Solve this equation in real numbers:
(x+y)^{3} = z
(y+z)^{3} = x
(z+x)^{3} = y
If x=y=z = t then (2t)^3 = t
t(8*t^2  1) = 0
t = {0, √2/4, √2/4}
Three solutions are:
(x,y,z) = {(0,0,0), (√2/4,√2/4,√2/4), (√2/4,√2/4,√2/4)}
There may be other solutions, but I consider it feasible that these are the only solutions.
Reasoning: Looking at any one of the equations, in 3D it looks similar to the 2 dimensional equation y=x^3 except the pattern is extended above and below "the paper", as if you took a flat plane then bent it into a y=x^3 shape when viewed on end. Only rotated relative to the x,y,z axes.
Now intersect 2 orthogonal versions of that structure and you would get a line curving through 3 dimensions.
Intersect that with one more orthogonal structure and I think you would get a finite number of points.
Obviously there may be more than just these 3 points, I look forward to more solutions or a proof that these are the full solution set.

Posted by Larry
on 20240110 11:18:15 