I'm interested in whether $$ 2x^3-1=z^3 $$ has any positive integer solutions. It has solutions mod $m$ for $m\le10^7$ so it seems unlikely that modular means would suffice to prove that there are not solutions. On the other hand I can't find any solutions for $x<10^9$ either.
I'm sure this is a routine problem but I don't know the techniques. Help?
Application: I was reading about near-Fermat triples $x^3+y^3=z^3+1$ where the restriction $x<y<z$ was given rather than $x\le y<z$ and I wondered if the excluded case was possible.