How find all positive $a^3=b^2+2000000$
We are looking for the positive integer solutions of the Mordell-Bachet Diophantine equation $$ y^2=x^3-d, $$ with $d=2000000$. This has positive integer solutions (one coming from $y^2=x^3-2$, which is $(3,5)$, yielding $(300,5000)$). To see this, one can proceed as shown in Theorem $3.4$ of K. Conrad's article http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf. Note that $x^3=(y+\sqrt{-d})(y-\sqrt{-d})$. The factors on the RHS are relatively prime. If we have unique factorisation, this can be used for a nice proof.