For any integer k and any positive integer n, let $\sigma _k(n)=\sum _{d\mid n}d^k$. For any prime p and any positive integer m, let $\nu _p(m)$ be the largest integer $\alpha $ such that $p^\alpha \mid m$ and let $\lceil x\rceil $ denote the least integer not less than x. In 2021, Amdeberhan et al. [‘Arithmetic properties of the sum of divisors’, J. Number Theory 223 (2021), 325–349] proved that $\nu _2(\sigma _1(n))\le \lceil \log _2n\rceil $ for any positive integer n and that $\nu _p(\sigma _1(n))\le \lceil \log _pn\rceil $ for any odd prime p if n satisfies some conditions. Recently, Zhao and Chen [‘p-adic valuation of the sum of divisors’, Front. Math. 20(4) (2025), 795–827] proved this unconditionally. We generalise these results to all k: for any prime p, any n and any $k\ge 2$, $\nu _p(\sigma _k(n))\le \lceil k\log _p n\rceil .$ Let $p^\star $ be an odd prime. We also prove that there are an integer $k\ge 2$ and a prime q satisfying $\nu _{p^\star }(\sigma _k(q))=\lceil k\log _{p^\star }q\rceil $ if and only if $p^\star $ is a Fermat prime.

Let C and W be two sets of integers. If $C+W=\mathbb {Z}$, then C is called an additive complement to W. We further call C a minimal additive complement to W if no proper subset of C is an additive complement to W. Answering a problem of Nathanson in part, we give sufficient conditions to show that W has no minimal additive complements. Our result extends a result of Chen and Yang [‘On a problem of Nathanson related to minimal additive complements’, SIAM J. Discrete Math. 26 (2012), 1532–1536].

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.

We prove two results related to the solutions of norm form equations. First, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Second, we give a finiteness result concerning solutions of norm form equations representable as sums of S-units with a fixed number of terms. To prove these results, we use a deep result concerning the finiteness of the solutions of polynomial-exponential equations and S-unit equations.

Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn’s criterion applies to a polynomial f if it applies to the number field $K=\mathbb {Q}[x]/(f)$ generated by f. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd (p-1,n)=1$. Let $\mathcal {F}_{p,n}$ denote the collection of monic polynomials $f\in \mathbb {Z}[x]$ of degree n that are Eisenstein at the prime p. Order the polynomials by the natural height $\operatorname {\mathrm {Ht}}(f)$. Define $\delta _{p,n}(X)$ to be the proportion of polynomials $f\in \mathcal {F}_{p,n}$ with $\operatorname {\mathrm {Ht}}(f)\leq X$ for which Heilbronn’s criterion applies. Then,

$$ \begin{align*} \liminf_{X\to\infty} \delta_{p,n}(X)\geq \max\{\tfrac{2}{27},\;1-\varepsilon(p)\}, \end{align*} $$

where $\varepsilon (p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to \infty $ uniformly in n. We also give a version of this result where we weaken the condition on $\gcd (p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree n fail to generate norm-Euclidean fields.

Given a nonzero integer n, Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull., to appear] classified all polynomials $x^2+ax+b\in {\mathbb {Z}}[x]$ for which the polynomial Pell equation $P^2-(x^2+ax+b)Q^2=n$ has solutions ${P,Q\in {\mathbb {Z}}[x]}$ with $Q\neq 0$. We generalise their work to the equation $P^2-(f^2+af+b)Q^2=nR$, where f is a fixed polynomial in ${\mathbb {Z}}[x]$. As an application of our results, we study the equation $P^2-D(f)Q^2=n$, where D is a monic, quartic and non square-free polynomial in ${\mathbb {Z}}[x]$. This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].

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 $6$.

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.

For a prime p, let $\mathcal {N}_p(G)$ denote the intersection of the normalisers of all non-p-nilpotent subgroups of a finite group G and set $\mathcal {N}_p(G)=G$ if G itself is p-nilpotent. We give some properties of $\mathcal {N}_p(G)$ and investigate the influence of $\mathcal {N}_p(G)$ on G.

For a group G and $m\ge 1$, $G^m$ denotes the subgroup generated by the elements $g^m$, where g runs through G. The subgroups not of the form $G^m$ are called nonpower subgroups. We extend the classification of groups with few nonpower subgroups from groups with at most nine nonpower subgroups to groups with at most 13 nonpower subgroups.

