Journal of the Ramanujan Mathematical Society
Volume 41, Issue 1, March 2026 pp. 15–18.
Four quadratic forms whose cubes sum to zero
Authors:
Tathagata Basak
Author institution:Department of Mathematics, Iowa State University, Ames, IA 50011.
Summary:
Let u be an even integer not divisible by 3. For each z = (z{1}, z{2}, z{3}) ∈ Z{3} satisfying z{2} ≡ 0 mod 2 and
Q{u}(z) = z{2}{1} − z{2}{2} − ((u{6} − 1)/3)z{2}{3}
= 0, we find integral binary quadratic forms s{z}, t{z} such that s{z}(x, y){3} + t{z}(x, y){3} =
s{z}(y, x){3} + t{z}(y, x){3}. The coefficients of s{z}, t{z} are integer linear combinations of (z{1}, z{2}, z{3}). This yields a three parameter
family of 4-tuples of binary quadratic forms spanning infinitely many GL2(Z) orbits, whose cubes add up to zero.
The first couple of examples go back to Ramanujan.
Contents
Full-Text PDF