Conjecture: $N=x_1^2+x_2^2-y^3$ has integer solutions for all $N$, with $x_1>0$,$x_2>0$ and $y>0$ Background.
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 $-y$.  
Question.
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.
My efforts.
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.
$$0=2^2+11^2-5^3$$
$$11=6^2+10^2-5^3$$
$$-3=5^2+6^2-4^3$$
$$999999=40^2+1718^2-125^3$$
$$-999999=8^2+1365^2-142^3$$

This was a problem posed by Noam Elkies and Irving Kaplansky in the January 1995 M.A.A. Monthly. They did not ask that your $y$ be positive, but that is how it comes out anyway, with a very few exceptions for the variable $x$ below being, for example $0,1,2.$ These can then be fixed by a quick search.
Later, Kap and I published a little paper where we pointed out that $x^2 + y^2 - z^9$ fails to represent infinitely many numbers 
The closest looking thing to your original that fails to represent infinitely many numbers would be $x^2 + 27 y^2 - 7 z^3.$
Here is a solution by Andrew Adler, in the June-July 1997 issue:
$$ 2x+1 = \left( x^3 - 3 x^2 + x \right)^2 + \left(x^2 - x - 1 \right)^2 -\left( x^2 - 2 x\right)^3   $$
$$ 4x+2 = \left(2 x^3 - 2 x^2 - x \right)^2 + \left(2 x^3 - 4 x^2 - x + 1 \right)^2 -\left( 2 x^2 - 2 x - 1 \right)^3   $$
$$ 8x+4 = \left(x^3 + x + 2 \right)^2 + \left(x^2 - 2x - 1 \right)^2 -\left(x^2 + 1 \right)^3   $$
$$ 16x+8 = \left( 2 x^3 - 8 x^2 + 4 x + 2 \right)^2 + \left(2 x^3 - 4 x^2 - 2 \right)^2 -\left(2 x^2 - 4 x \right)^3   $$
$$ 16x = \left(x^3 + 7 x - 2 \right)^2 + \left(x^2 + 2 x + 11 \right)^2 -\left( x^2 + 5 \right)^3   $$
