A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$-regular if R induces a $\kappa $-regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$-regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$-regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$, $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$-regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$-regular set of G if and only if it is a $(0,1)$-regular set of G.

Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$, then $F_{t2}(G)\leq (\Delta-1)n/\Delta$, with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$.

For an integer $b\geq 2$, a positive integer is called a b-Niven number if it is a multiple of the sum of the digits in its base-b representation. In this article, we show that every arithmetic progression contains infinitely many b-Niven numbers.

We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb {Q}(\sqrt {d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and ${\lvert d \rvert < 100}$.

Arithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of $\mathbb {N}$, including the asymptotic density, the Banach density and the analytic density. Let $B \subseteq \mathbb {N}$ be a nonempty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty $ (for example, the primes or the perfect powers). We show that, for each $\alpha \in [0,1]$, there is a set $A\subseteq \mathbb {N}$ such that, for every arithmetic quasidensity $\mu $, both A and the sumset $A+B$ are in the domain of $\mu $ and, in addition, $\mu (A + B) = \alpha $. The proof relies on the properties of a little known density first considered by Buck [‘The measure theoretic approach to density’, Amer. J. Math. 68 (1946), 560–580].

Let $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$. Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function

$$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$

which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function.

Lin introduced the partition function $\text {PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\text {PDO}_t(n)$, and conjectured certain congruences modulo $3^{k+2}$ for $k\geq 0$. He proved the conjecture for $k=0$ and $k=1$ [‘The number of tagged parts over the partitions with designated summands’, J. Number Theory 184 (2018), 216–234]. We prove the conjecture for $k=2$. We also study the lacunarity of $\text {PDO}_t(n)$ modulo arbitrary powers of 2 and 3. Using nilpotency of Hecke operators, we prove that there exists an infinite family of congruences modulo any power of 2 satisfied by $\text {PDO}_t(n)$.

Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$, $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where $\lambda =\lceil {k}/{\beta }\rceil $, then $kA\cap S\neq \emptyset $ for $2<\beta \leq 3$ and $k\geq {2\beta }/{(\beta -2)}$ or for $\beta>3$ and $k\geq 3$.

Let $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit [‘Counterexamples to a conjecture of Dombi in additive number theory’, Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon>0$, then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$. We also give an example to show that this bound is best possible.

Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow 1$. For $A=\{a_1<a_2<\cdots \}$, let $A(x)$ denote the counting function of A and let $a^*(x)$ denote the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa [‘Exact additive complements’, Quart. J. Math. 68 (2017) 227–235] and Chen and Fang [‘Additive complements with Narkiewicz’s condition’, Combinatorica 39 (2019), 813–823], we prove that, for exact additive complements $A,B$ with ${a_{n+1}}/ {na_n}\rightarrow \infty $,

$$ \begin{align*}A(x)B(x)-x\geqslant \frac{a^*(x)}{A(x)}+o\bigg(\frac{a^*(x)}{A(x)^2}\bigg) \quad\mbox{as } x\rightarrow +\infty.\end{align*} $$

We also construct exact additive complements $A,B$ with ${a_{n+1}}/{na_n}\rightarrow \infty $ such that

$$ \begin{align*}A(x)B(x)-x\leqslant \frac{a^*(x)}{A(x)}+(1+o(1))\bigg(\frac{a^*(x)}{A(x)^2}\bigg)\end{align*} $$

for infinitely many positive integers x.

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa 30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$. We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$. As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$. Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$.

Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$. We show that for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

The integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.

Let F be a subfield of the complex numbers and $f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$ an irreducible polynomial. We give an elementary characterisation of the Galois group of $f(x)$ as a transitive subgroup of $S_6$. The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.

Leonetti and Luca [‘On the iterates of the shifted Euler’s function’, Bull. Aust. Math. Soc., to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$, where $x_1,x_2\geq 1$, $k\geq 0$ and $2 \mid k$, is bounded by $4^{X^{3^{k+1}}}$, where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$, where $X=x_1+x_2+2k$.

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group $G.$ A new concept of strict $\alpha $-regularity is introduced and its basic properties are investigated. To illustrate the potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible $\alpha _N$-characters of N equals the number of $\alpha $-regular conjugacy classes of G contained in $N,$ where N is a normal subgroup of $G.$

Let p be a prime and let r, s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$, $x,y,m,n \in {\mathbb {N}}$, $\min \{x,y\}>1$, $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics.

By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun [‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7]. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.

We establish the linear independence of values of the q-analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q-series.

Let H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$-cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $-regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $-regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$-regular element of a normal Hall subgroup H is $\alpha $-regular in $G.$