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.

This paper is motivated by two conjectures proposed by Bender et al. [‘Complemented zero-divisor graphs associated with finite commutative semigroups’, Comm. Algebra 52(7) (2024), 2852–2867], which have remained open questions. The first conjecture states that if the complemented zero-divisor graph $ G(S) $ of a commutative semigroup $ S $ with a zero element has clique number three or greater, then the reduced graph $ G_r(S) $ is isomorphic to the graph $ G(\mathcal {P}(n)) $. The second conjecture asserts that if $ G(S) $ is a complemented zero-divisor graph with clique number three or greater, then $ G(S) $ is uniquely complemented. We construct a commutative semigroup $ S $ with a zero element that serves as a counter-example to both conjectures.

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].

Let A be a set of natural numbers. A set B of natural numbers is an additive complement of the set A if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. We establish that if $A=\{a_i: i\in \mathbb {N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb {N}$ and $\liminf _{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb {N}$ such that $B\cap A = \varnothing $ and B is a sparse additive complement of the set A.

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.

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.

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.

We study sums of squares of integers except for a fixed one. For any nonnegative integer n, we find the minimum number of squares of integers except for n whose sums represent all positive integers that are represented by a sum of squares except for n. This problem could be considered as a generalisation of the result of Dubouis [‘Solution of a problem of J. Tannery’, Intermédiaire Math. 18 (1911), 55–56] for the case when $n=0$.

For any integer $t \geq 2$, we prove a local limit theorem (LLT) with an explicit convergence rate for the number of parts in a uniformly chosen t-regular partition. When $t = 2$, this recovers the LLT for partitions into distinct parts, as previously established in the work of Szekeres [‘Asymptotic distributions of the number and size of parts in unequal partitions’, Bull. Aust. Math. Soc. 36 (1987), 89–97].

Let K be an infinite field. If $\alpha $ and $\beta $ are algebraic and separable elements over K, then by the primitive element theorem, it is well known that $\alpha +u\beta $ is a primitive element for $K(\alpha , \beta )$ for all but finitely many elements $u\in K$. If we let

$$ \begin{align*}\xi_K(\alpha, \beta) = \{u\in K : K(\alpha, \beta) \ne K(\alpha+u\beta)\}\end{align*} $$

be the exceptional set, then by the primitive element theorem, $|\xi _K(\alpha , \beta )| < \infty $. Dubickas [‘An effective version of the primitive element theorem’, Indian J. Pure Appl. Math. 53(3) (2022), 720–726] estimated the size of this set when $K = \mathbb {Q}$. We take K to be a finite extension over $\mathbb {Q}$ or $\mathbb {Q}_p$, the field of p-adic numbers for some prime p, and estimate the size of the exceptional set.

This note provides an alternative proof of a theorem by Li et al. [‘On the primitivity of some trinomials over finite fields’, Adv. Math. (China) 44(3) (2015), 387–393] regarding the nonprimitivity of the trinomial $x^{n}+ax+b$ over $\mathbb {F}_{q^{m}}$ under the condition $a^{n}b^{1-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. We extend this result to the trinomial $x^{n}+a^{k}x^{k}+b^{k}$, showing its nonprimitivity over $\mathbb {F}_{q^{m}}$ when $ a^{n}b^{k-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. While the existing proof relies on the theory of linear recurrences over finite fields, our approach is short and self-contained, requiring no prior knowledge of this area.

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 $B < 2.1594$.

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].

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.

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 $n\leq 3$.

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.

We extend the definition of the X-Rokhlin property to countable discrete groups and prove some permanence properties. If the action of a countable discrete group on X is free and minimal and the action of this group on the separable simple $C^*$-algebra has the X-Rokhlin property, then the reduced crossed product is simple.

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.

In the 1950s, Morse defined the analogue of Morse functions for topological manifolds. In many instances, when mathematicians are using techniques on topological manifolds that appear to be Morse-theoretic in nature, there is a topological Morse function implicit in the argument. Topological Morse functions are known to inherit most of the familiar properties of the usual (smooth) Morse functions, with two crucial exceptions: existence and deformability. This paper gives a simple construction of continuous families of topological Morse functions.

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.