A nonzero rational number is called a cube sum if it can be written as a sum of two cubes of nonzero rational numbers. Determining which rational numbers are cube sums is very interesting and natural, which is closely related to the arithmetic of elliptic curves.

In this talk, we will present a recent result (predicted by the BSD conjecture) on this classical Diophantine problem, which is joint work with Shu Jie and Yin Hongbo. If time permits, we would also like to recall a surprising connection between the BSD conjecture and the O’Nan moonshine, which is due to Duncan, Mertens and Ono.