How to use Parikh's Theorem to show language is not context free
If $L$ were context free, then the set
$$ H = \{(m,n) : m &gt; n\} \cup \{(m,n) : m \leq n \text{ and } m \text{ is prime}\} $$
would be semilinear. The set $\{(m,n) : m \leq n\} = \mathbb{N} (1,1) + \mathbb{N} (0,1)$ is also semilinear. The intersection of semilinear sets is also semilinear, so the set
$$ H' = \{(m,n) : m \leq n \text{ and } m \text{ is prime}\} $$
is also semilinear. The projection of a semilinear set is semilinear, and so the set of primes is semilinear. But one-dimensional semilinear sets with asymptotic density zero are finite, whereas the set of primes is an infinite set with asymptotic density zero.