Prove this langage isn't regular using the pumping lemma

Question

How can I prove that this language isn't regular using the pumping lemma for regular language?

$$L = \{a^nb^m| n,m > 0, \gcd(n,m)=1\}$$

I've already checked this answer: Proof that a language involving $gcd$ is not context-free but don't understand this part.

"Let s' be $a^{(n+ik)}b^m \ with \ k \geq 1$. Then s' e L if $gcd(n + ik, m) =$ , however the modular equation $n+ik = 0(m)$ have a solution $i = nk^{(-1)(m)}$ ..."

I don't understand how he proved, $gcd(n+ik, m) \gt 1$, could someone explain me more clearly the answer please.

Thank you.

Answer 1

As you know, the Pumping Lemma for regular languages states that if $L$ is a regular language, then there is an integer $p$ such that is $s$ is any string in $L$ with length $|s|\ge p$ then we can write $s=xyz$ with

1. $|y|>0$,
2. $|xy|\le p$, and
3. $xy^iz\in L$ for all $i\ge 0$.

If we let $s=a^nb^m$ for distinct primes $n,m\ge p$ then certainly $s\in L$ since $\gcd(n,m)=1$. Now we know that in this case $y=a^k$ with $0<k\le p$ and so if we pump $y$ we'll have $xy^{i+1}z=a^{n+ik}b^m$. Now we'll try to get a contradiction by showing that there is some $i$ for which $\gcd(n+ik,m)\ne 1$. We can get this if we can find an $i$ such that $m$ divides $n+ik$, since then we'll have $\gcd(n+ik,m)=m$.

In other words, we want an $i$ such that $n+ik\equiv 0\pmod m$. In modular arithmetic land we'll then have the following chain of equivalences:

$$\begin{align} n+ik &\equiv 0\pmod m\\ ik &\equiv -n\pmod m\\ i &\equiv -n(k^{-1})\pmod m \end{align}$$ The last step follows from the fact that when the modulus is prime, any nonzero value has a multiplicative inverse.

Here's an example: Suppose $p=6$, then we could pick $n=11,m=7$ and pump the string $a^{11}b^7\in L$. If it happened that $k=4$ then we could use $i=-11\cdot 2$, where the $2$ came from the fact that $4\cdot 2\equiv 1\pmod 7$. In other words, we'd have

$$ i=-11\cdot 2\equiv -22\equiv -1\equiv 6\pmod 7 $$ and then the pumped string would be $a^{11+6\cdot 4}b^7=a^{35}b^7$ and this is clearly not in $L$, since $\gcd(35,7)=7\ne 1$ $$ $$

Answer 2

An easy way to prove that this language isn't regular is using the Myhill–Nerode theorem. Let $p_i$ be an enumeration of all primes. The words $a^{p_i}$ are pairwise inequivalent, since $a^{p_i} b^{p_i} \notin L$ whereas $a^{p_j} b^{p_i} \in L$ when $i \neq j$. Since we found infinitely many pairwise inequivalent words, it follows that $L$ is not regular.