Find a combination of three different positive integers x, y, and z such that:
x3 + y3 = z4
Hint: there’s a better way than brute force / trial & error.
Solution
There are infinitely many solutions, so here’s a key insight to easily find one:
It’s trivial to find some three different positive integers a, b, and c such that:
a3 + b3 = z
Then multiple both sides by z3:
(az)3 + (bz)3 = z4
Let x = az, y = bz, and voila!
For example, 23 + 33 = 35, so 703 + 1053 = 354.