Recently I learned more about the Diophantine equation x3 + y3 = z2 and its history. I summarized this in a March 24, 2005 message to the sequence fans mailing list. Below is that message transcribed into html:
March 24, 2005
To Sequence Fans Mailing list:
A while back integer solutions to x3 + y3 = z2 were a topic here to which I contributed. I have since learned that a complete solution was given by Louis J. Mordell in his book "Diophantine equations", Academic Press, London-New York 1969.
This information comes from Dario Alpern's site where he provides explicit solutions to this and several other Diophantine equations.
The following online papers (some mentioned by Dario) are pertinent:
Nils Bruin: 0311002.pdf (see page 2).
Several OEIS sequences are related to this equation.
(see parent page, bottom)