Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$

Does the exponential diophantine equation

$$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$

with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have solutions?

Any solution will give a good $abc$ triple. After browsing the list of $abc$ triples I did not find a solution in the list. Any heuristics?

Update nonsense relating conic to the $abc$ triple $7^2+2^{17} 181^2=3^8 809^2$ deleted as pointed out by several commenters.

Ideas about searching for solutions?

edited Jun 17 2011 at 11:15

asked Jun 17 2011 at 6:53

joro
4,270●1●10●31

Of course, your last equation has infinitely many solutions, take $y=t=1$ and the rest a primitive pythagorean triple. – Gjergji Zaimi Jun 17 2011 at 7:12

Thank you Gjergji. Edited the question disallowing $\pm 1$ – joro Jun 17 2011 at 7:27

For any fixed y and t this is still a rational curve and has an infinite number of solutions. – Taylor Dupuy Jun 17 2011 at 8:14

Taylor thank you. Added "integer" solutions, coprime was in the OQ. – joro Jun 17 2011 at 8:44

The question is whether the sum of two coprime 4-full numbers can be 4-full. The first 4-full numbers are tabulated at oeis.org/A036967 – Gerry Myerson Jun 17 2011 at 12:15

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