Abstract The notion of $\theta $ -congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $ -congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $ -congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103 (2) (2021), 218–229] established criteria for numbers to be $\theta $ -congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $ -congruent and properly $\theta $ -congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$ .

Abstract The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

Abstract Jabłonowski [‘On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots’, Preprint, 2025, arXiv:2505.14724] proved that the knot quandles of Suciu’s n -knots, which share isomorphic knot groups, are mutually nonisomorphic, and Yasuda [‘Knot quandles distinguish Suciu’s ribbon knots’, Preprint, 2025, arXiv:2508.15129] later gave a different proof. We present yet another proof of this result by analysing the conjugacy classes of certain automorphisms of the free group of rank two.

Abstract We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p .

Abstract 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 p th 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.

Abstract Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$ , let $R_2(A,n)$ and $R_3(A,n)$ be the number of solutions of the equation $n=a_1+a_2$ with $a_1<a_2, a_1,a_2\in A$ and with $a_1\le a_2, a_1,a_2\in A$ , respectively. If $-N\le g\le N$ , Yan [‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152] showed that there is a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$ for all integers $n\ge 2N-1$ , where N is a positive integer. In this paper, we prove that if $g_1,g_2$ are nonnegative integers with $g_1\neq g_2$ , then there does not exist $A\subseteq \mathbb {N}$ such that $R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$ and $R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$ for all sufficiently large integers n .

Abstract 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.

Abstract A family of $n\times n$ matrices over a field $\mathbb {F}$ is irreducible if it has no common nontrivial invariant subspace, and minimally irreducible if it is irreducible but has no proper irreducible subfamily. If $\mathbb {F}$ is algebraically closed and $n\ge 2$ , a minimally irreducible family has at most $2n-1$ elements. We show that for complex $n\times n$ matrices, $n\ge 3$ , a family of minimally irreducible (i) matrix units, (ii) rank one projections, (iii) unicellular matrices and (iv) orthoatomic matrices has k elements where respectively (i) $n\le k\le 2n-2$ , (ii) $k=n$ , (iii) $2\le k\le n-1$ and (iv) $2\le k\le n-1$ . All of the values of k in these ranges are attained. If $n=2$ , each such minimally irreducible family has $2$ elements.

Abstract Several authors have investigated the structure of groups in which each subgroup satisfies a property $\mathcal {X}$ or a property which is antagonistic to $\mathcal {X}$ . This point of view will be adopted here, considering groups in which each subgroup is either nearly normal or contranormal.