Somnath Jha: On some cases of the cube sum problem

An integer $n$ is said to be a rational cube sum or simply a cube sum if n can be written as a sum of cubes of two rational numbers. For example,  $6 = (17/21)^3 +(37/21)^3$. 

 

A cube-free integer $n > 2$ is a cube sum if and only if the ‘elliptic curve’ $y^2 = x^3 − 432n^2$ has infinitely many solutions over the rational numbers. A recent result of AlpogeBhargava-Shnidman-Burungale-Skinner shows that a positive proportion of positive integers are cube sums and a positive proportion of integers are not. We will discuss the cube sum problem for some special family of integers.

 

This talk is based on joint works with Das, Majumdar, Shingavekar and Sury.