International Journal of Number Theory

Let $\mathbb{N}$ denote the set of all nonnegative integers and $A$ be a subset of $\mathbb{N}$. The set $A$ is called an asymptotic basis of order $2$ if every sufficiently large integer can be written as the sum of two elements of $A$. Otherwise, $A$ is called an asymptotic nonbasis of order $2$. Let $r_A(n)$ denote the number of representations of $n$ in the form $n=a+a^{\prime }$, where $a,a^{\prime }\in A$ and $a\le a^{\prime }$. An asymptotic nonbasis $A$ of order $2$ is called a maximal asymptotic nonbasis of order $2$ if $A\cup \{b\}$ is an asymptotic basis of order $2$ for every $b\notin A$. In this paper, a maximal asymptotic nonbasis $A$ is constructed satisfying $r_A(N_k)=0$ for all $k$ and $r_A(n)\to \infty$ as $n\to \infty ,n\ne N_k$, where $N_1,N_2,\dots$ is an increasing sequence of $\mathbb{N}$.

In this paper, we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum. We call such pairs of polynomials Davenport–Zannier pairs (DZ-pairs). The paper is devoted to the study of DZ-pairs with rational coefficients. In our earlier paper [F. Pakovich and A. K. Zvonkin, Minimum degree of the difference of two polynomials over $\mathbb{Q}$, and weighted plane trees, Selecta Math., (N.S.) 20(4) (2014) 1003–1065], in the framework of the theory of dessins d’enfants, we established a correspondence between DZ-pairs and weighted bicolored plane trees. These are bicolored plane trees whose edges are endowed with positive integral weights. When such a tree is uniquely determined by the set of black and white degrees of its vertices, it is called unitree, and the corresponding DZ-pair is defined over $\mathbb{Q}$. In our cited paper above, we classified all unitrees. In this paper, we compute all the corresponding polynomials. We also present some additional material concerning the Galois theory of DZ-pairs and weighted trees.

An explicit formula for the height-one multiple zeta values (MZVs) was proved by Kaneko and the second author. We give an alternative proof of this result and its generalization. We also prove its counterpart for the finite multiple zeta values (FMZVs).

Andrews defined the combinatorial objects called singular overpartitions denoted by ${\overline{C}}_{k,i}(n)$, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i(modk)$ may be overlined. In this paper, we investigate the arithmetic properties of Andrews singular overpartition pairs. Let ${\overline{A}}_{i,j}^{\delta }(n)$ be the number of overpartition pairs of $n$ in which no part is divisible by $\delta$ and only parts $\equiv \pm i,\pm j(mod\delta )$ may be overlined. We will prove a number of Ramanujan like congruences and infinite families of congruences for ${\overline{A}}_{1,2}^6(n)$ modulo $3,9,18$ and $36$, infinite families of congruences for ${\overline{A}}_{2,4}^8(n)$ modulo $4$ and $8$, infinite families of congruences for ${\overline{A}}_{1,5}^{12}(n)$ modulo $6$ and $9$.

For $a\in \mathbb{Z}$ and $b\in \mathbb{N}$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in \mathbb{N}$, with $b_i$ tending to infinity as $i$ grows, such that $s(a_i,b_i)=s(a,b)$ for all $i\in \mathbb{N}$.

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod $p$ less than $p^{1/2+3/(4\sqrt{e})+𝜖}$.

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on ${({𝔽}_p)}^2$. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

In 1862, Wolstenholme proved that the numerator of the $(p-1)$th harmonic number is divisible by $p^2$ for any prime $p\ge 5$. A variation of this theorem was shown by Alkan and Leudesdorf. Motivated by these results, we prove a congruence modulo some odd primes for some generalized harmonic type sums.

Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity ${\sum }_{k=1}^n(k-1,n)\chi (k)=\phi (n)\tau (n/d)$, which reduces to Menon’s identity if $\chi$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums.

