Find a counterexample to the statement that every positive integer can be written as the sum of thesquares of three integers.1

Sure, let's find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.

The statement is not true for the number 7.

Let's see why:

The squares of integers are 0, 1, 4, 9, 16, 25, 36, and so on.

If we try to express 7 as the sum of three of these numbers, we find that it's not possible.

For example, if we use 1 and 4, the smallest squares after 0, the third number would have to be 2 (because 7 - 1 - 4 = 2), but 2 is not a square of an integer.

Therefore, 7 is a counterexample to the statement.