Let n be a positive integer and $\underline {n}=\{1,2,\ldots ,n\}$. A conjecture arising from certain polynomial near-ring codes states that if $k\geq 1$ and $a_{1},a_{2},\ldots ,a_{k}$ are distinct positive integers, then the symmetric difference $a_{1}\underline {n}\mathbin {\Delta }a_{2}\underline {n}\mathbin {\Delta }\cdots \mathbin {\Delta }a_{k}\underline {n}$ contains at least n elements. Here, $a_{i}\underline {n}=\{a_{i},2a_{i},\ldots ,na_{i}\}$ for each i. We prove this conjecture for arbitrary n and for $k=1,2,3$.

One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by

$$ \begin{align*}\mathcal{C}_k(q)=\bigg[\begin{matrix}{2k}\\ {k}\end{matrix}\bigg]_q-q \bigg[\begin{matrix}{2k}\\ {k+1} \end{matrix}\bigg]_q.\end{align*} $$

Liu [‘On a congruence involving q-Catalan numbers’, C. R. Math. Acad. Sci. Paris 358 (2020), 211–215] studied congruences of the form $\sum _{k=0}^{n-1} q^k\mathcal {C}_k$ modulo the cyclotomic polynomial $\Phi _n(q)^2$, provided that $n\equiv \pm 1\pmod 3$. Apparently, the case $n\equiv 0\pmod 3$ has been missing from the literature. Our primary purpose is to fill this gap. In addition, we discuss a certain fascinating link to Dirichlet character sum identities.

Let $\alpha $ be a complex valued $2$-cocycle of finite order of a finite group $G.$ The nth Frobenius–Schur indicator of an irreducible $\alpha $-character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $-character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$. A formula, involving the real $\alpha $-regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $-characters of G under the additional assumption that these characters are class functions.

Let n be a nonzero integer. A set S of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. We show the contribution of intermediate elements is $O(\log \log |n|)$, improving a result by Dujella [‘Bounds for the size of sets with the property $D(n)$’, Glas. Mat. Ser. III 39(59)(2) (2004), 199–205]. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$, improving the best known upper bound on $M_n$ by Becker and Murty [‘Diophantine m-tuples with the property $D(n)$’, Glas. Mat. Ser. III 54(74)(1) (2019), 65–75].

Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$, where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$. In particular, if $r>p$, then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

We settle the noninner automorphism conjecture for finite p-groups ($p> 2$) with certain conditions. Also, we give an elementary and short proof of the main result of Ghoraishi [‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc. 89(2) (2014) 202–209].

Using properties of Ramanujan’s theta functions, we give an elementary proof of Hirschhorn’s conjecture on $2^n$-dissection of Euler’s product $E(q):=(q;q)_\infty $.

Let $\mathbb {Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product $\mathbb {Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that $\mathbb {Z}\times S$ has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

Cardona and Lario [‘Twists of the genus 2 curve $y^2 = x^6+1$’, J. Number Theory 209 (2020), 195–211] gave a complete classification of the twists of the curve $y^2 = x^6+1$. In this paper, we study the twists of the curve whose automorphism group is defined over a biquadratic extension of the rationals. If the twists are of type B or C in the Cardona–Lario classification, we find a pair of elliptic curves whose product is isogenous with the Jacobian of the twist.

We study the $L^p$-boundedness of the Berezin transform on the generalised Hartogs triangles which are defined by

$$ \begin{align*}H_k:=\{(z, w)\in\mathbb C^n\times\mathbb C: |z_1|^2+\cdots+|z_n|^2<|w|^{2k}<1\},\end{align*} $$

where $z=(z_1, \ldots , z_n)$ and $k\in \mathbb N$. We prove that the Berezin transform is bounded on $L^p(H_k)$ if and only if $p>nk+1$.

We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let $B_1,\dots ,B_r\in \mathrm {GL}_m(K)$ be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of $\mathrm {GL}_m(K)$ not passing through zero. Then there exist only finitely many elements of $\mathrm {GL}_m(K)$ of the form $B_1^{n_1}\cdots B_r^{n_r}$ (as we vary $n_1,\dots ,n_r$ in $\mathbb {Z}$) lying on the subvariety V.

Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi $ be the unique quadratic multiplicative character of the finite field $\mathbb {F}_q$. If the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$, then we prove that

$$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$

This confirms a conjecture of Zhi-Wei Sun.

Let G and H be two vertex disjoint graphs. The join $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup \{xy\;|\; x\in V(G), y\in V(H)\}$. A (finite) linear forest is a graph consisting of (finite) vertex disjoint paths. We prove that for any finite linear forest F and any nonnull graph H, if $\{F, H\}$-free graphs have a $\chi $-binding function $f(\omega )$, then $\{F, K_n+H\}$-free graphs have a $\chi $-binding function $kf(\omega )$ for some constant k.

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary $\alpha ,\beta \in \mathrm {GL}(V)$, we consider the semidirect products $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $, and show that if $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $ are isomorphic, then $\alpha $ must be similar to a power of $\beta $ that generates the same subgroup as $\beta $; that is, if H and K are cyclic subgroups of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then H and K must be conjugate subgroups of $\mathrm {GL}(V)$. If we remove the cyclic condition, there exist examples of nonisomorphic, let alone nonconjugate, subgroups H and K of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that noncyclic subgroups H and K of $\mathrm {GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with H and K nonconjugate in $\mathrm {GL}(V)$, but in this case, H and K must at least be isomorphic. If we replace V by a free module U over ${\mathbb {Z}}/p^m{\mathbb {Z}}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for nonconjugate cyclic subgroups of $\mathrm {GL}(U)$. If we completely abandon our requirements on V, a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of $\mathrm {Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

A cycle C of a graph G is dominating if $V(C)$ is a dominating set and $V(G)\backslash V(C)$ is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on $n\geq 3$ vertices with $k\geq 2$ is dominating if the degree sum is more than $(k+1)(n+1)/3$ for any $k+1$ pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy $(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ or $2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$

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. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, Utilitas Math. 94 (2014), 67–81] characterised the trees and graphs of minimum degree 2 with semitotal domination number half their order. In this paper, we characterise all graphs whose semitotal domination number is half their order.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

We establish a q-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a q-congruence of Guo and Zeng [‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory 145 (2014), 301–316]. The important ingredients in the proof include Andrews’ $_4\phi _3$ terminating identity.

Let $k\geq 4$ be an integer. We prove that the set $\mathcal {O}$ of all nonzero generalised octagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function $f_k$ satisfies the condition

$$ \begin{align*} f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{align*} $$

for arbitrary $x_1,\ldots ,x_k\in \mathcal {O}$, then $f_k$ is the identity function $f_k(n)=n$ for all $n\in \mathbb {N}$. We also show that $f_2$ and $f_3$ are not determined uniquely.