Using the Pumping Lemma to prove the language, $,$ is not

context$-$free, $=\{|$ n is a prime number$\}$

$($PLEASE use the following template to do the proof.$)$

$[$ANSWER$]$

Statement I want to prove:

Fill in the language

Is not a context$-$free language

Proof:

Let $=$ Fill in the language again

Assume is context$-$free

Let me be the number from the pumping lemma for CFL

Let s $=$ Fill in some cleverly picked string here that is at least m

characters long

Since s E and $|$s$|$ m$,$ the pumping lemma must apply,

CPSC $4370/5370-01$ and $9$S$1$ Spring $2024$

$2$

Specifically: s $=$ uvwxy where

$|$vx$|1$

$|$vwx$|$ m

uwy is in L for all k $0$

Say something interesting about the structure of v and$/$or x here

By the pumping lemma for context$-$free languages

Say something interesting about what happens when you $$pump$$

your string

Contradiction

Thus, our assumption must not be true

Thus, must not be a context$-$free language

QED