Let $L=\{b^ma^n \mid \text{$m$ and $n$ are coprime} \}$. Using Nerode's theorem, prove that $L$ is irregular.
From Nerode's theorem I know that $L$ is regular if and only if the number of equivalence classes of $R_L$ (the relation defined in Nerode's theorem) is finite, so I need to prove that there are infinitely many equivalence classes.
The first thing that came to mind from $L$'s definition is using Dirichlet's theorem, hence I tried:
Let $w_{m, i}=b^ma^i$, ($m,i$ are coprime), and I prove that for $j\ne i$, ($m, j$ coprime), $$w_{m, i} \not R_L w_{m, j}$$ Let $z=a^{m+ni}$, ($n$ an integer promised by Dirichlet's theorem such that $m+ni$ and $m$ are coprimes), so $$w_{m, i}z = b^ma^{m+ni+i}= b^ma^{m+(n+1)i}\in L$$ and $$w_{m, j}z = b^ma^{m+ni+j}\not\in L$$ But this isn't necessarily true as $m$ and $m+(n+1)i$ might not be coprime and $m$ and $m+ni+j$ might be.
I know from previous exercises that I need to find $w_i$ and show that for a word $z$ $$w_iz\in L\text{ and }w_jz\not\in L \space\space (i\ne j)$$ and therefore there are infinitely many equivalence classed, but I find coprimality difficult to handle.
