Related papers: Primitive divisors of elliptic divisibility sequen…

We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective…

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite…

Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The…

In this note we compute a constant $N$ that bounds the number of non--primitive divisors in elliptic divisibility sequences over function fields of any characteristic. We improve a result of Ingram--Mah{\'e}--Silverman--Stange--Streng,…

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over…

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree…

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be…

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(\zeta))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of…

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this…

We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive…

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a…

Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the…

Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence $\{B_n\}_{n\in \mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive…

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the…

Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $\alpha\in \text{End}(E)$, define $B_\alpha$ to be the $\mathcal{O}_K$-integral ideal generated by the denominator of $x(\alpha(P)+Q)$. Let…

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of…

Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{\epsilon, a} n^{2-\epsilon}$$ for any $\epsilon>0$. We study analogues…

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 $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…

We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical orbits, the Zsigmondy set is explicitly…