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I am trying to use the pumping lemma to prove that the language consisting of the set of $0$'s and $1$'s, beginning with a $1$, such that when interpreted as an integer, that integer is prime, is not regular.

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Pumping lemma is the key. If the language were regular, thered be strings $u,v,w$ such that $uv^*w\subseteq L$ and $|v|>0$ and wlog $|u|>0$, i.e. $u\in 1\{0,1\}^*$ If $U,V,W$ are the numbers represented by $u,v,w$ (where $v,w$ may have leading zeroes), then the number represented by $uv^kw$ is $U\cdot 2^{|w|+k|v|} + 2^{|w|}\cdot > \frac{2^{k|v|}-1}{2^{|v|}-1}\cdot V+W$, where $U\ge 1$. Show that these cannot all be prime.

From what I understand, you break the binary string into $U,V$ and $W$. The pumping lemma says that for a regular language, for every $i >= 0$, $UV^iW \in L$. You simply then choose an i and show that $UV^iW$ is not a member of the language.

However, I don't understand why it works that you can represent the number as $U\cdot 2^{|w|+k|v|} + 2^{|w|}\cdot \frac{2^{k|v|}-1}{2^{|v|}-1}\cdot V+W$, where $U\ge 1$

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