Brain Teasers and Puzzles

Sum of Powers

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.


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.

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