Related papers: Modular periodicity of the Euler numbers and a seq…

We establish noncommutative Kn\"{o}rrer periodicity for projective-module factorizations over an arbitrary ring, using the equivariantization theory with respect to various actions by a cyclic group of order two. We obtain an explicit…

In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form $x^2+p x=q$ with integer $p$ and $q$, $p^2+q^2\le R^2$. Our results concern the…

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.

In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…

Let $k\ge 1,a\ge 1,b\ge 0$ and $ c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by…

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by…

It is shown that the number $f_\lambda$ of free subgroups of index $6\lambda$ in the modular group $\PSL_2(\Z)$, when considered modulo a prime power $p^\al$ with $p\ge5$, is always (ultimately) periodic. In fact, an analogous result is…

We consider the linear combinations of elements of two sequences: the first one a priory given nonnegative sequence and the second random sequence from the unit interval. We investigate the expected value of the smallest natural number such…

Silverman and Stange define the notion of an aliquot cycle of length L for a fixed elliptic curve E defined over the rational numbers, and conjecture an order of magnitude for the function which counts such aliquot cycles. In the present…

We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.

We find an enumeration formula for a $(t,q)$-Euler number which is a generalization of the $q$-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the $(t,q)$-Euler number and find another…

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that,…

On any closed Riemannian 3-manifold which is not a torus bundle, every nonvanishing analytic solution of the stationary Euler equations has a periodic trajectory. This result is originally due to A. Rechtman (arXiv:0904.2719) and K.…

For any $\epsilon>0$, there exists $q_0(\epsilon)$ such for any $q\ge q_0(\epsilon)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three…

Hofstadter's Q-sequence remains an enigma fifty years after its introduction. Initially, the terms of the sequence increase monotonically by 0 or 1 at a time. But, Q(12)=8 while Q(11)=6, and monotonicity fails shortly thereafter. In this…

We extend the study of \emph{melonic} quartic tensor models to models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated…

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) -…

This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results…

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones…

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…