Nope, the list is infinite.
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All prime numbers are rational.
There is no specific name. Such numbers include all irrational numbers, all non-integral rational numbers, 0 and 1.
There is no need to do prime factorization as prime numbers are already prime.
All prime numbers are not odd numbers. 2 is an even prime number, and it is the only even prime number because all other even numbers have 2 as one of their factors.
It is impossible to list the infinite number of prime numbers and composite numbers.