Assume that B is a large real number and let $c_1, c_2, c_3$ be three randomly chosen integers in the box $[−B,B]^3$. Consider the probability that the “random” curve $c_1X^2 +c_2Y^2 +c_3Z^2 =0$ has a non-zero solution $(X,Y,Z)$ in the integers. Serre showed in the 90ś that this probability is $\ll (\log B)^{−3/2}$ while Hooley and Guo later proved that it is $\gg (\log B)^{−3/2}$. In joint work with Nick Rome we prove an asymptotic $\sim c (\log B)^{-3/2}$, where $c$ is a positive absolute constant.
By Jean-Louis Colliot-Thélène
By Will Sawin
By David Harari
By Pierre Le Boudec
By Cyril Demarche
By Arne Smeets
By Julia Hartmann
By Olivier Wittenberg
By Antoine Chambert-Loir
