We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.

Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree $d$, algebraic numbers of degree $d$ that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

We study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

We study the differentiability of the limiting distribution function associated to the normalized Euler function defined on the shifted primes.

We describe how the Hardy–Ramanujan–Rademacher formula can be implemented to allow the partition function p(n) to be computed with softly optimal complexity O(n1/2+o(1)) and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate p(1019), an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of p(n), where our implementation of the Hardy–Ramanujan–Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver’s tabulation of 76 065 congruences. Supplementary materials are available with this article.

For a class of multiplicative integer-valued functions $f$ the distribution of the sequence $f(n)$ in restricted residue classes modulo $N$ is studied. We consider a property weaker than weak uniform distribution and study it for polynomial-like multiplicative functions, in particular for $\varphi (n)$ and $\sigma (n)$.

We describe the period structure of the optimal continued fraction expansion of a quadratic surd, in terms of the period of its nearest square continued fraction expansion. The analysis results in a faster algorithm for determining the optimal continued fraction expansion of a quadratic surd.

Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.

We construct a family of ideals representing ideal classes of order two in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

We obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

We consider stacks of filtered $\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize $p$-adic Galois representations of the absolute Galois group of a $p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

In this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

For $k\geq 2$ and $r\geq 1$, we prove that the number of odd $k$-perfect numbers with $r$ distinct prime factors is at most $4^{r^2}(k-1)^{2r^2+3}$.

We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χ≠χ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.

Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.

From power series expansions of functions on curves over finite fields, one can obtain sequences with perfect or almost perfect linear complexity profile. It has been suggested by various authors to use such sequences as key streams for stream ciphers. In this work, we show how long parts of such sequences can be computed efficiently from short ones. Such sequences should therefore be considered to be cryptographically weak. Our attack leads in a natural way to a new measure of the complexity of sequences which we call expansion complexity.

The chromatic polynomial P(G,λ) gives the number of ways a graph G can be properly coloured in at most λ colours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separable θ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.