Use the Pumping Lemma to show that $L$ is not regular: $$ L = \{{a^{n^3} | \ge 0}\}$$
I feel like I have a good intuition of what the Pumping Lemma states; strings that belong to a regular language have a part or parts that repeat over and over again.
Example strings:
$a^{0^3} = a^0 = $ "" ($\epsilon$)
$a^{1^3} = a^1 = $ "a"
$a^{2^3} = a^8 = $ "aaaaaaaa"
$a^{3^3} = a^{27} = $ "aaaaaaaaaaaaaaaaaaaaaaaaaaa"
I understand that this proof is done via contradiction, and looking at the examples it becomes obvious that constructing a DFA is difficult. But how do I prove that's impossible?!
