For any integers x and y, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y, respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the ath power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as the $(i,j)$-entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divisor chain (namely, there is a permutation $\sigma $ of order n such that $x_{\sigma (1)}\mid \cdots \mid x_{\sigma (n)}$). In this paper, we show that if $a\mid b$ and S is factor closed, then $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and p-adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.

Given two graphs G and H, the Ramsey number $R(G,H)$ is the smallest positive integer N such that every graph of order N contains G or its complement contains H as a subgraph. Let $C_n$ denote the cycle on n vertices and let $tW_{2m+1}$ denote the disjoint union of t copies of the $(2m+2)$-vertex wheel $W_{2m+1}$. We show that for integers $m\ge 1$, $t\ge 2$ and $n\ge (6m+3)t-6m+999$,

$$ \begin{align*} R(C_n, tW_{2m+1})=3n+t-3. \end{align*} $$

This result extends several previous results and settles a conjecture posed by Sudarsana [‘A note on the Ramsey number for cycle with respect to multiple copies of wheels’, Electron. J. Graph Theory Appl. 9(2) (2021), 561–566].

The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note, we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the nth run, is $2$-synchronised and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates and Arnold [‘The summed paperfolding sequence’, Bull. Aust. Math. Soc. 110 (2024), 189–198] in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

A finite group is said to be n-cyclic if it contains n cyclic subgroups. For a finite group G, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group G and is denoted by $\textit {cdeg} (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which solves a problem posed by Tărnăuceanu and Tóth [‘Cyclicity degrees of finite groups’, Acta Math. Hungar. 145(2) (2015), 489–504].

It is shown that the Fourier sine transform, $\mathcal{F}_S [f(t)](\omega )$ on $\mathbb {R}_0^+$, of any given real-valued function $f(t)$ that does not vanish at $t=0$ or has a nonvanishing even-order derivative at $t=0$, has a definite sign at least for $\omega> \omega _0$, where $\omega _0$ can be estimated. Similarly, the cosine transform, $\mathcal{F}_C [f(t)](\omega )$, of functions with a nonvanishing odd-order derivative at zero also has a definite sign for sufficiently large $\omega $. Several examples are given.

In this paper, we generalise to the family of Fermat quartics $X^4 + Y^4 = 2^m, m \in \mathbb {Z}$, a result of Aigner [‘Über die Möglichkeit von $x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely $\mathbb {Q}(\sqrt {-7})$, that contains solutions to the Fermat quartic $X^4 + Y^4 = 1$. The $m \equiv 0 \pmod 4$ case is due to Aigner. The $m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation $X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases $m \equiv 1, 3 \pmod 4$, classifying for $m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for $m \equiv 3 \pmod 4$ that $\mathbb {Q}(\sqrt {2})$ and $\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.

Existing multivariate versions of the logistic probability distribution generally lack some of the useful properties of the univariate logistic distribution, such as its bounded score function or the tractability of its density function, or lack the rotational symmetry necessary for many applications. This paper clarifies some of the properties of such distributions and proposes a multivariate distribution closely related to the univariate logistic that has a tractable density, including the necessary normalising constant, bounded score function and elliptical symmetry. Some properties of its marginal distributions are explored, particularly in the bivariate case.

For each $n\geq 1$, let $FT_n$ be the free tree monoid of rank n and $E_n$ the full extensive transformation monoid over the finite chain $\{1, 2, \ldots , n\}$. It is shown that the monoids $FT_n$ and $E_{n+1}$ satisfy the same identities. Therefore, $FT_n$ is finitely based if and only if $n\leq 3$.

The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$-dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$. We also derive divisibility properties for the number of $(r+s)$-dimensional king walks of length p (an odd prime) through group actions.

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

A subset of a finite set of filling curves on a surface is not necessarily filling. However, when a filling set spans homology and curves intersect pairwise at most once, it is shown that one can always add a curve and subtract a different curve to obtain a filling set that spans homology. A motivation for filling sets of curves that span homology comes from the Thurston spine and the Steinberg module of the mapping class group.

A subgroup X of a group G is said to be transitively normal if X is normal in any subgroup Y of G such that $X\leq Y$ and X is subnormal in Y. We investigate the structure of generalised soluble groups with dense transitively normal subgroups, that is, groups in which every nonempty open interval in their subgroup lattice contains a transitively normal subgroup. In particular, it will be proved that nonperiodic generalised soluble groups with dense transitively normal subgroups are abelian.

We investigate semigroups S which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on S. We call such an S a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups, including periodic, stable, inverse and several well-known types of simple semigroups, the only DSC members are groups. However, it turns out that there exist nongroup DSC semigroups, which we obtain by utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.

Brun’s constant is the summation of the reciprocals of all twin primes, given by

$$ \begin{align*}B=\sum_{p \in P_2}{\bigg( \frac{1}{p} + \frac{1}{p+2}\bigg)}.\end{align*} $$

While rigorous unconditional bounds on B are known, we present the first rigorous bound on Brun’s constant under the assumption of GRH, yielding $B < 2.1594$.