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As far as I’m aware, I’ve found the only known primitive solution to $x^6+y^6+z^6=w^2$ with $7|x$ and $7|y$, and I would like to generate further solutions from this one. That solution is

$$40425^6+45990^6+40802^6=135794767970233^2$$

I’m especially interested to see if the${\pmod 7}$ properties are preserved in any generated solutions.

My question

Can this solution be used to produce other non-trivial solutions?

If so, what are those solutions, and, in the simplest terms, how are they found.

Remarks

$$7^63^6|w+z^3$$ $$5^6|w-z^3$$ $$40425=105\cdot385$$ $$45990=105\cdot438$$

These links are the reason I suspect there might be family of solutions

http://www.maroon.dti.ne.jp/fermat/dioph149e.html

http://www.maroon.dti.ne.jp/fermat/dioph150e.html

but I’ve not been able the follow the method.

These are my earlier relevant questions

The Diophantine equation $x_1^6+x_2^6+y^6=z^2$ where both $(x_i)\equiv 0{\pmod 7}$.

The Diophantine equation $x_1^6+x_2^6+x_3^6=z^2$ where exactly one $(x_i)\equiv 0{\pmod 7}$.

As I’m very far from my comfort zone with this, I apologise in advance if this is a stupid question.

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  • $\begingroup$ Any integer like $n^6$ is a perfect square too. so multiplying both sides of your relation by $n^6$ gives new solution. $\endgroup$ – sirous May 17 '18 at 14:42
  • $\begingroup$ No, I don’t think it does, but by all means show me a numerical example, I could be wrong. Although $(kx,ky,kz,k^3w)$ does give more solutions, none are primitive. Thanks for your interest. $\endgroup$ – Old Peter May 17 '18 at 15:55

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