Inspired by work of Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254] on the minimal excludant or ‘mex’ of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

A graph G is called an $[s,t]$-graph if any induced subgraph of G of order s has size at least $t.$ We prove that every $2$-connected $[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing results. There are $2$-connected $[4,2]$-graphs which do not satisfy the Chvátal–Erdős condition on Hamiltonicity. We also determine the triangle-free graphs among $[p+2,p]$-graphs for a general $p.$

Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

For an integer $k \geq 2$, let $P_{n}^{(k)}$ be the k-generalised Pell sequence, which starts with $0, \ldots ,0,1$ (k terms), and each term thereafter is given by the recurrence $P_{n}^{(k)} = 2 P_{n-1}^{(k)} +P_{n-2}^{(k)} +\cdots +P_{n-k}^{(k)}$. We search for perfect powers, which are sums or differences of two k-generalised Pell numbers.

Let G and H be two vertex disjoint graphs. The join $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup \{xy\;|\; x\in V(G), y\in V(H)\}$. A (finite) linear forest is a graph consisting of (finite) vertex disjoint paths. We prove that for any finite linear forest F and any nonnull graph H, if $\{F, H\}$-free graphs have a $\chi $-binding function $f(\omega )$, then $\{F, K_n+H\}$-free graphs have a $\chi $-binding function $kf(\omega )$ for some constant k.

Let s be a fixed positive integer constant and let $\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set ${\mathcal M}\subseteq \mathbb {F}_p^*$ of size $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer $H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$

When $s=1$, we show that, for almost all primes p, if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$

Let X be a smooth projective variety defined over a number field K. We give an upper bound for the generalised greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over K. To prove the result, we establish a rather uniform Riemann–Roch-type inequality.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

We develop a generalisation of the square sieve of Heath-Brown and use it to give an alternate proof of one of the large sieve inequalities in our previous paper [‘A large sieve inequality for characters to quadratic moduli’, Preprint, https://web.maths.unsw.edu.au/~ccorrigan/preprint6.pdf].

We prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka, Dharmendra and Kumar [‘Divisibility properties for overcubic partition triples’, Integers 24 (2024), Article no. a80, 9 pages]. We also generalise overcubic partition triples to overcubic partition k-tuples and prove arithmetic properties for these partitions.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.

We consider the relationship between the Mahler measure $M(f)$ of a polynomial f and its separation $\operatorname {sep}(f)$. Mahler [‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262] proved that if $f(x) \in \mathbb {Z}[x]$ is separable of degree n, then $\operatorname {sep}(f) \gg _n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relationship and led to questions about the optimal exponent on $M(f)$. However, there has been relatively little study concerning upper bounds on $\operatorname {sep}(f)$ in terms of $M(f)$. We prove that if $f(x) \in \mathbb {C}[x]$ has degree n, then $\operatorname {sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$; for example, if it has only real roots.

We use deformations and mutations of scattering diagrams to show that a scattering diagram with initial functions $f_1=(1+tx)^\mu $ and $f_2=(1+ty)^\nu $ has a dense region. This answers a question asked by Gross and Pandharipande [‘Quivers, curves, and the tropical vertex’, Port. Math. 67(2) (2010), 211–259] which had been proved only for the case $\mu =\nu $.

Let E be an elliptic curve defined over $\mathbb {Q}$ with good ordinary reduction at a prime $p\geq 5$ and let F be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell–Weil group of E over the $\mathbb {Z}_{p}^2$-extension of F is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb {Z}_{p}^2$.

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb {F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a three-dimensional linear system $\mathcal {L}$ with at most one geometrically reducible $\mathbb {F}_q$-member.

We establish explicit constructions of Mahler’s p-adic $U_{m}$-numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

Let $[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]$ be the continued fraction expansion of $x\in [0,1)$ and $q_n(x)$ be the denominator of its nth convergent. The irrationality exponent and Khintchine exponent of x are respectively defined by

$$ \begin{align*} \overline{v}(x)=2+\limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)} \quad \text{and}\quad \gamma(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log a_i(x). \end{align*} $$

We study the multifractal spectrum of the irrationality exponent and the Khintchine exponent for continued fractions with nondecreasing partial quotients. For any $v>2$, we completely determine the Hausdorff dimensions of the sets $\{x\in [0,1): a_1(x)\leq a_2(x)\leq \cdots , \overline {v}(x)=v\}$ and

$$ \begin{align*}\bigg\{x\in[0,1): a_1(x)\leq a_2(x)\leq\cdots, \lim\limits_{n\to\infty}\frac{\log a_1(x)+\log a_2(x)+\cdots+\log a_n(x)}{\psi(n)}=1\bigg\},\end{align*} $$

where $\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ is a function satisfying $\psi (n)\to \infty $ as $n\to \infty $.

Let $\mathcal {H}$ be the class of all analytic self-maps of the open unit disk $\mathbb {D}$. Denote by $H^n f(z)$ the nth-order hyperbolic derivative of $f\in \mathcal H$ at $z\in \mathbb {D}$. We develop a method allowing us to calculate higher-order hyperbolic derivatives in an expeditious manner. We also generalise certain classical results for variability regions of the nth derivative of bounded analytic functions. For $z_0\in \mathbb {D}$ and $\gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$, let ${\mathcal H} (\gamma ) = \{f \in {\mathcal H} : f (z_0) = \gamma _0,H^1f (z_0) = \gamma _1,\ldots ,H^{n-1}f (z_0) = \gamma _{n-1} \}$. We determine the variability region $\{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$ to prove a Schwarz–Pick lemma for the nth derivative. We apply this result to establish an nth-order Dieudonné lemma, which provides an explicit description of the variability region $\{h^{(n)}(z_0): h\in \mathcal {H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,\ldots , h^{(n-1)}(z_0)=w_{n-1}\}$ for given $z_0$, $w_0$, $w_1,\ldots ,w_{n-1}$. Moreover, we determine the form of all extremal functions.