  In the current paper we are seeking P1(y),P2(y),P3(y) with the highest
possible degree polynomials with integer coefficients, and Q(y) via the lowest
possible degree polynomial, such that P1(y)^3+P2(y)^3+P3(y)^3=Q(y). Actually,
the solution of this problem has close relation with the problem of the sum of
three cubes a^3+b^3+c^3=d, since deg Q(y)=0 case coincides with above mentioned
problem. It has been considered estimation of possibility of minimization of
deg Q(y). As a conclusion, for specific values of d we survey a new algorithm
for finding integer solutions of a^3+b^3+c^3=d.
