14.5.19 ** The numbers 1 and 2 are written on the blackboard. Now, new numbers are written on the blackboard according to the following rule: if the numbers $a$ and $b$ are already on the blackboard, then the number $ab + a + b$ can be written. Can the following numbers be obtained by such a rule:
(1) 13121, (2) 12131?
Parse: On the blackboard, 13121 can be obtained, but 12131 cannot. If $c=a b+a+b$, then $c+1=(a+1)(b+1)$. Adding 1 to the numbers appearing on the blackboard, the numbers 1, 2 become 2, 3. From $c+1=(a+1)(b+1)$, we know that all new numbers appearing later, when 1 is added, must have the form $2^{n} \cdot 3^{m} (n, m$ being natural numbers), and numbers of the form $2^{n} \cdot 3^{m}-1$ can certainly appear. Note that: $13121+1=2 \cdot 3^{8}, 12131+1=2^{2} \cdot 3^{2} \cdot 337$. Therefore, the blackboard has 13121, but not 12131.