I constructed some code for Algorithm for diophantine equation and decided to reuse it to investigate $N=x_1^2+x_2^2+z^3$ with $z$ integer. Negative values for $z$ seemed to produce plentiful small $N$ values, so I elected to concentrate my efforts here, and changed $z$ to $-y$.
Can anyone prove or disprove this conjecture, or help me find a method to do this, please? I would also appreciate any useful background information.
Under nine minutes brute-force to find a solution for each $N=-10^6$ to $10^6$.
I’ve searched around on the net to find similar solutions. Perhaps a method based on this link http://www.dms.umontreal.ca/~mlalin/Lagrange.pdf would do?
Examples. $$0=2^2+11^2-5^3$$ $$11=6^2+10^2-5^3$$ $$-3=5^2+6^2-4^3$$ $$999999=40^2+1718^2-125^3$$ $$-999999=8^2+1365^2-142^3$$