The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note, we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the nth run, is $2$-synchronised and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates and Arnold [‘The summed paperfolding sequence’, Bull. Aust. Math. Soc. 110 (2024), 189–198] in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

For any integers x and y, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y, respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the ath power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as the $(i,j)$-entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divisor chain (namely, there is a permutation $\sigma $ of order n such that $x_{\sigma (1)}\mid \cdots \mid x_{\sigma (n)}$). In this paper, we show that if $a\mid b$ and S is factor closed, then $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and p-adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.

Martin, Mossinghoff, and Trudgian [19] recently introduced a family of arithmetic functions called “fake $\mu $’s,” which are multiplicative functions for which there is a $\{-1,0,1\}$-valued sequence $(\varepsilon _j)_{j=1}^{\infty }$ such that $f(p^j) = \varepsilon _j$ for all primes p. They investigated comparative number-theoretic results for fake $\mu $’s and, in particular, proved oscillation results at scale $\sqrt {x}$ for the summatory functions of fake $\mu $’s with $\varepsilon _1=-1$ and $\varepsilon _2=1$. In this article, we establish new oscillation results for the summatory functions of all nontrivial fake $\mu $’s at scales $x^{1/2\ell }$ where $\ell $ is a positive integer (the “critical index”) depending on f; for $\ell =1$ this recovers the oscillation results in [19]. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $\mu $’s.

We show that the set of Liouville numbers has a rich set-theoretic structure: it can be partitioned in an explicit way into an uncountable collection of subsets, each of which is dense in the real line. Furthermore, each of these partitioning subsets can be similarly partitioned, and the process can be repeated indefinitely.

Weighted sieves are used to detect numbers with at most S prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilise the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes.

As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

Étant donnée une suite $A = (a_n)_{n\geqslant 0}$ d’entiers naturels tous au moins égaux à 2, on pose $q_0 = 1$ et, pour tout entier naturel n, $q_{n+1} = a_n q_n$. Tout nombre entier naturel $n\geqslant 1$ admet une unique représentation dans la base A, dite de Cantor, de la forme

$$ \begin{align*} m = \sum_{j\geqslant 0}\varepsilon_j(m)q_j\quad \text{avec} \quad 0\leqslant \varepsilon_j(m) < a_j = \frac{q_{j+1}}{q_j}. \end{align*} $$

$$ \begin{align*} S = \sum_{n \leqslant x}\Lambda(n) f(n) \end{align*} $$

$\Lambda $

$(a_n)_{n\geqslant 0}$

Nous étudions plusieurs exemples dans la base $A = (j+2)_{j\geqslant 0}$, dite factorielle. En particulier, si $s_A$ désigne la fonction somme de chiffres dans cette base et p parcourt la suite des nombres premiers, nous montrons que la suite $(s_A(p))_{p\in \mathcal {P}}$ est bien répartie dans les progressions arithmétiques, et que la suite $(\alpha s_A(p))_{p\in \mathcal {P}}$ est équirépartie modulo $1$ pour tout nombre irrationnel $\alpha $.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

We show that for any set D of at least two digits in a given base b, almost all even integers taking digits only in D when written in base b satisfy the Goldbach conjecture. More formally, if $\mathcal {A}$ is the set of numbers whose digits base b are exclusively from D, almost all elements of $\mathcal {A}$ satisfy the Goldbach conjecture. Moreover, the number of even integers in $\mathcal {A}$ which are less than X and not representable as the sum of two primes is less than $|\mathcal {A}\cap \{1,\ldots ,X\}|^{1-\delta }$.

Let $k \geqslant 2$ and $b \geqslant 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots , b - 1\}$ are distinct and coprime. Let $\mathcal {S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then every sufficiently large integer is a sum of at most $b^{160 k^2}$ numbers of the form $x^k$, $x \in \mathcal {S}$.

Let X be a smooth projective variety defined over a number field K. We give an upper bound for the generalised greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over K. To prove the result, we establish a rather uniform Riemann–Roch-type inequality.

Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

For fixed m and a, we give an explicit description of those subsets of ${\mathbb F}_{q}$, q odd, for which both x and $mx+a$ are quadratic residues (and other combinations). These results extend and refine results that date back to Gauss.

One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by

$$ \begin{align*}\mathcal{C}_k(q)=\bigg[\begin{matrix}{2k}\\ {k}\end{matrix}\bigg]_q-q \bigg[\begin{matrix}{2k}\\ {k+1} \end{matrix}\bigg]_q.\end{align*} $$

Liu [‘On a congruence involving q-Catalan numbers’, C. R. Math. Acad. Sci. Paris 358 (2020), 211–215] studied congruences of the form $\sum _{k=0}^{n-1} q^k\mathcal {C}_k$ modulo the cyclotomic polynomial $\Phi _n(q)^2$, provided that $n\equiv \pm 1\pmod 3$. Apparently, the case $n\equiv 0\pmod 3$ has been missing from the literature. Our primary purpose is to fill this gap. In addition, we discuss a certain fascinating link to Dirichlet character sum identities.

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

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.