International Journal of Number Theory

We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous methods of Dani–Nogueira.

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma \in \overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_{\gamma }(x)=a_n{x}^n+\cdots +a_0$. Denote $e(\gamma )=gcd(a_n,a_{n-1},\dots ,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[\gamma ]\cap \mathbb{Q}=\{\alpha \in \mathbb{Q}|\{p|v_p(\alpha )<0\}\subseteq \{p:p|e(\gamma )\}\}$. Given a number field $K$ and $\gamma \in \overline{\mathbb{Q}}$, we show that there is a subset $X(K,\gamma )\subseteq \text{Spec}({{𝒪}}_K)$, for which ${{𝒪}}_K[\gamma ]\cap K=\{\alpha \in K|\{𝔭|v_{𝔭}(\alpha )<0\}\subseteq X(K,\gamma )\}$. We prove that ${{𝒪}}_K[\gamma ]\cap K$ is a principal ideal domain if and only if the primes in $X(K,\gamma )$ generate the class group of ${{𝒪}}_K$. We show that given $\gamma \in \overline{\mathbb{Q}}$, we can find a finite set $S\subseteq \overline{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\gamma )=\{𝔭\in \text{Spec}({{𝒪}}_K)|p\cap S\ne \varnothing \}$. We study how this set $S$ relates to the ring $\overline{\mathbb{Z}}[\gamma ]$ and the ideal ${𝔇}_{\gamma }=\{a\in \overline{\mathbb{Z}}|a\gamma \in \overline{\mathbb{Z}}\}$ of $\overline{\mathbb{Z}}$. We also show that ${\gamma }_1,{\gamma }_2\in \overline{\mathbb{Q}}$ satisfy ${𝔇}_{{\gamma }_1}={𝔇}_{{\gamma }_2}$ if and only if $X(K,{\gamma }_1)=X(K,{\gamma }_2)$ for all number fields $K$.

Let $\mathbb{Z}$ be the set of integers and $\mathbb{N}$ the set of positive integers. For a nonempty set $A$ of integers and any integers $n$, $h$ with $h\ge 2$, define $r_{A,h}(n)$ as the number of solutions of $n=a_1+\cdots +a_h$, where $a_1\le \cdots \le a_h$ and $a_i\in A$ for $i=1,\dots ,h.$ A set $A$ of integers is defined as a basis of order $h$ for $\mathbb{Z}$ if $r_{A,h}(n)\ge 1$ for every integer $n$. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function $f:\mathbb{Z}\to \mathbb{N}$, there exists a bounded basis of order $h$ for $\mathbb{Z}$ such that $r_{A,h}(n)=f(n)$ for every integer $n$.

In 2004, Corteel and Lovejoy introduced the notion of overpartitions in order to give a combinatorial proof of several celebrated $q$-series identities. Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Many scholars have been investigated subsequently congruence properties modulo powers of $2$ satisfied by $\overline{p}(n)$. Congruence properties modulo powers of 2 for $\overline{pp}(n)$ were also considered by several scholars, where $\overline{pp}(n)$ denotes the number of overpartition pairs of $n$. In this paper, utilizing some $q$-series identities and iterative computations, we prove several internal congruences and congruences modulo powers of 2 enjoyed by $\overline{p}(n)$ and $\overline{pp}(n)$. Moreover, we conjecture that these internal congruences and congruences are initial cases in the corresponding internal congruence families and congruence families. Finally, we pose a related conjecture and some questions that merit further investigation.

In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${\{g_d(n)\}}_{d\ge 0,n\ge 1}$ be the double sequences ${\sigma }_d(n)={\sum }_{\ell |n}{\ell }^d$ or ${\psi }_d(n)=n^d$. We associate double sequences $\{p^{g_d}(n)\}$ and $\{q^{g_d}(n)\}$, defined as the coefficients of

For a number field $k$ having at least one embedding into the real numbers we give an algorithm to obtain a Shintani domain for the action of the totally positive units of $k$ under its geometric embedding. Our algorithm modifies a known signed, or virtual, fundamental domain until it becomes a true one. We examine the results of extensive runs of our algorithm for cubic, quartic and quintic number fields. We also give some data for totally real sextic fields where our algorithm is still often practical.

