Show that $x^3+2y^3 = 7(z^3+2w^3)$ has no solutions in coprime integers $x,y,z,w$.
I've been stuck on this problem for a bit and haven't made any progress or found any strategies that might work. Any help would be greatly appreciated.
2026-05-03 11:09:07.1777806547
Show a Cubic Diophantine Equation has no solutions in coprime integers
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Oh: if there is any solution (not all zero), there is such a solution in coprime integers, just divide out by the greatest common divisor. This is the "infinite descent" in brief. We assume that we have a not-all-zero solution in coprime integers.
it's just mod 7. Cubes are $0,1,-1 \pmod 7.$ Check, the only way to get this expression $0 \pmod 7$ is to have both $x,y \equiv 0 \pmod 7.$
Next, the left hand side is currently divisible by $343.$ The reult is that $z^3 + 2 w^3$ must also be divisible by $7,$ at which point we have all four $x,y,z,w \equiv 0 \pmod 7.$ All divisible by $7.$ This contradicts the assumption of coprime variables.