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We define an
$(a,b)$
-Wall–Sun–Sun prime to be a prime p such that
$\gcd (p,b)=1$
and
$\pi (p^2)=\pi (p),$
where
$\pi (p):=\pi _{(a,b)}(p)$
is the length of the period of
$\{U_n\}_{n\ge 0}$
modulo p. When
$(a,b)=(1,1)$
, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing
$a^2+4b$
is an
$(a,b)$
-Wall–Sun–Sun prime.
We discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect numbers.
In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field
$\mathbb {F}_p$
, with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic
$\mathbb {F}_p^\omega $
-systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for
$k\leq p+1$
the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for
$k< p$
. We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on
$\mathbb {F}_p^n$
.
Let
$(x_n)_{n\geq 0}$
be a linear recurrence of order
$k\geq 2$
satisfying
$x_n=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$
for all integers
$n\geq k$
, where
$a_1,\ldots ,a_k,x_0,\ldots , x_{k-1}\in \mathbb {Z},$
with
$a_k\neq 0$
. Sanna [‘The quotient set of k-generalised Fibonacci numbers is dense in
$\mathbb {Q}_p$
’, Bull. Aust. Math. Soc.96(1) (2017), 24–29] posed the question of classifying primes p for which the quotient set of
$(x_n)_{n\geq 0}$
is dense in
$\mathbb {Q}_p$
. We find a sufficient condition for denseness of the quotient set of the kth-order linear recurrence
$(x_n)_{n\geq 0}$
satisfying
$ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$
for all integers
$n\geq k$
with initial values
$x_0=\cdots =x_{k-2}=0,x_{k-1}=1$
, where
$a_1,\ldots ,a_k\in \mathbb {Z}$
and
$a_k=1$
. We show that, given a prime p, there are infinitely many recurrence sequences of order
$k\geq 2$
whose quotient sets are not dense in
$\mathbb {Q}_p$
. We also study the quotient sets of linear recurrence sequences with coefficients in certain arithmetic and geometric progressions.
with initial data
$(x_1,x_2,x_3)=(0,x,1)$
, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion
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.
Let
$h \geq 2$
be a positive integer. We introduce the concept of minimal restricted asymptotic bases and obtain some examples of minimal restricted asymptotic bases of order h.
Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem’s strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.
Let
$\mathcal {A}$
be the set of all integers of the form
$\gcd (n, F_n)$
, where n is a positive integer and
$F_n$
denotes the nth Fibonacci number. Leonetti and Sanna proved that
$\mathcal {A}$
has natural density equal to zero, and asked for a more precise upper bound. We prove that
for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.
A Sidon set is a subset of an Abelian group with the property that the sums of two distinct elements are distinct. We relate the Sidon sets constructed by Bose to affine subspaces of
$ \mathbb {F} _ {q ^ 2} $
of dimension one. We define Sidon arrays which are combinatorial objects giving a partition of the group
$\mathbb {Z}_{q ^ 2} $
as a union of Sidon sets. We also use linear recurring sequences to quickly obtain Bose-type Sidon sets without the need to use the discrete logarithm.
We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including
We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.
Recently, Lin and Liu [‘Congruences for the truncated Appell series
$F_3$
and
$F_4$
’, Integral Transforms Spec. Funct.31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series
$F_3$
. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.
When a page, represented by the interval
$[0,1]$
, is folded right over left
$n $
times, the right-hand fold contains a sequence of points. We specify these points using two different representation techniques, both involving binary signed-digit representations.
We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form
$\{m,m+[p_n^a], m+[p_n^b]\}$
, where
$a,b$
are positive nonintegers and
$p_n$
denotes the nth prime, a property that fails if a or b is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences, and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime k-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.
Let n and k be positive integers with
$n\ge k+1$
and let
$\{a_i\}_{i=1}^n$
be a strictly increasing sequence of positive integers. Let
$S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$
. In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull.20 (1978), 117–118] confirmed a conjecture of Erdős by showing that
$S_{n,1}\le 1-{1}/{2^{n-1}}$
. Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar.160 (2020), 360–375] improved Borwein’s upper bound to
$S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$
and derived optimal upper bounds for
$S_{n,2}$
and
$S_{n,3}$
. In this paper, we present a sharp upper bound for
$S_{n,4}$
and characterise the sequences
$\{a_i\}_{i=1}^n$
for which the upper bound is attained.
We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s
$_{8}\phi _{7}$
transformation, Andrews’ terminating q-analogue of Watson’s
$_{3}F_{2}$
summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci.2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math.72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo
$p^4$
for any prime
$p\equiv 1\pmod {4}$
. Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for
$p\equiv 3 \pmod 4$
and
$p>3$
. In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo
$p^3$
for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358].
Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the
$\mathbb{C}$
-algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.