Yes, 0 and 1 are neithe prime nor composite.
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The number 1 is neither composite nor prime.
Zero and one are neither composite nor prime.
There is no name for numbers which are neither prime nor composite.
They are neither prime numbers nor composite numbers. It is because, they do not have two factors at least.
Yes. All prime numbers and composite numbers are positive integers, or whole counting numbers. That leaves infinitely many numbers that are neither prime nor composite. If you intended to narrow the scope of your question to the whole counting numbers or to the positive integers, then there are NO such numbers that are neither. A counting number, however large, will be either prime or composite.