Speaker: Tathagata Basak
Title: Sums of cubes
We’ll talk about writing numbers as sums of cubes. Much less is known here and and many basic questions are open. In particular, we’ll talk about the equation
a^3 + b^3 = c^3 + d^3
made famous by Ramanujan’s taxicab observation:
1729 = 9^3 + 10^3 = 1^3 + 12^3.
Ramanujan also observed that, if
a(x, y) = -5*x^2 – 5*x*y + 3*y^2;
b(x, y) = 6*x^2 + 4*x*y + 4*y^2;
c(x, y) = 3*x^2 – 5*x*y – 5*y^2;
d(x, y) = 4*x^2 + 4*x*y + 6*y^2;
then
a(x, y)^3 + b(x, y)^3 – c(x, y)^3 – d(x, y)^3 = 0.
We’ll describe a completely elementary result which produces lots of four tuples of binary quadratic forms whose cubes add up to zero. The first nontrivial example Ramanujan’s four tuple given above. This talk will be completely elementary.