Let n,r,k\in \mathbb{N}$n,r,k\in \mathbb{N}$. An r$r$-colouring of the vertices of a regular n$n$-gon is any mapping \unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An r$r$-ary necklace of length n$n$ is an equivalence class of r$r$-colourings of \mathbb{Z}_{n}$\mathbb{Z}_{n}$. We say that a colouring is k$k$-alternating if all k$k$ consecutive vertices have pairwise distinct colours. We compute the smallest number r$r$ for which there exists a k$k$-alternating r$r$-colouring of \mathbb{Z}_{n}$\mathbb{Z}_{n}$ and we count, for any r$r$, 2-alternating r$r$-colourings of \mathbb{Z}_{n}$\mathbb{Z}_{n}$ and 2-alternating r$r$-ary necklaces of length n$n$.

We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.

As usual, P_{n}$P_{n}$ (n\geq 1$n\geq 1$) denotes the path on n$n$ vertices. The gem is the graph consisting of a P_{4}$P_{4}$ together with an additional vertex adjacent to each vertex of the P_{4}$P_{4}$. A graph is called (P_{5}$P_{5}$, gem)-free if it has no induced subgraph isomorphic to a P_{5}$P_{5}$ or to a gem. For a graph G$G$, \unicode[STIX]{x1D712}(G)$\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and \unicode[STIX]{x1D714}(G)$\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in G$G$. We show that \unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every (P_{5}$P_{5}$, gem)-free graph G$G$.

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted {\mathcal{S}}_{n}${\mathcal{S}}_{n}$, and the star graph, denoted K_{1,m}$K_{1,m}$, is considered for the first time. It is proved that the crossing number of {\mathcal{S}}_{n}\Box K_{1,2}${\mathcal{S}}_{n}\Box K_{1,2}$ is n$n$, and the crossing number of {\mathcal{S}}_{n}\Box K_{1,3}${\mathcal{S}}_{n}\Box K_{1,3}$ is 3n$3n$. An upper bound for the crossing number of {\mathcal{S}}_{n}\Box K_{1,m}${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

Let n$n$ be a positive integer and a$a$ an integer prime to n$n$. Multiplication by a$a$ induces a permutation over \mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo p$p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary k$k$th power residues modulo p$p$ and primitive roots modulo a power of p$p$.

We prove that, given a positive integer m$m$, there is a sequence \{n_{i}\}_{i=1}^{k}$\{n_{i}\}_{i=1}^{k}$ of positive integers such that

\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray} $$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$

\{1/n_{i}\}_{i=1}^{k}$\{1/n_{i}\}_{i=1}^{k}$

We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].

For a graph G$G$, let f(G)$f(G)$ denote the maximum number of edges in a bipartite subgraph of G$G$. Given a fixed graph H$H$ and a positive integer m$m$, let f(m,H)$f(m,H)$ denote the minimum possible cardinality of f(G)$f(G)$, as G$G$ ranges over all graphs on m$m$ edges that contain no copy of H$H$. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph H$H$, there exists an \unicode[STIX]{x1D716}(H)>0$\unicode[STIX]{x1D716}(H)>0$ such that f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph W_{2k}$W_{2k}$ of 2k$2k$ spokes, there exists c(k)>0$c(k)>0$ such that f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for W_{4}$W_{4}$ and give general lower bounds for W_{2k+1}$W_{2k+1}$.

In this note, we provide refined estimates of two sums involving the Euler totient function,

\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray} $$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$

[x]$[x]$

x$x$

We prove that for every m\geq 0$m\geq 0$ there exists an \unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if 0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and x$x$ is sufficiently large in terms of m$m$ and \unicode[STIX]{x1D706}$\unicode[STIX]{x1D706}$, then

\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray} $$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$

\unicode[STIX]{x1D700}(m)$\unicode[STIX]{x1D700}(m)$

\unicode[STIX]{x1D706}$\unicode[STIX]{x1D706}$

m$m$

m$m$

\unicode[STIX]{x1D706}$\unicode[STIX]{x1D706}$

x$x$

\mathbb{P}$\mathbb{P}$

We establish an error term in the Sato–Tate theorem of Birch. That is, for p$p$ prime, q=p^{r}$q=p^{r}$ and an elliptic curve E:y^{2}=x^{3}+ax+b$E:y^{2}=x^{3}+ax+b$, we show that

\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray} $$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$

I\subseteq [0,\unicode[STIX]{x1D70B}]$I\subseteq [0,\unicode[STIX]{x1D70B}]$

\unicode[STIX]{x1D703}_{a,b}$\unicode[STIX]{x1D703}_{a,b}$

2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$

\unicode[STIX]{x1D707}_{ST}(I)$\unicode[STIX]{x1D707}_{ST}(I)$

I$I$

We establish finite analogues of the identities known as the Aoki–Ohno relation and the Le–Murakami relation in the theory of multiple zeta values. We use an explicit form of a generating series given by Aoki and Ohno.

A Ducci sequence is a sequence of integer n$n$-tuples generated by iterating the map

\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray} $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$

P(n)$P(n)$

n$n$

n$n$

p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$

2$2$

x^{2}-py^{2}=-4$x^{2}-py^{2}=-4$

x$x$

y$y$

Let p$p$ be a prime. If an integer g$g$ generates a subgroup of index t$t$ in (\mathbb{Z}/p\mathbb{Z})^{\ast },$(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that g$g$ is a t$t$-near primitive root modulo p$p$. We point out the easy result that each coprime residue class contains a subset of primes p$p$ of positive natural density which do not have g$g$ as a t$t$-near primitive root and we prove a more difficult variant.

A cyclotomic polynomial \unicode[STIX]{x1D6F7}_{k}(x)$\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of f(x)\in \mathbb{Z}[x]$f(x)\in \mathbb{Z}[x]$ if \unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of k$k$ is less than or equal to the number of terms of f.$f.$ We show that if a monic polynomial with coefficients from \{-1,0,1\}$\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

We study the singularity at the origin of \mathbb{C}^{n+1}$\mathbb{C}^{n+1}$ of an arbitrary homogeneous polynomial in n+1$n+1$ variables with complex coefficients, by investigating the monodromy characteristic polynomials \unicode[STIX]{x1D6E5}_{l}(t)$\unicode[STIX]{x1D6E5}_{l}(t)$ as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case n=2$n=2$, we give a description of \unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$ in terms of the multiplier ideal.

Let D$D$ be a positive nonsquare integer, p$p$ a prime number with p\nmid D$p\nmid D$ and 0<\unicode[STIX]{x1D70E}<0.847$0<\unicode[STIX]{x1D70E}<0.847$. We show that there exist effectively computable constants C_{1}$C_{1}$ and C_{2}$C_{2}$ such that if there is a solution to x^{2}+D=p^{n}$x^{2}+D=p^{n}$ with p^{n}>C_{1}$p^{n}>C_{1}$, then for every x>C_{2}$x>C_{2}$ with x^{2}+D=p^{n}m$x^{2}+D=p^{n}m$ we have m>x^{\unicode[STIX]{x1D70E}}$m>x^{\unicode[STIX]{x1D70E}}$. As an application, we show that for x\neq \{5,1015\}$x\neq \{5,1015\}$, if the equation x^{2}+76=101^{n}m$x^{2}+76=101^{n}m$ holds, then m>x^{0.14}$m>x^{0.14}$.

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the k$k$th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each k$k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.

Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, a_{1}(5n+4)=0$a_{1}(5n+4)=0$, where a_{1}(n)$a_{1}(n)$ is defined by

\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray} $$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$