Related papers: Perfect powers in elliptic divisibility sequences

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation…

Let $u_k$ be a Lucas sequence. A standard technique for determining the perfect powers in the sequence $u_k$ combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to…

Let $E/\mathbb Q$ be an elliptic curve, let $P\in E(\mathbb Q)$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. We study when a product \[ \prod_{i=1}^k D_{n_i} \] can be a $\rho$-th power, where $\rho$ is…

Let $E_\lambda$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots , P_n \in E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)$ we prove that there are at most finitely many complex numbers…

An elliptic divisibility sequence (EDS) is a sequence of integers W_0,W_1,W_2,... generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of W_n for unbounded…

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $ n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear…

For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is…

Starting from the classical division polynomials we construct homogeneous polynomials $\alpha_n$, $\beta_n$, $\gamma_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP =…

The subject matter of this work is the set of integral points(i.e. points with both coordinates integers) on the graphs of rational functions of the form f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic…

One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach---weaker in strength of evidence but more broadly applicable---to suggesting that concrete~NP…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the…

A conjecture by Sun states that the partition function $p(n)$, for $n>1$, is never a perfect power. Recent work by Merca et al. proposes generalizations of perfect-power repulsion for $p(n)$. In this note, we prove these generalizations for…

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal…

In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it verifies the recurrence relation…

For every n there exists an elliptic curve E over the rational numbers and an n-dimensional subspace V of non-zero rationals modulo squares such that for all v in V, the quadratic twist of E by v has positive rank.

Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and…

Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding…

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation…