Correct me if i am wrong, in the finite field ℤ/pℤ there exist a primitive root of order p-1 which is also a (p-1)th root of unity modulo n. Thus to evaluate a N degree polynomial, where N is a power of 2, at (p-1)th roots of unity modulo n, we require p-1 = N = 2^k, or expressed in another way,

p = 2^k + 1.

But some online sources cite, p = m * (2^k) + 1 for m > 0.

Question- Now i don’t understand the purpose and benefit of putting ‘m’ into the equation, why not just p = 2^k + 1, how putting a ‘m’ other than 1 would help for our purpose of calculating NTT at various point