Let $L\subseteq \{0, 1\}^*$ be the set of natural numbers in binary notation, which together compose the infinite arithmetic progression with first term 4 and common difference 3. Is it true that $L$ is regular?
I tried to tackle this down using $a^{-1}L$ but failed to accomplish much.
Any help is very much appreciated.
Essentially you are looking for all of the binary numbers greater than 1 that have a remainder of 1 modulo 3. If you think about how appending a 0 or 1 to the end of a binary number changes its modulus modulo 3, you shouldn't have any trouble building a FSM for this language.