Imprimitive solutions to $x^2+y^3=z^7$

Question

Poonen, Schaefer, & Stoll give the primitive solutions to $x^2+y^3=z^7$:

$$ (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),\\ (±2213459, 1414, 65), (±15312283, 9262, 113), (±21063928, −76271, 17). $$

I'm looking for all the solutions with $1\le z\le\ell$ for some fixed $\ell$.

Clearly the primitive solutions yield infinitely many imprimitive solutions via multiplication by $\operatorname{lcm}(2,3,7)=42$nd powers, but this doesn't find them all. For example, $250^2+25^3 = 5^7$ and $832^2+112^3 = 8^7$.

How can I find all the solutions? My first instinct was to deal with each special case on its own but I'm afraid I'll miss cases that way.

[1] Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of X(7) and primitive solutions to x^2+y^3=z^7, Duke Math. J. 137:1 (2007), pp. 103-158.

Answer 1

You can find the solutions for any given $z$ by looking for the integral points on the elliptic curve $$x^2 = (-y)^3 + z^7$$ (which would usually be written $y^2 = x^3 + z^7$). The curve is isomorphic to the curve obtained by replacing $z^7$ with $z$, so the computation is feasible for reasonable values of $z$.

Magma (for example, but also SAGE) has an implementation of a procedure that finds all these points (there are finitely many in each case) in quite reasonable time. For example, there are exactly 990 solutions (up to a sign change in $x$) for $1 \le z \le 1000$. Here is the Magma code (replace ell by $\ell$):

list := [];
for z := 1 to ell do
  pts := IntegralPoints(EllipticCurve([0, z^7]));
  list cat:= [<Abs(pt[2]), -pt[1], z> : pt in pts];
end for;
list;

If you do not have access to Magma, you can try it out with the online Magma calculator. With ell=100, it takes about 13 seconds there. The computation for $\ell = 1000$ on my computer took about as long as writing this answer.