Question

Disprove the statement that every positive integer is the sum of the cubes of eight nonnegative integers.

Short answer

The statement is not true for \(n = 23\).

Step-by-step solution

Describe the given information

Every positive integer is the sum of the cubes of eight nonnegative integers.

Disprove the given statement

To disprove the statement, it is required to find a positive integer that cannot be rewritten as the sum of the cubes of eight nonnegative integers.

The cubes of nonnegative integers are as follows:

\(\begin{array}{c}{0^3} = 0\\{1^3} = 1\\{2^3} = 8\\{3^3} = 27\\{4^3} = 64\\ \vdots \end{array}\)

The smallest nonnegative integer that cannot be written as the sum of eight terms out of the above list is \(23\).

\(\begin{array}{c}23 = 8 + 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1\\ = {2^3} + {2^3} + {1^3} + {1^3} + {1^3} + {1^3} + {1^3} + {1^3} + {1^3}\end{array}\)

Since the above sum contains \(9\) terms instead of \(8\) terms and the sum of 23 cannot be written with less term, the statement is not correct.

Therefore, the statement is not true for \(n = 23\).