On one hand, we can think of the prime factorization of positive integer $n$ as representing $n$ by a product of prime numbers, each being smaller than $n$. In that sense, a prime number $p_{k}$ cannot have a prime factorization, the "product" comprising only one number and that number not being smaller than the original number $p_{k}$. On the other hand, we can think of the prime factorization of $n$ as the representation of $n$ by an expression where all numbers in the expression must be primes and where the only operators that may appear in the expression are those denoting multiplication. In that more general sense, a prime number does have a prime factorization, where the number and its factorization are identical. (Indeed, I have even seen arguments that the prime factorization of 1 exists and is equal to 1 itself, the justification being that a product of no factors is vacuously 1).
What is the consensus, if there is a consensus, on whether $p_k$ is its own prime factorization? Will I be laughed out of the room if I refer to the prime factorization of a prime?