# p constant in pumping lemma for regular languages

We have a language

$L = \{ a^{n^{2}}, n < 12\}$ We have to find minimum constant P for pumping lemma.

What does p represent here? I thought p in pumping lemma represent number of states , yet the right answer is $p >= 122$ , i would understand if it was $p <= 122$.

So for example it would be correct when $p = 155$. But how could we create such automat with 155 states?

• Which pumping lemma? The meaning of $p$ becomes apparent from the proof. – Raphael Jan 27 '17 at 17:55
• The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! – Raphael Jan 27 '17 at 17:55

The key here is that $L$ is a finite language. Therefore it can be recognized by a DFA which has no cycles - in this case, a chain of 122 states, numbered 1 through 122, with transitions on $a$ from each state $i$ to state $i+1$ [except for a loop on $a$ from state 122 to itself], with states 1, 4, 9, ..., 121 being final. For P=122, this automaton satisfies the Pumping Lemma - i.e all strings in $L$ of length at least P can be pumped - in a purely vacuous sense, because there are no strings in $L$ of length at least P.