This person has found an actual solution:
https://mlzeng.com/an-interesting-equation.html
x = 36875131794129999827197811565225474825492979968971970996283137471637224634055579
y = 154476802108746166441951315019919837485664325669565431700026634898253202035277999
z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036
This is apparently the smallest solution.
The derivation of the solution itself requires an understanding of Diophantine equations and elliptical curves, which I will not pretend to understand well enough to summarize.
From the same article, this is a solution when negative integers are allowed:
{{x -> 11, y -> 9, z -> -5}}
🍎=376556443707
🍌=50000000000
🍍=50000000000
no, no I can’t
Is it the intersection of 3 curves?
Am I being trolled?
Some discussion here
Ew please don’t link Reddit without warning.
The solutions don’t quite fancy me as positive whole values, see for yourself.
An integer solution does exist.
