Given a large integer n, determining the relative size of each of its prime divisors as well as the spacings between these prime divisors has been the focus of several studies. Here, we examine the spacings between particular types of prime divisors of n, such as prime divisors in certain congruence classes of primes and various other subsets of the set of prime numbers.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let K_{\operatorname {cyc}}$K_{\operatorname {cyc}}$ be the cyclotomic \mathbb {Z}_p$\mathbb {Z}_p$ -extension of K and K_n$K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of E(K_n)$E(K_n)$ is equal to the rank of E(K)$E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

Let p_t(a,b;n)$p_t(a,b;n)$ denote the number of partitions of n such that the number of t-hooks is congruent to a \bmod {b}$a \bmod {b}$ . For t\in \{2, 3\}$t\in \{2, 3\}$ , arithmetic progressions r_1 \bmod {m_1}$r_1 \bmod {m_1}$ and r_2 \bmod {m_2}$r_2 \bmod {m_2}$ on which p_t(r_1,m_1; m_2 n + r_2)$p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males and Ono [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma 10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t-cores and t-quotients.

A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if \langle Q,g\rangle $\langle Q,g\rangle$ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that G/Q$G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that G/N$G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

The Euler–Mascheroni constant \gamma =0.5772\ldots \!$\gamma =0.5772\ldots \!$ is the K={\mathbb Q}$K={\mathbb Q}$ example of an Euler–Kronecker constant \gamma _K$\gamma _K$ of a number field K.$K.$ In this note, we consider the size of the \gamma _q=\gamma _{K_q}$\gamma _q=\gamma _{K_q}$ for cyclotomic fields K_q:={\mathbb Q}(\zeta _q).$K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

\begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$\begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*}$$

In other words, under EH, the \gamma _q /\!\log q$\gamma _q /\!\log q$ in these ranges converge to the one point distribution at 1$1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime q.$q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

We give a sufficient and necessary condition for a Praeger–Xu graph to be a Cayley graph.

We prove a result on the existence and uniqueness of the solution of a new feature-preserving nonlinear nonlocal diffusion equation for signal denoising for the one-dimensional case. The partial differential equation is based on a novel diffusivity coefficient that uses a nonlocal automatically detected parameter related to the local bounded variation and the local oscillating pattern of the noisy input signal.

We define a graph encoding the structure of contact surgery on contact 3$3$ -manifolds and analyse its basic properties and some of its interesting subgraphs.

Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic \mathbb {Z}_p$\mathbb {Z}_p$ extension of \mathbb {Q}$\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

We show that the system of equations

\begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$\begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*}$$

has appreciably fewer solutions in the subcritical range s < \tfrac 12k(k+1)$s < \tfrac 12k(k+1)$ than its homogeneous counterpart, provided that a_{\ell } \neq 0$a_{\ell } \neq 0$ for some \ell \leqslant k-1$\ell \leqslant k-1$ . Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

We present a family of counterexamples to a question proposed recently by Moretó concerning the character codegrees and the element orders of a finite solvable group.

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on \mathbb {R}$\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of \mathbb {R}$\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set \mathcal {L}^{2}$\mathcal {L}^{2}$ , where \mathcal {L}$\mathcal {L}$ is the set of all Liouville numbers, and the fat set is \mathbb {R}$\mathbb {R}$ itself. Finally, it is shown that \mathcal {L}$\mathcal {L}$ and \mathcal {L}^{2}$\mathcal {L}^{2}$ are both homeomorphic to \mathbb {P}$\mathbb {P}$ , the space of all irrational numbers.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial D \in \mathbb {Z}[x]$D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation D(x) \equiv 0 \pmod {p}$D(x) \equiv 0 \pmod {p}$ has a solution.

Let \eta (G)$\eta (G)$ be the number of conjugacy classes of maximal cyclic subgroups of G. We prove that if G is a p-group of order p^n$p^n$ and nilpotence class l, then \eta (G)$\eta (G)$ is bounded below by a linear function in n/l$n/l$ .

We prove that if s\ge 2$s\ge 2$ is a fixed integer, then the equation ns^n+1=(b^m-1)/(b-1)$ns^n+1=(b^m-1)/(b-1)$ has only finitely many positive integer solutions (n,b,m)$(n,b,m)$ with b\ge 2$b\ge 2$ and m\ge 3$m\ge 3$ . When s=2$s=2$ , it has no solution.

In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant -31$-31$ and Fourier coefficients of a weight 16$16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant -23$-23$ .

Let k\geq 2$k\geq 2$ be an integer. We prove that the 2-automatic sequence of odious numbers \mathcal {O}$\mathcal {O}$ is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$ for arbitrary x_1,\ldots ,x_k\in \mathcal {O}$x_1,\ldots ,x_k\in \mathcal {O}$ , then f is the identity function f(n)=n$f(n)=n$ for all n\in \mathbb {N}$n\in \mathbb {N}$ .

If G is permutation group acting on a finite set \Omega $\Omega$ , then this action induces a natural action of G on the power set \mathscr{P}(\Omega )$\mathscr{P}(\Omega )$ . The number s(G)$s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, \inf ({\log _2 s(G)}/{\log _2 |G|})$\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any {{\textrm {A}}}_l, l> 4${{\textrm {A}}}_l, l> 4$ , as a composition factor.

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and \dim G=\dim V$\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or \mathrm {GL}_2(\mathbb {Z})$\mathrm {GL}_2(\mathbb {Z})$.