Partitions related to mock theta functions were widely studied in the literature. Recently, Andrews et al. introduced two new kinds of partitions counted by $p_{\omega }(n)$ and $p_{\nu }(n)$, whose generating functions are $q\omega (q)$ and $\nu (-q)$, where $\omega (q)$ and $\nu (q)$ are two third mock theta functions. Meanwhile, they obtained some congruences for $p_{\omega }(n)$, $p_{\nu }(n)$, and the associated smallest parts function $sp{t}_{\omega }(n)$. Furthermore, Andrews et al. discussed the overpartition analogues of $p_{\omega }(n)$ and $sp{t}_{\omega }(n)$ which are denoted by ${\overline{p}}_{\omega }(n)$ and ${\overline{spt}}_{\omega }(n)$. In this paper, we derive more congruences for $p_{\omega }(n)$, $p_{\nu }(n)$, ${\overline{p}}_{\omega }(n)$, ${\overline{spt}}_{\omega }(n)$, and $\overline{spt}(n)$. Moreover, we establish some congruences for ${\overline{p}}_{\sigma }(n)$ and its associated smallest parts function ${\overline{spt}}_{\sigma }(n)$, where ${\overline{p}}_{\sigma }(n)$ denotes the number of overpartitions of $n$ such that all even parts are at most twice the smallest part, and in which the smallest part is always overlined.

The purpose of the present paper is to use the improvement of Runge’s theorem on diophantine equations due to Schinzel to prove that if $P$ has at least three different roots then the equation in the title has a finite number of solutions.

This paper discusses four formulae conjectured by Lara Rodríguez on certain power series in function fields, which yield a “harmonic product” formula for Thakur’s double zeta values. We prove affirmatively the first two formulae, detect and correct errors in the last two formulae, and prove the corrected ones.

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

We prove, in particular, that if $A,\mathrm{\Gamma }\subset {𝔽}_p^{\ast }$, $\left|\mathrm{\Gamma }\right|<p^{3/4}$ are two arbitrary multiplicative subgroups satisfying $A-A\subseteq \mathrm{\Gamma }\bigsqcup \{0\}$, then $\left|A\right|\ll |\mathrm{\Gamma }{|}^{1/3+o(1)}$. Also, we obtain that for any $𝜀>0$ and a sufficiently large subgroup $\mathrm{\Gamma }$ with $\left|\mathrm{\Gamma }\right|\ll p^{1/2-𝜀}$ there is no representation $\mathrm{\Gamma }$ as $\mathrm{\Gamma }=A+B$, where $A$ is another subgroup, and $B$ is an arbitrary set, $\left|A\right|,\left|B\right|>1$. Finally, we study the number of collinear triples containing in a set of ${𝔽}_p$ and prove a variant of sum–product estimate.

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 $x>1$.

In this paper, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [A. Aryasomayajula, Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight, Arch. Math. 106(2) (2016) 165–173; J. S. Friedman, J. Jorgenson and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, in Arbeitstagung Bonn 2013, Progress in Mathematics, Vol. 319 (Birkhäuser/Springer, Cham, 2016), pp. 127–154] to higher dimensions.

We consider for integers $k\ge 2$ the $k$-generalized Fibonacci sequences $F^{(k)}:={(F_n^{(k)})}_{n\ge 2-k}$, whose first $k$ terms are $0,\dots ,0,1$ and each term afterwards is the sum of the preceding $k$ terms. We give a lower bound for the largest prime factor of the sum of two terms in $F^{(k)}$. As a consequence of our main result, for every fixed finite set of primes $S$, there are only finitely many positive integers $k$ and $S$-integers which are a non-trivial sum of two $k$-Fibonacci numbers, and all these are effectively computable.

In this paper, we prove a multiple Abel summation formula and give some applications of this formula. We use these evaluations in tandem with some recent results to obtain various asymptotic evaluations for multiple sums.