Can we write CFG for this language? I searched multiple websites but I couldn't find any answers.
CFG for L = {a^mb^nc^k: k = m×n}
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We can prove this is not context-free using the pumping lemma for context-free languages, as follows.
Assume the language were context-free. Then we'd be able to write any string w in the language as w = uvxyz where |vxy| <= p, |vy| > 0 and for all integers k >= 0, u(v^k)x(y^k)z is also a string in the language.
Consider the string a^(2p) b^(3p) c^(6p^2). This is a string in our language since 2p x 3p = 6p^2. Now, consider the five cases for the placement of the substring vxy:
In all five possible cases we see the equality cannot be maintained while pumping down (choosing k = 0) and so our string cannot be pumped down. But then the language cannot be context-free.