(a) Use the Pumping Lemma for Context-Free Languages to prove that if B is any infinite context free language, then B contains strings whose lengths form an infinite arithmetic progression whose common difference is at most the pumping length p of B.

(b) Show that in the sequence of powers of two, 1, 2, 2 2 , 2 3 , . . ., there is no arithmetic progression of length 3 or more.

(c) Use parts (a) and (b) of this exercise to prove that the language B = {a 2 n : n 0} over = {a} is not context-free.