Results 11 to 20 of about 366 (33)

Equal sums of like polynomials [PDF]

, 2005
Let $f$ be a polynomial of degree $d>6$, with integer coefficients. Then the paucity of non-trivial positive integer solutions to the equation $f(a)+f(b)=f(c)+f(d)$ is established.
Browning, T. D.
core   +1 more source

Counting rational points on smooth cubic surfaces [PDF]

, 2016
We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.Comment: 11 pages, minor ...
Frei, Christopher, Sofos, Efthymios
core   +4 more sources

Density of solutions to quadratic congruences [PDF]

, 2016
A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M.
Prabhu, Neha
core   +2 more sources

Characterizing algebraic curves with infinitely many integral points

, 2009
A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have
Bilu, Yuri, Dimitrios, Poulakis, Paraskevas, Alvanos   +2 more
core   +2 more sources

On large $F$-Diophantine sets

, 2017
Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$.
El-Sissi, Nermine, Sadek, Mohammad
core   +1 more source

Average Analytic Ranks of Elliptic Curves over Number Fields

Forum of Mathematics, Sigma
We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the ...
Tristan Phillips
doaj   +1 more source

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

Forum of Mathematics, Sigma
The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.
Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi   +3 more
doaj   +1 more source

The density of twins of $k$-free numbers

, 2014
For $k \geq 2$, we consider the number $A_k(Z)$ of positive integers $n \leq Z$ such that both $n$ and $n+1$ are $k$-free. We prove an asymptotic formula $A_k(Z) = c_k Z + O(Z^{14/(9k)+\epsilon})$, where the error term improves upon previously known ...
Dietmann, Rainer, Marmon, Oscar
core   +1 more source

A Mean Value Theorem for the Diophantine Equation $axy-x-y=n$

, 2011
In this paper, we prove an asymptotic formula for the average number of solutions to the Diophantine equation $axy-x-y=n$ in which $a$ is fixed and and $n$ varies.Comment: 11 pages, to appear in Acta Math. Hungar. A couple of typos are corrected in the
Huang, Jing-Jing
core   +1 more source

Theta functions, fourth moments of eigenforms and the sup-norm problem II

Forum of Mathematics, Pi
Let f be an $L^2$ -normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of ...
Ilya Khayutin, Paul D. Nelson, Raphael S. Steiner   +2 more
doaj   +1 more source