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 by Alpoge, Bhargava, Shnidman, Burungale, and Skinner shows that a positive proportion of positive integers are cube sums, while a positive proportion of integers are not. We will discuss the cube sum problem for a special family of integers.

 

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