We devise schemes for producing, in the least possible time, p identical objects with n agents that work at differing speeds. This involves halting the process to transfer production across agent types. For the case of two types of agent, we construct schemes based on the Euclidean algorithm that seeks to minimize the number of pauses in production.

For any positive integer n, let $\sigma (n)$ be the sum of all positive divisors of n. We prove that for every integer k with $1\leq k\leq 29$ and $(k,30)=1,$

$$ \begin{align*} \sum_{n\leq K}\sigma(30n)>\sum_{n\leq K}\sigma(30n+k) \end{align*} $$

for all $K\in \mathbb {N},$ which gives a positive answer to a problem posed by Pongsriiam [‘Sums of divisors on arithmetic progressions’, Period. Math. Hungar. 88 (2024), 443–460].

For $g \geqslant 2$, we show that the number of positive integers at most X which can be written as sum of two base g palindromes is at most ${X}/{\log^c X}$. This answers a question of Baxter, Cilleruelo and Luca.

We establish a q-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a q-congruence of Guo and Zeng [‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory 145 (2014), 301–316]. The important ingredients in the proof include Andrews’ $_4\phi _3$ terminating identity.

Let $k\geq 4$ be an integer. We prove that the set $\mathcal {O}$ of all nonzero generalised octagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function $f_k$ satisfies the condition

$$ \begin{align*} f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{align*} $$

for arbitrary $x_1,\ldots ,x_k\in \mathcal {O}$, then $f_k$ is the identity function $f_k(n)=n$ for all $n\in \mathbb {N}$. We also show that $f_2$ and $f_3$ are not determined uniquely.

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

In this paper, we prove that the bound

\begin{equation*}\max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}}\end{equation*}

$A \subset \mathbb R$

\begin{equation*}\max \{ |16A|, |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c},\end{equation*}

$c\gt 0$

$\mathbb R$

$3/2$

\begin{equation*}|AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{\frac{2}{3}-c^{\prime}},\end{equation*}

$c,C \gt 0$

Liu [‘Supercongruences for truncated Appell series’, Colloq. Math. 158(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series $F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.

We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory 13(11) (2011), 2219–2238]. Let p be an odd prime. Then

$$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$

where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p-adic integers $a, b$.

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then

\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}

where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

The Thue–Morse sequence $\{t(n)\}_{n\geqslant 0}$ is the indicator function of the parity of the number of ones in the binary expansion of nonnegative integers n, where $t(n)=1$ (resp. $=0$) if the binary expansion of n has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E. Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt {\varphi }=1.272019\ldots $, the set of the numbers

$$\begin{align*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align*}$$

$\mathbb {Q}(\beta )$

$\varphi :=(1+\sqrt {5})/2$

$k\geqslant 1$

$a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta )$

$a_1t(n)+a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$

We prove $\times a \times b$ measure rigidity for multiplicatively independent pairs when $a\in \mathbb {N}$ and $b>1$ is a ‘specified’ real number (the b-expansion of $1$ has a tail or bounded runs of $0$s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $\times a$ invariant measure. The quantitative version together with the $\times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a-shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.

In this article, we give generalizations of the well-known Fermat’s Little Theorem, Wilson’s theorem, and the little-known Gegenbauer’s theorem.

Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$, we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $

For an integer $b\geq 2$, a positive integer is called a b-Niven number if it is a multiple of the sum of the digits in its base-b representation. In this article, we show that every arithmetic progression contains infinitely many b-Niven numbers.

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv _w$ introduced by Šobot is an equivalence relation on $\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv _w$ coincides with the strong congruence relation $\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient $(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv _w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat {{\mathbb Z}}$ of the integers.

Let $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$. Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function

$$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$

which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function.

We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative.

As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]

$p\equiv 1\pmod 3$

\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]

$f_n=\sum _{k=0}^n\binom {n}k^3$

$n$