Alladi Ramakrishnan Hall

An effective lower bound for the integer cube sum

Saunak Bhattacharjee

IISER Tirupati

For an integer m, let N(m) denotes the number of integer solutions to the equation x^3 + y^3 =m. The sum of cubes problem concerns with the question of how large N(m) can be. Analogously, for any cubic form f(x,y), let N_f(m) denote the number of integer solutions for f(x,y)=m. In 1983, Silverman obtained large values for N_f(m) by connecting this to construction of higher rank elliptic curves. In this talk, we will discuss an explicit lower bound attained by N(m) for infinitely many m.