We give the generating function of split $(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split $n$-colour partitions. We derive a combinatorial relation between the number of restricted split $n$-colour partitions and the function $\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with $d(a)$ copies of each part $a$ and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math 22(9) (1991), 737–743].

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

Let $q\geq 1$ be any integer and let $\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes $p$ satisfying

$$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$

$g$

$p$

$\text{gcd}(g,(p-1)/q)=1$

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$, let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $|G|=2|A|$, and (b) if $h$ is even and $|G|=2|A|$, then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$, then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$.

We prove two conjectural congruences on the $(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.

A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ never equals the sum of $l$ (not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a $(k,l)$-sum-free subset in $G$ for all $k$ and $l$ in the case when $G$ is cyclic by proving that it suffices to consider $(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’, Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a $(k,l)$-sum-free subset of an abelian group’, Int. J. Number Theory 5(6) (2009), 953–971].

Let $n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set $\mathbb{Z}/n\mathbb{Z}$. In particular, let $p$ be an odd prime and let $\unicode[STIX]{x1D6FC}$ be a positive integer. If $H_{k}$ is a subgroup of $(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index $k=p^{\unicode[STIX]{x1D6FD}}u$ such that $0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and $u\mid p-1$, then

$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$

For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except for an explicit list of exceptions and that $S$ is always ‘large’ in the sense that $|T|^{1/3}<|S|\leq |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that, for every $T$ and every prime divisor $r$ of $|T|$ with $r\neq p$, the order of the Sylow $r$-subgroup of $T$ is at most $|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$, where $\ell$ is the Lie rank of $T$.

One of the open questions in the study of Carmichael numbers is whether, for a given $R\geq 3$, there exist infinitely many Carmichael numbers with exactly $R$ prime factors. Chernick [‘On Fermat’s simple theorem’, Bull. Amer. Math. Soc.45 (1935), 269–274] proved that Dickson’s $k$-tuple conjecture would imply a positive result for all such $R$. Wright [‘Factors of Carmichael numbers and a weak $k$-tuples conjecture’, J. Aust. Math. Soc.100(3) (2016), 421–429] showed that a weakened version of Dickson’s conjecture would imply that there are an infinitude of $R$ for which there are infinitely many such Carmichael numbers. In this paper, we improve on our 2016 result by weakening the required conjecture even further.

Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$, let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$. There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers $n$. We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer $m$, exactly one of $2m+1$ and $2m$ is in $A$.

If $\unicode[STIX]{x1D703}$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\unicode[STIX]{x1D703}$-subgroups if it admits no infinite double sequences

$$\begin{eqnarray}\cdots <X_{-n}<\cdots <X_{-1}<X_{0}<X_{1}<\cdots <X_{n}<\cdots\end{eqnarray}$$

$\unicode[STIX]{x1D703}$

In this paper, we consider the Diophantine equations

$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$

$q,p\geq 2$

$\gcd (F_{n},F_{m})=1$

$F_{k}$

$q=2$

$q$

$q\equiv 3\;(\text{mod}\;4),3<q<1087$

$q=3$

We introduce an Almgren frequency function of the sub-$p$-Laplace equation on the Heisenberg group to establish a doubling estimate under the assumption that the frequency function is locally bounded. From this, we obtain some partial results on unique continuation for the sub-$p$-Laplace equation.

We improve some previously known deterministic algorithms for finding integer solutions $x,y$ to the exponential equation of the form $af^{x}+bg^{y}=c$ over finite fields.

We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

In this article, we prove the transcendence of certain infinite sums and products by applying the subspace theorem. In particular, we extend the results of Hančl and Rucki [‘The transcendence of certain infinite series’, Rocky Mountain J. Math.35 (2005), 531–537].

By using variational and some new analytic techniques, we prove the existence of ground state solutions for the quasilinear Schrödinger equation with variable potentials and super-linear nonlinearities. Moreover, we establish a minimax characterisation of the ground state energy. Our result improves and extends the existing results in the literature.