[ Evenness and Oddness ]

[ Factorization ]

Find all natural numbers that cannot be represented as the difference of squares of any natural numbers.
Let $x^{2}-y^{2}=n$, where $n$ is a given natural number. Since $x^{2}-y^{2}=(x-y)(x+y)$, and the numbers $x-y$ and $x+y$ have the same parity, then $n$ is either divisible by 4 or has the form $4 k \pm 1$.

We will prove that all numbers $n$ of the form $4 k$ and $4 k \pm 1$, except for the numbers 1 and 4, can be represented as $x^{2}-y^{2}$. Indeed, if $n=4 k$, then it is sufficient to take $x=k+1, y=k-1$ (the exception is $n=4$, since in this case $y=0$ is not a natural number); if $n=2 l+1$, then it is sufficient to take $x=l+1, y=l$ (the exception is $n=1$, since then $y=0$).

## Answer

1, 4, and all numbers of the form $4 k+2$.