This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph $\Gamma (R)$ for the Boolean ring $R = (\mathbb {Z}_2)^n$. The unique structure of this graph, characterised by its diameter of $3$ and lack of common neighbours, has hindered all standard approaches. We introduce a novel combinatorial method that constructs an explicit resolving set. Consequently, we provide a precise formula for $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$, closing a notable gap in the literature on metric dimensions of zero-divisor graphs. As an application, we compute the metric dimension of a zero-divisor graph of a ring with a Boolean factor.

A positive integer N is said to be an infinitary harmonic number (IHN) if the harmonic mean of its infinitary divisors is integral. It is still an open problem whether or not there exist infinitely many IHNs. Cohen and Hagis [‘Infinitary harmonic numbers’, Bull. Aust. Math. Soc. 41 (1989), 151–158] showed that there exist at most finitely many IHNs with a fixed number of I-components. We give an explicit Nielsen-type upper bound for the number of IHNs. We also consider a class of general infinitary amicable numbers and obtain the Borho-type upper bound for them.

Let A be a subset of an additive abelian semigroup S and let $hA$ be the h-fold sumset of A. The following question is considered. Let $(A_q)_{q=1}^{\infty }$ be a strictly decreasing sequence of sets in S and let $A = \bigcap _{q=1}^{\infty } A_q$. When does one have

$$ \begin{align*} hA = \bigcap_{q=1}^{\infty} hA_q \end{align*} $$

for some or all $h \geq 2$?

In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.

We investigate solutions to the matrix Diophantine equation $ X^n + Y^n = n!Z^n $ within specific classes of $2 \times 2$ integer matrices. Firstly, we introduce a class of $2\times 2$ matrices $G_2(d)$ and show that $X^n + Y^n = n!Z^n$ admits a solution in $G_2(d)$ if and only if the classical Diophantine equation $x^n + y^n = n!z^n$ has a solution in $\mathbb {Q}(\sqrt {d})$. We further present a constructive theorem for generating matrix solutions over a general number field K. Finally, we introduce a determinant-fixed class of matrices $G_2(d, l)$, and prove that the Fermat-type equation $X^n+Y^n=2Z^n$ has only the trivial solution in $G_2(d, l)$ whenever $n>2$.

Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$. Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16(1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$’, Math. Ann. 383(3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$, then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$, then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$, then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$.

We solve three problems raised in recent articles by Longstaff [Bull. Aust. Math. Soc. 102 (2020), 226–236; 103 (2021), 260–270; 104 (2021), 78–93]. We show that: (1) the length of an irreducible family of $n\times n$ rank-one matrices can take any value $1,2,\ldots ,n$; (2) the slot length of an irreducible pair of $n\times n$ matrices can exceed $2n-4$; and (3) we give bounds on the $1$-minimum spanning lengths of irreducible pairs.

In a recent work of Moreno-Fernandez, Wierstra and the present author [‘A recognition principle for iterated suspensions as coalgebras over the little cubes operad’, Preprint, 2022, arXiv:2210.00839], a coendomorphism operad in the category of pointed topological spaces endowed with the wedge sum was introduced. In this paper, we construct an analogue completely internal to the category of simplicial sets with the goal of defining simplicial coalgebras. As an application, we show that simplicial n-fold suspensions are coalgebras up to coherent homotopy over the Barratt–Eccles $E_n$-operad provided they have finitely many nondegenerate simplices.

It is established that the sporadic simple groups, the alternating groups and the projective special linear groups $L_2(p)$ for primes $p \geq 5$ are uniquely determined by their group orders and the minimum nontrivial codegrees of their irreducible characters.

In this paper, we prove a property of kernels of Brauer characters. We propose a candidate for the kernels of Isaacs’ partial characters, and we show that this candidate has the same property.

A subgroup R of a finite group G is called weakly subnormal in G if R is not subnormal in G but it is subnormal in every proper overgroup of R in G. In this paper, weak subnormality is used to construct a subgroup lattice of a finite soluble group containing the lattice of all subnormal subgroups. A new characterisation of Schmidt groups is also obtained: they are exactly those groups with all subgroups subnormal or weakly subnormal.

Let $\lambda (G)$ be the maximum number of subgroups in an irredundant cover of the finite group G. We establish bounds on the order, exponent and derived length of the group in terms of this invariant.

We define the concept of slenderness for arbitrary semigroups containing at least one idempotent element, and give necessary and sufficient conditions for semigroups to be slender. We describe the semigroup of all homomorphisms of the direct product of a nonempty family of groups into a slender rectangular abelian group.

We introduce a module-theoretic approach and a linear-programming method to compute the matricial dimensions of numerical semigroups. We compute the matricial dimension of every numerical semigroup with Frobenius number at most $10$ or genus at most $6$. Many of these evaluations are beyond the scope of previous techniques.

If A is in the p-Schatten class on ${\mathbb {R}}^n$, $1\leq p \leq {4n}/{(2n-1)}$, then the quantum translates of A are linearly independent. Moreover, there exists a nonzero operator in the p-Schatten class on ${\mathbb {R}}^n$, $p>{4n}/{(2n-1)}$, whose quantum translates are linearly dependent.

We prove a new characterisation of slices embedded into some warped products in terms of higher-order mean curvatures.

The planar Skorokhod embedding problem was first proposed and solved by Gross [‘A conformal Skorokhod embedding’, Electron. Commun. Probab. 24 (2019), 11 pages; doi:10.1214/19-ECP272]. Gross worked with probability distributions having finite second moment. Boudabra and Markowsky [‘Remarks on Gross’ technique for obtaining a conformal Skorokhod embedding of planar Brownian motion’, Electron. Commun. Probab. 25 (2020), 13 pages; doi:10.1214/20-ECP300] extended the solution to all distributions with a finite pth moment for $p>1$. The case $p=1$ has remained uncovered since then. In this note, we show that the planar Skorokhod embedding problem is solvable for $p=1$ when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

Let $(Z_{t})_{t \geq 0}$ be a planar Brownian motion running in some domain W and denote by $\tau _{W}$ the exit time of $Z_{t}$ from W. To establish the finiteness of $\mathbf {E}(\sup _{0\leq t\leq \tau _{W}}|Z_{t}|^{p})$ from the finiteness of $\mathbf {E}(|Z_{\tau _{W}}|^{p})$ for some $p>0$, Burkholder [‘Exit times of Brownian motion, harmonic majorization, and Hardy spaces’, Adv. Math. 26(2) (1977), 182–205] imposed an additional condition on the exit time $\tau _{W}$, namely the finiteness of $\mathbf {E}(\log (\tau _{W}))$. Such a condition is typically difficult to verify, since the law of the exit time is often delicate. In this paper, we revisit Burkholder’s condition and propose an alternative viewpoint. Our approach is purely analytic, weaker and formulated in terms of proper analytic maps rather than exit times themselves. This provides a more flexible framework for further applications.