## Task A-4.2.

A Gaussian integer is a complex number whose real and imaginary parts are integers. Determine the largest natural number $n$ for which there exists a set of $n$ Gaussian integers such that the squares of their absolute values are consecutive natural numbers.
## Solution.

If the complex number $z=x+y i$ is a Gaussian integer, then $|z|^{2}=x^{2}+y^{2}$ is the sum of the squares of two integers.

The square of an even integer is divisible by 4, while the square of an odd integer gives a remainder of 1 when divided by 4. Therefore, the sum of the squares of two integers can give remainders of 0, 1, or 2 when divided by 4.

If $n$ were greater than 3, then among any four consecutive natural numbers in the sequence

$$
\left|z_{1}\right|^{2},\left|z_{2}\right|^{2}, \ldots,\left|z_{n}\right|^{2}
$$

there would be a number that gives a remainder of 3 when divided by 4, which we have shown to be impossible. Therefore, $n$ is less than or equal to 3.

Notice that $2+2 i, 3, 3+i$ are Gaussian integers whose modulus squares sum to $8, 9, 10$ respectively. Thus, the largest natural number $n$ in the problem is equal to three.