No, because now we can consider "The Fundamental Theorem of Arithmetic", this Theorem states; "that every integer greater than 1 either is prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not"
Therefore there is only one prime factorization of a composite number.
It is actually not possible to not have any prime numbers, because even prime numbers are composed by only one prime number, so composite numbers have more them one prime number in them.