4. The numbers 1 and 2 are written on the board. It is allowed to add new numbers in the following way: If the numbers $a$ and $b$ are written on the board, then the number $a b+a+b$ can be written. Can the following numbers be obtained in this way:
a) 13121,
b) 12131.
Solution. We notice that if $c=a+b+a b$, then $c+1=(a+1)(b+1)$. Therefore, instead of the numbers written on the board, we can consider the numbers increased by 1. Then each new number is the product of two numbers already written on the board. Initially, we have the numbers $2=1+1$ and $3=2+1$, so all new numbers are of the form $2^{n} \cdot 3^{m}$, where $m, n \in \mathbb{N}$. However,

$$
\begin{aligned}
& 13121+1=13122=2 \cdot 3^{8} \text { and } \\
& 12131+1=12132=2^{2} \cdot 3^{2} \cdot 337
\end{aligned}
$$

which means that the number 13121 can be obtained in the described manner, while the number 12131 cannot be obtained in the described manner.

## II year