Pumping Lemma with Prime Number
Helpful property of prime numbers: The gaps between consecutive prime numbers become arbitrarily large. That means for every r, there is a prime p such that none of the numbers p+1 to p+r are primes. That means $a^nd^mb^k$ is not in the language for p+1 ≤ m ≤ p+r. 

That would generally work for "m is a member of an infinite set with arbitrarily large gaps between members of the set", like "m is a square, a cube, a factorial" and many others.