We investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$, with the exception of a set of density zero.

We consider finite simple graphs. Given a graph H and a positive integer $n,$ the Turán number of H for the order $n,$ denoted $\mathrm {ex}(n,H),$ is the maximum size of a graph of order n not containing H as a subgraph. Erdős asked: ‘For which graphs H is it true that every graph on n vertices and $\mathrm {ex}(n,H)+1$ edges contains at least two H’s? Perhaps this is always true.’ We solve this problem in the negative by proving that for every integer $k\ge 4$ there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size $\mathrm {ex}(n,H)+1$ which contains exactly one copy of $H.$ Denote by $C_4$ the $4$-cycle. We also prove that for every integer n with $6\le n\le 11$ there exists a graph of order n and size $\mathrm {ex}(n,C_4)+1$ which contains exactly one copy of $C_4,$ but, for $n=12$ or $n=13,$ the minimum number of copies of $C_4$ in a graph of order n and size $\mathrm {ex}(n,C_4)+1$ is two.

Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

In the 1993 Western Number Theory Conference, Richard Guy proposed Problem 93:31, which asks for integers n representable by ${(x+y+z)^3}/{xyz}$, where $x,\,y,\,z$ are integers, preferably with positive integer solutions. We show that the representation $n={(x+y+z)^3}/{xyz}$ is impossible in positive integers $x,\,y,\,z$ if $n=4^{k}(a^2+b^2)$, where $k,\,a,\,b\in \mathbb {Z}^{+}$ are such that $k\geq 3$ and $2\nmid a+b$.

Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in \mathbb {Z}$ be a primitive root modulo p. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo p in the interval $(0,p)$. Then clearly the sequence $a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$ is a permutation of the sequence $g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$. We determine the sign of this permutation.

For a set A of positive integers and any positive integer n, let $R_{1}(A, n)$, $R_{2}(A,n)$ and $R_{3}(A,n)$ denote the number of solutions of $a+a^{\prime }=n$ with $a, a^{\prime }\in A$ and the additional restriction that $a<a^{\prime }$ for $R_{2}$ and $a\leq a^{\prime }$ for $R_{3}$. We consider Problem 6 of Erdős et al. [‘On additive properties of general sequences’, Discrete Math. 136 (1994), 75–99] about locally small and locally large values of $R_{1}, R_{2}$ and $R_{3}$.

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$. For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$, $n\geq 3$, $x,\,y>0$ and $\gcd (x,\,y)=1$. Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

We answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J. 7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

Let $f(x)\in \mathbb {Z}[x]$ be a nonconstant polynomial. Let $n\ge 1, k\ge 2$ and c be integers. An integer a is called an f-exunit in the ring $\mathbb {Z}_n$ of residue classes modulo n if $\gcd (f(a),n)=1$. We use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,\ldots ,x_k)$ of the congruence $x_1+\cdots +x_k\equiv c\pmod n$ with all $x_i$ being f-exunits in the ring $\mathbb {Z}_n$. This extends a recent result of Anand et al. [‘On a question of f-exunits in $\mathbb {Z}/{n\mathbb {Z}}$’, Arch. Math. (Basel) 116 (2021), 403–409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.

In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$-numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$.

Necessary and sufficient conditions for the equality of the determinant and permanent for all powers of a given matrix are provided. The characterisation is based on a condition on merely one power.

Let $\mathcal {P}_s(n)$ denote the nth s-gonal number. We consider the equation

$$ \begin{align*}\mathcal{P}_s(n) = y^m \end{align*} $$

for integers $n,s,y$ and m. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number greater than 1 expressible as a perfect mth power with $m>1$ is $\mathcal {P}_{10}(3) = 3^3$.

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct. 30(7) (2019), 581–593].

Using a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math. 78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by $\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$ contains infinitely many consecutive square-free integers whenever $1<c<3/2$.

For any x in $[0,1)$, let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $. For any positive integers s and t, we study the set

$$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$

and determine its Hausdorff dimension.

Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$, where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

In this note, by introducing a new variant of the resonator function, we give an explicit version of the lower bound for $\log |L(\sigma ,\chi )|$ in the strip $1/2<\sigma <1$, which improves the result of Aistleitner et al. [‘On large values of $L(\sigma ,\chi )$’, Q. J. Math. 70 (2019), 831–848].