1. In the ring Z/2Z[x1,...,xn], n > 2, can x13 + ... + xn3 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,...,xn]) 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?