Let $F$ be a Hecke–Maass cusp form on $\mathrm{\text{SL}}(3,\mathbb{Z})$ and $\chi$ a primitive Dirichlet character of prime power conductor $𝔮=p^k$ with $p$ prime. In this paper, we will prove the following subconvexity bound $L(\frac{1}{2}+it,F\times \chi ){\ll }_{\pi ,𝜀}{p}^{3/4}{(𝔮(1+\left|t\right|))}^{3/4-3/40+𝜀},$ for any $𝜀>0$ and $t\in ℝ$.

Getz presented a family of level one modular forms $f_k$ for which all zeros lie on the unit circle in the fundamental domain. Expanding on work from Nozaki, Griffin et al., and Saha and Saradha, we show that the non-elliptic zeros of these $f_k$ satisfy two interlacing properties: standard interlacing, where the zeros of $f_k$ and $f_{k+a}$ alternate if and only if $a\in \{2,4,6,8,12\}$ for sufficiently large $k$; and Stieltjes interlacing, where for $\ell >k$ large enough, between any two zeros of $f_k$, there is a zero of $f_{\ell }$.

Covering systems were introduced by Erdős in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister et al. gave a simpler and more versatile proof of Hough’s result. Building upon their work, we show that there exists some absolute constant $c>0$ such that the $j$th smallest modulus of a minimal covering system with distinct moduli is $\le exp(c{j}^2/log(j+1))$.

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing ${\zeta }_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}({\zeta }_n)$ with $5\nmid n$, this proportion is at least $1-\epsilon$ with only finitely many exceptions of $n$, for any choice of $\epsilon >0$.

In this paper, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-d{Y}^2=\pm 1$, which is a difference of two Fibonacci numbers.

In this paper, we provide proofs of two ${}_5{\psi }_5$ summation formulas of Bailey using a ${}_5{\varphi }_4$ identity of Carlitz. We show that in the limiting case, the two ${}_5{\psi }_5$ identities give rise to two ${}_3{\psi }_3$ summation formulas of Bailey. Finally, we prove the two ${}_3{\psi }_3$ identities using a technique initially used by Ismail to prove Ramanujan’s ${}_1{\psi }_1$ summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised ${}_6{\psi }_6$ sum.

Recently, the growth of the products of consecutive partial quotients $a_i(x)$ in the continued fraction expansion of a real number $x$ was studied in connections with improvements to Dirichlet’s theorem. In this paper, for a non-decreasing positive measurable function $F(x_1,\dots ,x_m)$ and a function $\varphi :\mathbb{N}\to {ℝ}_{>0}$, we consider the set ${{ℰ}}_F(\varphi )=\{x\in [0,1]:F(a_n(x),\dots ,a_{n+m-1}(x))\ge \varphi (n)\text{ for infinitely many }n\in \mathbb{N}\}$, and obtain its Lebesgue measure ${ℒ}({{ℰ}}_F(\varphi )$). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when $F(x_1,\dots ,x_m)=x_1+\cdots +x_m$.

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation $A^3+B^3=C^3+D^3$. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation $A^4+B^4+C^4+D^4+E^4=F^4$ in the spirit of Ramanujan.

We prove that a nonzero Jacobi form of level $N$ (an odd integer) and square-free index $m_1{m}_2$ with $m_1|N$ and $(N,m_2)=1$ has a nonzero theta component $h_{\mu }$ with either $(\mu ,2m_1{m}_2)=1$ or $(\mu ,2m_1{m}_2)$$\nmid$$2m_2$. As an application, we prove that a nonzero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin–Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over $\mathbb{Q}$ with $12$-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of $12$-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves.

A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$ or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves.

In this paper, we study several topics on additive decompositions of primitive elements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy as well as Shparlinski.