In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by $\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if $G\neq N_2$ is a connected claw-free graph of order $n\geq 6$ with minimum degree $\delta (G)\geq 3$, then $\gamma _{t2}(G)\leq \frac{4}{11}n$ and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin. 33(5) (2017), 1119–1130].

It is proved that Tutte’s $3$-flow conjecture is true for Cayley graphs on groups of order $8p$ where p is an odd prime.

Let G be a graph with m edges, minimum degree $\delta $ and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. [‘Bisections of graphs without short cycles’, Combinatorics, Probability and Computing 27(1) (2018), 44–59] showed that if (i) G is $2$-connected, or (ii) $\delta \ge 3$, or (iii) $\delta \ge 2$ and the girth of G is at least 5, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$, where $e(V_i)$ denotes the number of edges of G with both ends in $V_i$. Let $s\ge 2$ be an integer. In this note, we prove that if $\delta \ge 2s-1$ and G contains no $K_{2,s}$ as a subgraph, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$.

For any positive integers $k_1,k_2$ and any set $A\subseteq \mathbb {N}$, let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. Let g be a fixed integer. We prove that if $k_1$ and $k_2$ are two integers with $2\le k_1<k_2$ and $(k_1,k_2)=1$, then there does not exist any set $A\subseteq \mathbb {N}$ such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=g$ for all sufficiently large integers n, and if $1=k_1<k_2$, then there exists a set A such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=1$ for all positive integers n.

The sequence $a( 1) ,a( 2) ,a( 3) ,\ldots, $ labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by: $a( n) $ is half of the $( n+1) $th component, that is, the $( n+2) $th term, of the continued fraction expansion of

$$ \begin{align*} \sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}. \end{align*} $$

Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.

We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory 13(11) (2011), 2219–2238]. Let p be an odd prime. Then

$$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$

where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p-adic integers $a, b$.

A subset of positive integers F is a Schreier set if it is nonempty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of F). For each positive integer k, we define $k\mathcal {S}$ as the collection of all the unions of at most k Schreier sets. Also, for each positive integer n, let $(k\mathcal {S})^n$ be the collection of all sets in $k\mathcal {S}$ with maximum element equal to n. It is well known that the sequence $(|(1\mathcal {S})^n|)_{n=1}^\infty $ is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence $(|(k\mathcal {S})^n|)_{n=1}^\infty $ satisfies a linear recurrence for every positive k.

Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting $a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field $\mathbb {F}_q$ containing q elements with $2\nmid q$, we give the explicit value of the determinant $D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then

$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$

where $(\frac {\cdot }{p})$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb {Q}(\sqrt {-p})$.

Lehmer [‘On certain character matrices’, Pacific J. Math. 6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math. 10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Given a set X of $n\times n$ matrices and a positive integer m, we consider the problem of estimating the cardinalities of the product sets $A_1 \cdots A_m$, where $A_i\in X$. When $X={\mathcal M}_n(\mathbb {Z};H)$, the set of $n\times n$ matrices with integer elements of size at most H, we give several bounds on the cardinalities of the product sets and solution sets of related equations such as $A_1 \cdots A_m=C$ and $A_1 \cdots A_m=B_1 \cdots B_m$. We also consider the case where X is the subset of matrices in ${\mathcal M}_n(\mathbb {F})$, where $\mathbb {F}$ is a field with bounded rank $k\leq n$. In this case, we completely classify the related product set.

A generalisation of the well-known Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$, $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n\ge 0$ is the k-generalised Pell sequence $\{P^{(k)}_n\}_{n\ge -(k-2)}$ whose first k terms are $0,\ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$. For the Pell sequence, the formula $P^2_n+P^2_{n+1}=P_{2n+1}$ holds for all $n\ge 0$. In this paper, we prove that the Diophantine equation

$$ \begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m \end{align*} $$

has no solution in positive integers $k, m$ and n with $n>1$ and $k\ge 3$.

We propose generating functions, $\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.

We show that there is a set $S \subseteq {\mathbb N}$ with lower density arbitrarily close to $1$ such that, for each sufficiently large real number $\alpha $, the inequality $|m\alpha -n| \geq 1$ holds for every pair $(m,n) \in S^2$. On the other hand, if $S \subseteq {\mathbb N}$ has density $1$, then, for each irrational $\alpha>0$ and any positive $\varepsilon $, there exist $m,n \in S$ for which $|m\alpha -n|<\varepsilon $.

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3) 101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.

We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193].

Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-$2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

Let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of an irrational number $x\in [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all $x\in [0,1)$,

$$ \begin{align*} \liminf_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n} = 0\quad \text{and}\quad \limsup_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n}=1. \end{align*} $$

We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.

We provide explicit bounds for the Riemann zeta-function on the line $\mathrm {Re}\,{s}=1$, assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.

We prove that any subset of $\overline {\mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $\mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].