No abstract in the paper. Here is an abstract written ten years later:

A. Wakulicz (Coll. Math. 5 (1957), 11-15) gave an elementary proof of the fact that x^3+y^3=2z^3 has no integral solutions besides x=y, x=-y. As a consequence (using the factorization (x-1)(x+1)=y^3) he obtained an elementary proof of the fact that x^2-y^3=1 has no integral solutions besides (+-1, 0), (0,-1) and (+-3,2). We give another elementary derivation of this result, starting with x^2=(y+1)(y^2-y+1) and reducing the problem to the Pell equation x^2-3y^2=1. In fact, this method works for the equations x^2-y^3=+-3^{3k}.