Given a group G and an automorphism $\varphi $ of G, two elements $x,y\in G$ are said to be $\varphi $-conjugate if $x=gy\varphi (g)^{-1}$ for some $g\in G$. The number $R(\varphi )$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi $ and the set $\{R(\varphi ) \mid \varphi \in \text {Aut}(G)\}$ is called the Reidemeister spectrum of G. We determine the Reidemeister spectrum of ZM-groups, extending some results of Senden [‘The Reidemeister spectrum of split metacyclic groups’, Preprint, 2022, arXiv:2109.12892].

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.

We consider a nonautonomous abstract evolution equation on Banach space. If the evolutionary operator and the evolutionary kernel are real analytic at the initial time there is an explicit formula for the coefficients of the operator in terms of the coefficients of the kernel. In this note we show that the converse is also true by finding an explicit formula for the coefficients of the kernel in terms of the coefficients of the operator.

We consider quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $-\Delta u+V(x)u-u\Delta (u^2)=g(u)$, where the potential V is allowed to be sign-changing and the nonlinearity g is sublinear at zero. Except for being subcritical, no additional condition is imposed on $g(u)$ for $|u|$ large. We obtain a sequence of solutions with negative energy and converging to zero via Clark’s theorem. We also obtain a similar result for fourth-order quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $\Delta ^2u-\Delta u+ V(x)u-u\Delta (u^2)=g(u)$.

We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals and give an application to Baire category results for functions of the digits.

Let $(\Omega , \mathfrak {M}, \mu )$ be a measure space. We show that if a surjective map $f: S_{L^1(\mu )} \to S_Y$ between the unit spheres of real $L^1(\mu )$ and of an arbitrary real normed space Y satisfies

$$ \begin{align*} \min\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\min\{\|x+y\|, \|x-y\|\},\quad x,y\in S_{L^1(\mu)}, \end{align*} $$

then there exists a phase function $\varepsilon : S_{L^1(\mu )} \to \{-1, 1\}$ such that $\varepsilon \cdot f$ is a surjective isometry from $S_{L^1(\mu )}$ onto $S_Y$, and furthermore, this isometry can be extended to a linear isometry on the whole space $L^1(\mu )$.

We study projection algorithms for finding fixed points of nonexpansive mappings in a Hilbert space. In previous work, the well-definedness of the iterates depended on the nonemptiness of the fixed point set. We first show that the existence of the iterates generated by a general projection algorithm is independent of the existence of fixed points. Next, we introduce a multi-step inertial projection algorithm and establish results on existence, strong convergence and iteration complexity. Our results generalise and improve projection algorithms discussed in the literature.

We study a discrete process on planar convex bodies in which, at each step, a body is replaced by a weighted Minkowski average of itself and its rotation by a fixed angle. Up to translation and uniform scaling, this produces a rigid averaging dynamical system. We give a complete classification of the limit shapes. If the angle is an irrational multiple of $2\pi $, the iterates converge to a disk. If the angle is rational, they converge to the average of finitely many rotated copies of the initial body. We also obtain sharp convergence rates. In the rational case, the decay is uniform and exponential with an explicit constant depending only on the weight and the denominator of the angle. For irrational angles, we prove quantitative rates under a mild number-theoretic condition that holds for almost every angle: low regularity inputs have polynomial decay up to a logarithmic factor, while real analytic inputs have stretched exponential decay. For angles with bounded continued fraction coefficients, we give matching lower bounds along subsequences. These results describe the global attractors of the dynamics and indicate the absence of chaotic behaviour.

Cromwell [‘Homogeneous links’, J. London Math. Soc. (2) 39(3) (1989), 535–552] proved that the minimum v-degree of the HOMFLY polynomial of a homogeneous link L is bounded above by $1-\chi (L)$, where $\chi (L)$ is the maximum Euler characteristic of Seifert surfaces of L. We prove its slice version, stating that the minimum v-degree of the HOMFLY polynomial of a homogeneous link L is bounded above by $1-\chi _4(L)$, where $\chi _4(L)$ is the maximum four-dimensional Euler characteristic of L. As a byproduct, we prove a conjecture of Stoimenow [‘Some inequalities between knot invariants’, Internat. J. Math. 13(4) (2002), 373–393] that for an alternating link, the minimum v-degree of the HOMFLY polynomial is smaller than or equal to its signature.

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.