Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 4G / b / iii / Solution

This language is not context-free. Let $p$ be a proposed pumping length and choose $n$ with $3^n\ge p$. In any pumping decomposition of the unary word $a^{3^n}$, pumping once more increases its length by an integer $d$ with $1\le d\le p$. Butso the pumped length is not a power of three, contrary to the pumping lemma for context-free languages. This is also an instance of the fact that a unary context-free language is regular, whereas the powers of three are not ultimately periodic.