31. The Pythagorean rule for calculating the sides of a right-angled triangle is based on the identity:

$$
(2 n+1)^{2}+\left(2 n^{2}+2 n\right)^{2}=\left(2 n^{2}+2 n+1\right)^{2}
$$

Using this identity, calculate the sides of the right-angled triangles for \( n=1,2,3,4,5 \).
31. According to the Pythagorean rule, for the smaller leg, we take an odd number $2 n+1$. If we square it, subtract one, and divide the remainder by two, we get the larger leg:

$$
(2 n+1)^{2}=4 n^{2}+4 n+1, \frac{4 n^{2}+4 n}{2}=2 n^{2}+2 n
$$

By adding one to the obtained result, we find the hypotenuse $2 n^{2}+2 n+1$. For example, if the smaller leg is 3, then the larger leg $=\frac{3^{2}-1}{2}=4$, and the hypotenuse $4+1=5$. The specified identity gives:

$$
\begin{aligned}
& \text { for } n=1 \text { respectively } 3,4,5 \\
& n==2 \quad n \quad 5,12,13 \\
& { }_{n} n=3 \quad n \quad 7,24,25 \\
& { }_{n} n=4 \quad n \quad 9,40,41 \\
& n=5 \quad \quad n \quad 11,60,61
\end{aligned}
$$

Plato (429-348 BCE), a philosopher, one of the founders of idealist philosophy, who enjoyed immense authority not only in antiquity but also in modern times. Like Pythagoras, 76

Plato was eagerly credited with a number of mathematical discoveries and the creation of new methods of proof.