We present a new version of a generalisation to elliptic nets of a theorem of Ward [‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948), 31–74] on symmetry of elliptic divisibility sequences. Our results cover all that is known today.

We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$. Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

Given a prime $p\geqslant 5$ and an integer $s\geqslant 1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$, such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$. We also provide some explicit formulas for the optimal $M$.

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$

$k=6$

$p$

$A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$

$g\in \{2,3,5\}$

$|A|\gg p/(\log p)^{3}$

$A\cdot A+A\cdot A$

$\mathbb{F}_{p}$

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where $\mathsf{D}(G)$ denotes the Davenport constant of $G$. The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

We prove that for each positive integer k in the range 2≤k≤10 and for each positive integer k≡79 (mod 120) there is a k-step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.

The sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let LEn and Lon be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that LEn ≠ Lon when n is even. Drisko showed that LEp+1 ≢ Lop+1 (mod p3) for prime p ≥ 3 and asked if similar congruences hold for orders of the form pk + 1, p + 3, or pq + 1. In this article we show that if t ≤ n, then LEn+1 ≢ L0n+1 (mod t3) only if t = n and n is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for Lo/LE, and propose a generalization of the Alon-Tarsi conjecture.

For a primitive root g modulo a prime p≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn, n=1,…,N, which is uniform over all integers a with gcd (a,p)=1.

Let x0 < x1 < x2 < ⋅⋅⋅ be an increasing sequence of positive integers given by the formula xn=⌊βxn−1 + γ⌋ for n=1, 2, 3, . . ., where β > 1 and γ are real numbers and x0 is a positive integer. We describe the conditions on integers bd, . . ., b0, not all zero, and on a real number β > 1 under which the sequence of integers wn=bdxn+d + ⋅⋅⋅ + b0xn, n=0, 1, 2, . . ., is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qxn+1 − pxn ∈ {0, 1, . . ., q−1}, n=0, 1, 2, . . ., where x0 is a positive integer, p > q > 1 are coprime integers and xn=⌈pxn−1/q⌉ for n=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 or S:x↦(3x+1)/2) in the 3x+1 problem is also given.

For a large integer $m$, we obtain asymptotic formulae for the number of solutions of certain congruences modulo $m$ with several variables, where the variables belong to special sets of residue classes modulo $m$. In particular, we obtain new information on the exceptional set of the multiplication table problem in the residue ring modulo $m$.

A lower and an upper bound are proved for the maximum size of a subset of $\mathbb{Z}/p\mathbb{Z}$ containing no solutions of a fixed linear equation.

Let $G$ be an additively written abelian group, and let $S$ be a sequence in $G \setminus \{0\}$ with length $|S| \ge 4$. Suppose that $S$ is a product of two subsequences, say $S = B C$, such that the element $g+h$ occurs in the sequence $S$ whenever $g \cdot h$ is a subsequence of $B$ or of $C$. Then $S$ contains a proper zero-sum subsequence, apart from some well-characterized exceptional cases. This result is closely connected with restricted set addition in abelian groups. Moreover, it solves a problem on the structure of minimal zero-sum sequences, which recently occurred in the theory of non-unique factorizations.

A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.

Let $p$ be prime and let $\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form

$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$

and prove that for any $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.