A generating set S for a group G is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in G for all $s \in S.$ We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G.

We discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect numbers.

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal {S}$ is a quotient of a numerical semigroup with k generators, we call $\mathcal {S}$ a k-quotient. We give a necessary condition for a given numerical semigroup $\mathcal {S}$ to be a k-quotient and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a k-quotient. We also examine the probability that a randomly selected numerical semigroup with k generators is a k-quotient.

Let $(x_n)_{n\geq 0}$ be a linear recurrence of order $k\geq 2$ satisfying $x_n=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$, where $a_1,\ldots ,a_k,x_0,\ldots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$. Sanna [‘The quotient set of k-generalised Fibonacci numbers is dense in $\mathbb {Q}_p$’, Bull. Aust. Math. Soc. 96(1) (2017), 24–29] posed the question of classifying primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$. We find a sufficient condition for denseness of the quotient set of the kth-order linear recurrence $(x_n)_{n\geq 0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\cdots =x_{k-2}=0,x_{k-1}=1$, where $a_1,\ldots ,a_k\in \mathbb {Z}$ and $a_k=1$. We show that, given a prime p, there are infinitely many recurrence sequences of order $k\geq 2$ whose quotient sets are not dense in $\mathbb {Q}_p$. We also study the quotient sets of linear recurrence sequences with coefficients in certain arithmetic and geometric progressions.

For any Liouville number $\alpha $, all of the following are transcendental numbers: ${e}^\alpha $, $\log _{e}\alpha $, $\sin \alpha $, $\cos \alpha $, $\tan \alpha $, $\sinh \alpha $, $\cosh \alpha $, $\tanh \alpha $, $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$, it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr. 49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2) 10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.

Given a Fell bundle $\mathscr C\overset {q}{\to }\Xi $ over the discrete groupoid $\Xi $, we study the symmetry of the associated Hahn algebra $\ell ^{\infty ,1}(\Xi \!\mid \!\mathscr C)$ in terms of the isotropy subgroups of $\Xi $. We prove that $\Xi $ is symmetric (respectively hypersymmetric) if and only if all of the isotropy subgroups are symmetric (respectively hypersymmetric). We also characterise hypersymmetry using Fell bundles with constant fibres, showing that for discrete groupoids, ‘hypersymmetry’ equals ‘rigid symmetry’.

Let a, b, c be fixed coprime positive integers with $\min \{ a,b,c \}>1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples, then $c>10^{18}$, and there are exactly two solutions $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$ with $2 \mid x_1$, $2 \mid y_1$, $z_1=1$, $2 \nmid y_2$, $z_2>1$, and, taking $a<b$, we must have $a=2$, $b \equiv 1 \bmod 12$, $c \equiv 5\, \mod 12$, with $(a,b,c)$ satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to $a^x + b^y = c^z$’, Publ. Math. Debrecen 88 (2016), 131–138].

In this note, we give an example of a domain whose d-balanced squeezing function is nonplurisubharmonic.

We provide a geometric interpretation of the well-known A3w condition for regularity in optimal transport.

In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$, the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$. Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$.

The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$. We show that the Ree group ${}^2G_2(q)$, where $q = 3^{2f+1}$, is determined up to isomorphism by its set of codegrees.

Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$, where $1\le k_1<k_2<\cdots <k_r=k$. Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$, we show that there is a paucity of nondiagonal solutions to the Diophantine system $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.

If G is a group with subgroup H and m, k are two fixed nonnegative integers, H is called an $(m,k)$ -subnormal subgroup of G if it has index at most m in a subnormal subgroup of G of defect less than or equal to k. We study the behaviour of uncountable groups of cardinality $\aleph $ where all subgroups of cardinality $\aleph $ are $(m,k)$ -subnormal.

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 $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ 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 $E(K_n)$ is equal to the rank of $E(K)$ . 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 $p_t(a,b;n)$ denote the number of partitions of n such that the number of t-hooks is congruent to $a \bmod {b}$ . For $t\in \{2, 3\}$ , arithmetic progressions $r_1 \bmod {m_1}$ and $r_2 \bmod {m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ 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 $ 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 $G/Q$ 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 $G/N$ 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 $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

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

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