Related papers: The Uniform Primality Conjecture for the Twisted F…

Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…

We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show…

Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and [F:Q] is odd. For delta \in F^times, let [delta] denote its class in F^times/F^{times p}. In this paper, we show Main Theorem. There are infinitely many classes…

We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes,…

We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the…

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…

We give a new sufficient condition which allows to test primality of Fermat's numbers. This characterization uses uniquely values at most equal to tested Fermat number. The robustness of this result is due to a strict use of elementary…

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…

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $\Gamma$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\in\Gamma$ if and only if $P\in…

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain…

We give examples of sequences of smooth non-isotrivial curves for every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence…

Classifications of twin primes are established and then applied to triplets that generalize to all higher multiplets. Mersenne and Fermat twins and triplets are treated in this framework. Regular prime number multiplets are related to…

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power…