1. 174 On the blackboard, the numbers 1 and 2 are written. Now, it is allowed to write a new number according to the following rule: if the numbers $a$ and $b$ are already written on the blackboard, then the number $a b + a + b$ can be written. Can the following numbers be obtained by such a rule:
(1) 13121;
(2) 12131?
[Solution] Let $C$ represent the new number $ab + a + b$. Then $c + 1 = ab + a + b + 1 = (a + 1)(b + 1)$. This means that if we do not consider the numbers already written on the board, but instead consider the numbers that are one greater than the written numbers, then each new number is equal to the product of two existing numbers. Since we start with the numbers 2 and 3, after several multiplications, we obtain numbers of the form $2^n \cdot 3^m$, where $n$ and $m$ are natural numbers. Clearly, all numbers of this form can be obtained. Therefore, in the original scenario, we can only obtain numbers of the form $2^n \cdot 3^m - 1$.

Since the number 13121 is of this form, while 12131 is not, we can obtain the number 13121, but it is impossible to obtain 12131.