Round 3: Tossup 5

Summing the first n natural numbers raised to this power gives the square of the nth triangular number, per Nicomachus’s theorem. The nth (15[1])taxicab number is the smallest that (15[1])can be written as the sum of two numbers (-5[1])raised to this power in n different ways. Multiplying by i this number of times is equivalent to a rotation (15[1])through 270 degrees. This is the (15[1])lowest number n for which the equation “a to the n plus b to the n equals c to the n” has (*) no integer solution, per Fermat’s last theorem. The derivative of x to this power is proportional to “x squared.” For 10 points, the (10[1])volume of a cube is found by raising the side length to what power? ■END■

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