Question

Disprove the statement that every positive integer is the sum of 36 fifth powers of nonnegative integers.

Short answer

The statement is not true for \(223\).

Step-by-step solution

Describe the given information

Every positive integer is the sum of 36 fifth powers of 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 fifth power of nonnegative integers is as follows:

\(\begin{array}{c}{0^5} = 0\\{1^5} = 1\\{2^5} = 32\\{3^5} = 243\\{4^5} = 1024\\ \vdots \end{array}\)

The smallest nonnegative integer that cannot be written as the sum of \(36\) terms out of the above list is \(223\). The next sum contains six times \(32\) and thirty-one times \(1\).

\(\begin{array}{c}223 = 32 + 32 + 32 + 32 + 32 + 32 + 1 + 1 + 1 + ... + 1\\ = 6\left( {32} \right) + 31\left( 1 \right)\\ = 6\left( {{2^5}} \right) + 31\left( {{1^5}} \right)\\ = {2^5} + {2^5} + {2^5} + {2^5} + {2^5} + {1^5} + {1^5} + {1^5} + ... + {1^5}\end{array}\)

Since the above sum contains \(37\) terms instead of \(36\) terms and the sum of \(223\) cannot be written with less terms, the statement is not correct.

Therefore, the statement is not true for \(223\) and thus it disproven the given statement.