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

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one.…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all…

In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical…

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an…

Let $C: y^2=ax^4+bx^2+c$, be an elliptic curve defined over $\mathbb Q$. A set of rational points $(x_i,y_i) \in C(\mathbb Q)$, $i=1,2,\cdots,$ is said to be a sequence of consecutive squares if $x_i= (u + i)^2$, $i=1,2,\cdots$, for some…

Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over the finite field F_p is prime. This is an analogue of the Hardy and…

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case.…

Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to…

Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of F_p-points of E is prime. However, the constant occurring in his asymptotic…

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

We show that every smooth cubic hypersurface X in P^{n+1}, n> 1 is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured…

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number…

This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $\beta$, as…

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…