  In the ring R=K[X,Y,Z]/(X^3+Y^3+Z^3), where K is a field of prime
characteristic p other than 3, determining the tight closure of the ideal (X^2,
Y^2, Z^2)R had existed as a classic example of the difficulty involved in tight
closure computations. We settle this question, compute the Frobenius closure of
this ideal, and generalize these results to the diagonal hypersurfaces
K[X_1,...,X_n]/(X_1^n + ... + X_n^n).
