Matched at Every Power

Find two sets of six whole numbers, each below $100$, sharing no member, such that the two sets have the same sum, the same sum of squares, the same sum of cubes, the same sum of fourth powers and the same sum of fifth powers.

Solution

The two sets $$\{0, 5, 6, 16, 17, 22\} \qquad \text{and} \qquad \{1, 2, 10, 12, 20, 21\}$$ agree at every power from the first to the fifth: $$\begin{array}{ll} \text{sum} = 66, & \text{squares} = 1090,\\ \text{cubes} = 19998, & \text{fourth powers} = 385234,\\ \text{fifth powers} = 7632966, & \end{array}$$ and they have no number in common. Such a matched pair is called a Prouhet-Tarry-Escott pair, and this is the smallest one that holds all the way up to fifth powers.