Open Problems

1. In the ring Z/2Z[x₁,...,x_n], n > 2, can  x₁³  + ... +  x_n³   be written as a sum of less than  n  cubes? Of three cubes?

2.  Is therre  an number n  and a matrix m in  SL₂(Z[x₁,...,x_n])  such that every matrix in  SL₂(Z)   can be obtained by plugging integral values for variables into  m ? (F.Beukers, CRM Proc & Lecture Notes 19, p.390)

3. Is 30 sum of three integral cubes?

4. Is every integer a sum of four  integral cubes?