We investigate the relationship between lower bounds for the Mahler measure and splitting of primes, and prove various lower bounds for the Mahler measure of algebraic integers in terms of the least common multiples of all inertia degrees of primes. The results generalise work of the second author and Kumar [‘Lehmer’s problem and splitting of rational primes in number fields’, Acta Math. Hungar. 169(2) (2023), 349–358].

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

Recently, Alanazi et al. [‘Refining overpartitions by properties of nonoverlined parts’, Contrib. Discrete Math. 17(2) (2022), 96–111] considered overpartitions wherein the nonoverlined parts must be $\ell $-regular, that is, the nonoverlined parts cannot be divisible by the integer $\ell $. In the process, they proved a general parity result for the corresponding enumerating functions. They also proved some specific congruences for the case $\ell =3$. In this paper we use elementary generating function manipulations to significantly extend this set of known congruences for these functions.

Let $[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]$ be the continued fraction expansion of $x\in [0,1)$ and $q_n(x)$ be the denominator of its nth convergent. The irrationality exponent and Khintchine exponent of x are respectively defined by

$$ \begin{align*} \overline{v}(x)=2+\limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)} \quad \text{and}\quad \gamma(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log a_i(x). \end{align*} $$

We study the multifractal spectrum of the irrationality exponent and the Khintchine exponent for continued fractions with nondecreasing partial quotients. For any $v>2$, we completely determine the Hausdorff dimensions of the sets $\{x\in [0,1): a_1(x)\leq a_2(x)\leq \cdots , \overline {v}(x)=v\}$ and

$$ \begin{align*}\bigg\{x\in[0,1): a_1(x)\leq a_2(x)\leq\cdots, \lim\limits_{n\to\infty}\frac{\log a_1(x)+\log a_2(x)+\cdots+\log a_n(x)}{\psi(n)}=1\bigg\},\end{align*} $$

where $\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ is a function satisfying $\psi (n)\to \infty $ as $n\to \infty $.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

We investigate the set $T_{n}$ of the possible number of triangles in a graph on n vertices. The first main result says that every natural number less than $\binom {n}{3} - (\sqrt {2}+o(1) )n^{3/2} $ belongs to $T_{n}$. On the other hand, we show that there is a number $m = \binom {n}{3}-( \sqrt {2} +o(1))n^{3/2}$ that is not a member of $T_{n}$. In addition, we prove that there are two interlacing sequences $\binom {n}{3} - ( \sqrt {2} + o(1) )n^{3/2} = c_{1} \le d_{1} \le c_{2} \le d_{2} \le \cdots \le c_{s} \le d_{s} = \binom {n}{3}$ with $| c_{t} - d_{t}| = n-2-\binom {s - t +1}{2}$ such that $( c_{t}, d_{t} ) \cap T_{n} = \emptyset $ for all t. Moreover, we obtain a generalisation of these results for the set of the possible number of copies of a connected graph H in a graph on n vertices.

We establish explicit constructions of Mahler’s p-adic $U_{m}$-numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

Let A be an F-central simple algebra of degree $m^2=\prod _{i=1}^k p_i^{2\alpha _i}$ and G be a subgroup of the unit group of A such that $F[G]=A$. We prove that if G is a central product of two of its subgroups M and N, then $F[M]\otimes _F F[N]\cong F[G]$. Also, if G is locally nilpotent, then G is a central product of subgroups $H_i$, where $[F[H_i]:F]=p_i^{2\alpha _i}$, $A=F[G]\cong F[H_1]\otimes _F \cdots \otimes _F F[H_k]$ and $H_i/Z(G)$ is the Sylow $p_i$-subgroup of $G/Z(G)$ for each i with $1\leq i\leq k$. Additionally, there is an element of order $p_i$ in F for each i with $1\leq i\leq k$.

Let $\mathcal {S}$ denote the class of univalent functions in the open unit disc $\mathbb {D}:=\{z\in \mathbb {C}:\, |z|<1\}$ with the form $f(z)= z+\sum _{n=2}^{\infty }a_n z^n$. The logarithmic coefficients $\gamma _{n}$ of $f\in \mathcal {S}$ are defined by $F_{f}(z):= \log (f(z)/z)=2\sum _{n=1}^{\infty }\gamma _{n}z^{n}$. The second Hankel determinant for logarithmic coefficients is defined by

$$ \begin{align*} H_{2,2}(F_f/2) = \begin{vmatrix} \gamma_2 & \gamma_3 \\ \gamma_3 & \gamma_4 \end{vmatrix} =\gamma_2\gamma_4 -\gamma_3^2. \end{align*} $$

We obtain sharp upper bounds of the second Hankel determinant of logarithmic coefficients for starlike and convex functions.

Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith. 99(1) (2000), 61–65].

A complete Reidemeister characterisation of welded links is a long-standing open problem. We present a Reidemeister theorem for a related class of four-dimensional links: solid ribbon torus links, that is, immersed solid tori in $\mathbb {R}^4$ with only ribbon singularities, considered up to generalised ribbon isotopy.

Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers.

We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialised case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

We investigate when a group of the form $G\times \mathbb {Z}^m\ (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\mathrm {FGFP_a}$), by using the BNS-invariant, and provide some partial answers and nontrivial examples.

For fixed m and a, we give an explicit description of those subsets of ${\mathbb F}_{q}$, q odd, for which both x and $mx+a$ are quadratic residues (and other combinations). These results extend and refine results that date back to Gauss.

If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

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.