The solutions for all long operators of the N=4 SYM theory at one loop and of its dual, the classical superstring theory on AdS5xS5 space, will be presented in terms of their algebraic curves encoding all conservation laws. The SYM theory is treated by the Bethe ansatz equations in the thermodynamical limit, whereas the classical superstring is solved by the finite gap method, using the integrability. In a special (Frolov-Tseytlin) limit both curves coincide, demonstrating the full AdS/CFT correspondence among states/operators, including fermions.