If G is a graph, then $X\subseteq V(G)$ is a general position set if for every two vertices $v,u\in X$ and every shortest $(u,v)$-path P, no inner vertex of P lies in X. We propose three algorithms to compute a largest general position set in G: an integer linear programming algorithm, a genetic algorithm and a simulated annealing algorithm. These approaches are supported by examples from different areas of graph theory.

A fundamental extremality result due to Sidorenko [‘A partially ordered set of functionals corresponding to graphs’, Discrete Math. 131(1–3) (1994), 263–277] states that among all connected graphs G on k vertices, the k-vertex star maximises the number of graph homomorphisms of G into any graph H. We provide a new short proof of this result using only a simple recursive counting argument for trees and Hölder’s inequality.

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime p. The conjectures state that: (i) $J_p$ is always finite and of the order $O(p^2(\log \log p)^{2+\epsilon })$; (ii) the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; (iii) no harmonic number is divisible by $p^4$. We prove parts (i) and (iii) for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to $50\times 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately $30$ and $50$, respectively.

Let p be a fixed odd prime. We prove the following results for positive integer solutions $(x,m,n)$ of the equation $(*)\ x^2=2^m+p^n$. (i) If $p \equiv 3 \pmod 8$, then $(*)$ has only the solution $(p,x,m,n)=(3,5,4,2)$. (ii) If $p \equiv 5 \pmod 8$, then $(*)$ has only the solution $(p,x,m,n)=(5,3,2,1)$. (iii) If $p \equiv 7 \pmod 8$, then $(*)$ has at most one solution $(x,m,n)$, except for $p=7$, $(x,m,n)=(3,1,1)$ and $(9,5,2)$. Moreover, if $p=2^q-1$ is a Mersenne prime with $p>7$, where q is an odd prime with $q>3$, then $(*)$ has exactly one solution $(x,m,n)=(2^q+1,q+2,2)$. If $p \equiv 7\pmod 8$, p is not a Mersenne prime and either $p<1.5\times 10^{12}$ or $p>C$, where C is an effectively computable absolute constant, then $(*)$ has only the solutions $p=a^2-2$, where a is an odd positive integer, $(x,m,n)=(a,1,1)$. (iv) If $p \equiv 1\pmod 8$ with $p \ne 17$, then $(*)$ has at most two solutions $(x,m,n)$.

A square-full number is a positive integer for which all prime divisors divide the number at least twice. For a relatively prime pair $(a,b)\in \mathbb N\times \mathbb N\cup \{0\}$ and an affine polynomial $f(x)=ax+b$, the number of $n\leqslant N$ such that $f(n)$ is square-full is of order $N^{{1}/{2}}$. For $\varepsilon>0$ and an admissible quadratic polynomial $f(x)\in \mathbb Z[x]$, we show that the number of $n\leqslant N$ such that $f(n)$ is square-full is at most $O_{\varepsilon ,f}(N^{\varpi +\varepsilon })$ for some absolute constant $\varpi <1/2$. Under the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon ,f}(N^\varepsilon )$.

Bergelson and Richter [‘Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions’, Duke Math. J. 171(15) (2022), 3133–3200] established a new dynamical generalisation of the prime number theorem (PNT) and the PNT for arithmetic progressions. Let $h\ge 1, k\ge 2$. Mirsky [‘Note on an asymptotic formula connected with r-free integers’, Quart. J. Math. Oxford Ser. 18 (1947), 178–182] showed that the numbers n such that $n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers $l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

Andrews and El Bachraoui [‘On two-colour partitions with odd smallest part’, Preprint, arXiv:2410.14190] explored many integer partitions in two colours, some of which are generated by the mock theta functions of third order of Ramanujan and Watson. They also posed questions regarding combinatorial proofs for these results. In this paper, we establish bijections to provide a combinatorial proof of one of these results and a companion result. We give analytic proofs of further companion results.

Amdeberhan et al. [‘Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime’, Aequationes Math. (2024), doi:10.1007/s00010-024-01116-7] defined the generalised cubic partition function $a_c(n)$ as the number of partitions of n whose even parts may appear in $c\geq 1$ different colours and proved that $a_3(7n+4)\equiv 0\pmod {7}$ and $a_5(11n+10)\equiv 0\pmod {11}$ for all $n\geq 0$ via modular forms. Recently, the author [‘A note on congruences for generalized cubic partitions modulo primes’, Integers 25 (2025), Article no. A20] gave elementary proofs of these congruences. We prove in this note two infinite families of congruences modulo $5$ for $a_c(n)$ given by

$$ \begin{align*} a_{25c+4}(25n+18)&\equiv 0\pmod{5},\\ a_{25c+9}(25n+8)&\equiv 0\pmod{5} \end{align*} $$

for all $c\geq 0$ and $n\geq 0$ using elementary q-series manipulations.

