No.
It's the non-factorability of primes that is important.
It's the non-factorability of primes that is important.
Primes by definition have two, and only two factors.
The difficulty of factoring a number which is the product of two large primes is the important bit.
Also Möbius becomes strange with infty factors.
Having no proper factors is the definition of an irreducible element.
The two definitions agree for the integers and nice algebraic structure (UFDs) but there are algebraic structures in which they are not the same which explains why there are two differenr definitions and names in abstract algebra
The only unknown is which two primes.