Narrow Results

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang–Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang–Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.

The Cauchy equation for automorphisms of the unit interval is fulfilled only by the identity mapping. We consider two weakened forms of this Cauchy equation: the reciprocity equation and the n-divisibility equation. Although the solution sets of both functional equations are quite large, requiring that an automorphism is both reciprocal and n-divisible, drastically increases the set of obligatory fixed points. However, increasing n also enlarges the domain on which the automorphism can be chosen freely. We also describe the solution sets of two functional equations that arise by composing the reciprocity and n-divisibility property.

Let $a$, $b$ and $n$ be positive integers with$n\ge 2$, $f$ be an integer-valued arithmetic function, and the set $S=\{x_1,\dots ,x_n\}$ of $n$ distinct positive integers be a divisor chain such that $x_1|x_2|\cdots |x_n$. We first show that the matrix $(f^a(S))$ having $f$ evaluated at the $a$th power ${(x_i,x_j)}^a$ of the greatest common divisor of $x_i$ and $x_j$ as its $i,j$-entry divides the GCD matrix $(f^b(S))$ in the ring $M_n(\mathbf{Z})$ of $n\times n$ matrices over integers if and only if $f^{b-a}(x_1)\in \mathbf{Z}$ and $(f^a(x_i)-f^a(x_{i-1}))$ divides $(f^b(x_i)-f^b(x_{i-1}))$ for any integer $i$ with $2\le i\le n$. Consequently, we show that the matrix $(f^a[S])$ having $f$ evaluated at the $a$th power ${[x_i,x_j]}^a$ of the least common multiple of $x_i$ and $x_j$ as its $i,j$-entry divides the matrix $(f^b[S])$ in the ring $M_n(\mathbf{Z})$ if and only if $f^{b-a}(x_n)\in \mathbf{Z}$ and $(f^a(x_i)-f^a(x_{i-1}))$ divides $(f^b(x_i)-f^b(x_{i-1}))$ for any integer $i$ with$2\le i\le n$. Finally, we prove that the matrix $(f^a(S))$ divides the matrix $(f^b[S])$ in the ring $M_n(\mathbf{Z})$ if and only if $f^a(x_1)|f^b(x_i)$ and $(f^a(x_i)-f^a(x_{i-1}))|(f^b(x_i)-f^b(x_{i-1}))$ for any integer $i$ with $2\le i\le n$. Our results extend and strengthen the theorems of Hong obtained in 2008.

In this paper, we give some heuristics suggesting that if (u_n)_n≥0 is the Lucas sequence given by u_n = (aⁿ - 1)/(a - 1), where a > 1 is an integer, then ω(u_n) ≥ (1 + o(1))log n log log n holds for almost all positive integers n.

Given positive integers a,b,c and d such that c and d are coprime, we show that the primes p ≡ c (mod d) dividing a^k+b^k for some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ₊ be fixed. Does there exist a positive integer k such that

The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that

The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems.

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight $k\ge 0$, which is related to the weight of Borcherds lifts when $k=0$. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for $\mathbb{Q}(\sqrt{5})$. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime $p$ and positive integer $m=o(p^{1/2}{(\mathrm{\text{log}}p)}^{-4})$, outputs an elliptic curve $E$ over the finite field ${𝔽}_p$ for which the cardinality of $E({𝔽}_p)$ is divisible by $m$. The running time of the algorithm is $m{p}^{1/2+o(1)}$, and this leads to more efficient constructions of rational functions over ${𝔽}_p$ whose image is small relative to $p$. We also give an unconditional version of the algorithm that works for almost all primes $p$, and give a probabilistic algorithm with subexponential time complexity.

We prove that if ${(u_n)}_{n\ge 0}$ is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers $n$ does not exceed $x$ and such that $n$ divides $u_n$ is less than

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version of Linnik’s theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers.

Let $m\ge 1$ be an integer and $p$ be an odd prime. We study sums and lacunary sums of $m$th powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the $p$-adic convergence of some subsequences and that in every step we gain at least one or three more $p$-adic digits of the limit if $m=1$ or $m\ge 2$, respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.

In this paper, we consider two particular binomial sums

Let $m\ge 2$ be an even integer and $p$ be an odd prime. We study Franel-like sums and alternating sums, as well as lacunary sums of $m$th powers of binomial coefficients from the point of view of arithmetic properties. This paper complements the author’s prior work on the cases with $m\ge 1$ odd although, it uses a different approach. It develops new supercongruences and determines the $p$-adic order of these sums as well as of generalized harmonic sums restricted to particular remainder classes modulo $p$.

Let ${𝒜}$ be the set of all positive integers $n$ such that $n$ divides the central binomial coefficient $\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)$. Pomerance proved that the upper density of ${𝒜}$ is at most $1-log2$. We improve this bound to $1-log2-0.05551$. Moreover, let ${ℬ}$ be the set of all positive integers $n$ such that $n$ and $\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)$ are relatively prime. We show that $\#({ℬ}\cap [1,x])\ll x/\sqrt{logx}$ for all $x>1$.

Let $a,n,m\in {\mathbb{Z}}^{+}$ with $(a,p)=1$ and $p$ be an odd prime. We find a supercongruence for ${\sum }_{p|k}{\left(\genfrac{}{}{0ex}{}{a{p}^n}{k}\right)}^m$ and related sums of powers of binomial coefficients. These results complement prior results for ${\sum }_{k\equiv i(\mathrm{\text{mod}}p)}{\left(\genfrac{}{}{0ex}{}{a{p}^n}{k}\right)}^m$ with $i=1,2,\dots ,p-1$ obtained recently by the author.

In this paper, we establish some new divisibility results involving the Franel numbers $f_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^3$$(n=0,1,2,\dots )$ and the polynomials $g_n(x)={\sum }_{k=0}^n{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)x^k$$(n=0,1,2,\dots )$. For example, we show that for any positive integer $n$ we have

Let $a$ and $n$ be positive integers and let $S=\{x_1,\dots ,x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_S(x)=\{d\in S:d<x,d|x\mathrm{\text{and}}(d|y|x,y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by $(S^a)$ (respectively, $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (respectively, the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In this paper, we show that for arbitrary positive integers $a$ and $b$ with $a|b$, the $b$th power matrices $(S^b)$ and $[S^b]$ are both divisible by the $a$th power matrix $(S^a)$ if $S$ is a gcd-closed set (i.e. $gcd(x_i,x_j)\in S$ for all integers $i$ and $j$ with $1\le i,j\le n$) such that ${max}_{x\in S}\{|G_S(x)|\}=1$. This confirms two conjectures of Shaofang Hong proposed in 2008.

Let $U={(U_n)}_{n\ge 0}$ be a Lucas sequence and, for every prime number $p$, let ${\rho }_U(p)$ be the rank of appearance of $p$ in $U$, that is, the smallest positive integer $k$ such that $p$ divides $U_k$, whenever it exists. Furthermore, let $d$ be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes $p\le x$ such that $d$ divides ${\rho }_U(p)$, as $x\to +\infty$.