We provide two new bounds on the number of visible points on exponential curves modulo a prime for all choices of primes. We also provide one new bound on the number of visible points on exponential curves modulo a prime for almost all primes.

In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc. 5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$ . We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$ . Then $\min (S)=1$ . Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and  $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$ -entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$ , respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$ , where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$ . Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$ . In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$ . Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$ , then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$ .

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$ . In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s $\unicode[STIX]{x1D717}$ -function.

Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum

$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$

$p$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}$

$k\in \mathbb{Z}$

$g:\mathbb{N}\rightarrow \mathbb{Z}$

$b$

$\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross–Koblitz formula. In this paper we prove four supercongruences between two truncated hypergeometric series $_{2}F_{1}$ . The results generalise the four Rodriguez-Villegas supercongruences.

A seminal result due to Wall states that if $x$ is normal to a given base $b$ , then so is $rx+s$ for any rational numbers $r,s$ with $r\neq 0$ . We show that a stronger result is true for normality with respect to the continued fraction expansion. In particular, suppose $a,b,c,d\in \mathbb{Z}$ with $ad-bc\neq 0$ . Then if $x$ is continued fraction normal, so is $(ax+b)/(cx+d)$ .

We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.

We prove an asymptotic formula for squarefree numbers in arithmetic progressions, improving previous results by Prachar and Hooley. As a consequence we improve a lower bound of Heath-Brown for the least squarefree number in an arithmetic progression.

Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$ -ary expansion and the $s$ -ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$ , and we show that this bound is best possible.

A lattice walk with all steps having the same length $d$ is called a $d$ -walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$ -walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$ , we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$ . To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$ , there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$

$k=6$

$p$

$A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$

$g\in \{2,3,5\}$

$|A|\gg p/(\log p)^{3}$

$A\cdot A+A\cdot A$

$\mathbb{F}_{p}$

In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base- $b$ expansion. This partially answers a question of Bugeaud.

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

Let $q\in (1,2)$ . A $q$ -expansion of a number $x$ in $[0,1/(q-1)]$ is a sequence $({\it\delta}_{i})_{i=1}^{\infty }\in \{0,1\}^{\mathbb{N}}$ satisfying

$$\begin{eqnarray}x=\mathop{\sum }_{i=1}^{\infty }\frac{{\it\delta}_{i}}{q^{i}}.\end{eqnarray}$$

${\mathcal{B}}_{\aleph _{0}}$

$q$

$x$

$q$

${\mathcal{B}}_{1,\aleph _{0}}$

$q$

$1$

$q$

$1=\sum _{i=1}^{\infty }q^{-n_{i}}$

$\min {\mathcal{B}}_{\aleph _{0}}=\min {\mathcal{B}}_{1,\aleph _{0}}=(1+\sqrt{5})/2$

${\mathcal{B}}_{\aleph _{0}}\cap ((1+\sqrt{5})/2,q_{1}]=\{q_{1}\}$

$q_{1}\,({\approx}1.64541)$

$x^{6}-x^{4}-x^{3}-2x^{2}-x-1=0$

${\mathcal{B}}_{1,\aleph _{0}}$

$q_{3}\,({\approx}1.68042)$

$x^{5}-x^{4}-x^{3}-x+1=0$

${\mathcal{B}}_{\aleph _{0}}\cap (q_{1},q_{3}]=\{q_{2},q_{3}\}$

$q_{2}\,({\approx}1.65462)$

$x^{6}-2x^{4}-x^{3}-1=0$