For a graph G, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of G. Given a positive integer m and a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. We prove bounds on $f(m,H)$ for some bipartite graphs H and give a bound for a conjecture of Alon et al. [‘MaxCut in H-free graphs’, Combin. Probab. Comput. 14 (2005), 629–647] concerning $f(m,K_{4,s})$.

Let G be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited while maintaining the general position property at all times. The mobile general position number of G is the cardinality of a largest mobile general position set of G. We give bounds on the mobile general position number and determine exact values for certain common classes of graphs, including block graphs, rooted products, unicyclic graphs, Kneser graphs $K(n,2)$ and line graphs of complete graphs.

The diamond is the complete graph on four vertices minus one edge; $P_n$ and $C_n$ denote the path and cycle on n vertices, respectively. We prove that the chromatic number of a $(P_6,C_4,\mbox {diamond})$-free graph G is no larger than the maximum of 3 and the clique number of G.

For a fixed integer h, the standard orthogonality relations for Ramanujan sums $c_r(n)$ give an asymptotic formula for the shifted convolution $\sum _{n\le N} c_q(n)c_r(n+h)$. We prove a generalised formula for affine convolutions $\sum _{n\le N} c_q(n)c_r(kn+h)$. This allows us to study affine convolutions $\sum _{n\le N} f(n)g(kn+h)$ of arithmetical functions $f,g$ admitting a suitable Ramanujan–Fourier expansion. As an application, we give a heuristic justification of the Hardy–Littlewood conjectural asymptotic formula for counting Sophie Germain primes.

We first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.

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.

For a graph G and a family of graphs $\mathcal {F}$, the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$-free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.

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.

Let a and b be positive integers and let $\{U_n\}_{n\ge 0}$ be the Lucas sequence of the first kind defined by

$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$

We define an $(a,b)$-Wall–Sun–Sun prime to be a prime p such that $\gcd (p,b)=1$ and $\pi (p^2)=\pi (p),$ where $\pi (p):=\pi _{(a,b)}(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo p. When $(a,b)=(1,1)$, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing $a^2+4b$ is an $(a,b)$-Wall–Sun–Sun prime.

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 $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$-extensions of F known as constant $\mathbb {Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$, where K is any field of characteristic $\ell $, showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.

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].

We investigate the large values of the derivatives of the Riemann zeta function $\zeta (s)$ on the 1-line. We give a larger lower bound for $\max _{t\in [T,2T]}|\zeta ^{(\ell )}(1+{i} t)|$, which improves the previous result established by Yang [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510].

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $-approximable points and for the set of inhomogeneous multiplicative $\psi $-approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math. 740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$, to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

We prove some zero density theorems for certain families of Dirichlet L-functions. More specifically, the subjects of our interest are the collections of Dirichlet L-functions associated with characters to moduli from certain sparse sets and of certain fixed orders.

Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for $c_o(n)$, the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by $c_o(n)$. We confirm the conjecture and evaluate $c_o(4n)$ modulo $8$ by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.

We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math. 31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$. We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$, and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$.

Motivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers $(u_n)_{n\geq 0}$. We show that if $(u_n)_{n\geq 0}$ is a sequence of integers with $u_n = O(n)$ which can be p-adically interpolated by an analytic function $f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$, then $f(x)$ is a polynomial function of degree at most one. The case $u_n=O(n^d)$ with $d>1$ is also considered with additional conditions. Moreover, if X and Y are subsets of $\mathbb {Z}$ dense in $\mathbb {Z}_p$, we prove that there are uncountably many p-adic analytic injective functions $f:\mathbb {Z}_p\to \mathbb {Q}_p$, with rational coefficients, such that $f(X)=Y$.

Let k be a field of characteristic zero and $k^{[n]}$ the polynomial algebra in n variables over k. The LND conjecture concerning the images of locally nilpotent derivations arose from the Jacobian conjecture. We give a positive answer to the LND conjecture in several cases. More precisely, we prove that the images of rank-one locally nilpotent derivations of $k^{[n]}$ acting on principal ideals are MZ-subspaces for any $n\geq 2$, and that the images of a large class of locally nilpotent derivations of $k^{[3]}$ (including all rank-two and homogeneous rank-three locally nilpotent derivations) acting on principal ideals are MZ-subspaces.