finding integer solutions 
$x,y$ to the exponential equation of the form 
$af^{x}+bg^{y}=c$ over finite fields.
For a positive integer 
$n$ let where 
$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$
$p$ runs over primes and 
$s_{p}(n)$ is the sum of the base 
$p$ digits of 
$n$ . For all 
$n$ we prove that 
$\mathfrak{P}_{n}$ is divisible by all “small” primes with at most one exception. We also show that 
$\mathfrak{P}_{n}$ is large and has many prime factors exceeding 
$\sqrt{n}$ , with the largest one exceeding 
$n^{20/37}$ . We establish Kellner’s conjecture that the number of prime factors exceeding 
$\sqrt{n}$ grows asymptotically as 
$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$ for some constant 
$\unicode[STIX]{x1D705}$ with 
$\unicode[STIX]{x1D705}=2$ . Further, we compare the sizes of 
$\mathfrak{P}_{n}$ and 
$\mathfrak{P}_{n+1}$ , leading to the somewhat surprising conclusion that although 
$\mathfrak{P}_{n}$ tends to infinity with 
$n$ , the inequality 
$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$ is more frequent than its reverse.
