Proving that $L=\{w\in \Sigma^*: |w|_a= 2^n +273$, $n\in \mathbb{N} \}$ is irregular.

Question

I am trying to prove that $L=\{w\in \Sigma^*: |w|_a= 2^n +273$, $n\in \mathbb{N} \}$ is irregular, whereas: $\Sigma=\{a,b\}$.
I tried to use the pumping lemma with no success. I have also tried to prove that there are infinite equivalency classes (thus, $L$ doesn't have a finite automata which accepts it) but I got stuck, any help?

Answer 1

The pumping lemma should work. Let $w = a^m \in L$ with $m$ greater than the pumping constant (this will happen when $m = 2^n+273$ for some $n$, so just choose large enough $n$). Then if $L$ were regular, there is some $k$ such that for all $c\in \mathbb{N}$, all of $a^{m+ck} \in L$. Now show that for any $k>0$ and $m$, not all numbers of the form $m+ck$ can be of the form $2^n+273$.