how to prove {(a^m)(b^n)(c^k): m!=k and m,n,k ∈ N} is non-regular?

Question

It's a question in CS courses "Theory of Computation", about proofs of regular or non-regular language.

How to prove {(a^m)(b^n)(c^k): m!=k and m,n,k ∈ N} is non-regular?

I try to solve it by pumping theorem but didn't work out.

Answer 1

Suppose you have your machine. There are two prefixes a^i and a^j, i≠j, that have to both go to the the same state, since you only have a finite number of states. But then your machine can't tell the difference between a^ibc^i and a^jbc^i, one of which should pass and one of which shouldn't.