We establish two new truncated identities for theta series. Their partition-theoretic interpretations are also given.

One of the important problems in algebraic number theory is to study the monogenity of number fields. Monogenic number fields arise from the roots of monogenic polynomials. In this article, we deal with the problem of monogenity of the composition of two monic polynomials having integer coefficients. We provide necessary and sufficient conditions for the composition to be monogenic together with a further sufficient condition. At the end of the paper, we construct an infinite tower of monogenic number fields.

We prove the conjecture of Franceschini and Lorenzini [‘Fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace’, J. Pure and Appl. Algebra 160 (2001), 169–182] about the regularity index of fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace.

Let G be a finite group and p be a prime. We prove that if G has three codegrees, then G is an M-group. We prove for some prime p that if the degree of every nonlinear irreducible Brauer character of G is a prime, then for every normal subgroup N of G, either $G/N$ or N is an $M_p$-group.

We strengthen two results of Moretó. We prove that the index of the Fitting subgroup is bounded in terms of the degrees of the irreducible monomial Brauer characters of the finite solvable group G and it is also bounded in terms of the average degree of the irreducible Brauer characters of G that lie over a linear character of the Fitting subgroup.

For a group G, a subgroup $U \leqslant G$ and a group A such that $\mathrm {Inn}(G) \leqslant A \leqslant \mathrm {Aut}(G)$, we say that U is an A-covering group of G if $G = \bigcup _{a\in A}U^a$. A theorem of Jordan (1872), implies that if G is a finite group, $A = \mathrm {Inn}(G)$ and U is an A-covering group of G, then $U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1990) conjectured that, more generally, there is an integer function f such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U| \leqslant f(|A:\mathrm {Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger [‘Kronecker classes of fields and covering subgroups of finite groups’, J. Aust. Math. Soc. 57 (1994), 17–34], which asserts that there is a two-variable integer function g such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U|\leqslant g(|A:\mathrm {Inn}(G)|,c)$, where c is the number of A-chief factors of G. Unfortunately, the proof of this theorem contains an error. In this paper, using a different argument, we give a correct proof of the theorem.

Let g be an element of a group G. For a positive integer n, let $R_n(g)$ be the subgroup generated by all commutators $[\ldots [[g,x],x],\ldots ,x]$ over $x\in G$, where x is repeated n times. Similarly, $L_n(g)$ is defined as the subgroup generated by all commutators $[\ldots [[x,g],g],\ldots ,g]$, where $x\in G$ and g is repeated n times. In the literature, there are several results showing that certain properties of groups with small subgroups $R_n(g)$ or $L_n(g)$ are close to those of Engel groups. The present article deals with orderable groups in which, for some $n\geq 1$, the subgroups $R_n(g)$ are polycyclic. Let $h\geq 0$, $n>0$ be integers and G be an orderable group in which $R_n(g)$ is polycyclic with Hirsch length at most h for every $g\in G$. It is proved that there are $(h,n)$-bounded numbers $h^*$ and $c^*$ such that G has a finitely generated normal nilpotent subgroup N with $h(N)\leq h^*$ and $G/N$ nilpotent of class at most $c^*$. The analogue of this theorem for $L_n(g)$ was established in 2018 by Shumyatsky [‘Orderable groups with Engel-like conditions’, J. Algebra 499 (2018), 313–320].

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.

In this paper, we characterise all maximal elements in the semigroup $S(X,Y)=\{f\in T(X) : f(Y)\subseteq Y\}$ with respect to the natural partial order. Our results correct an error in the work of Sun and Wang [‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc. 87(1) (2013), 94–107].

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 $\mathcal {G}$.

Kam Cheong Au [‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, Journal of Symbolic Computation 130 (2025), Article no. 102241] discovered a powerful methodology for finding new Wilf–Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of $\pi $ and their duals for Dirichlet L-values. In this note, we explain how a modification of Au’s WZ pairs together with a classical analytic argument yields simpler proofs of these results. We illustrate our method with examples elaborated with assistance of Maple code that we have developed.

Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let $\{T_s\}$ be an asymptotic nonexpansive semigroup of nonlinear mappings acting within C. Consider the implicit iteration process defined by the sequence of equations:

$$ \begin{align*} x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k,\end{align*} $$

where each $c_k \in (0,1)$ and the initial point $x_0 \in C$ is arbitrarily chosen. In this context, we investigate the conditions under which the sequence $\{x_k\}$ converges, either weakly or strongly, to a common fixed point of the semigroup $\{T_s\}$. We also touch upon the question of the stability of such processes.