This is correct. To be fair, though, the standard terminology is confusing: calling a number only divisible by 1 and itself a "prime" already assumes the FTA.In a more abstract setting, "p is prime" means that if p|ab, then p|a or p|b, and "irreducible" means only divisible by itself or a unit (in this case 1). The FTA corresponds to unique factorization into irreducibles, and the fact that irreducible and prime are the same thing is a consequence of unique factorization. (In an integral domain, every prime is irreducible; in a unique factorization domain, the converse is also true).

 `````` ... the standard terminology is confusing: calling a number only divisible by 1 and itself a "prime" already assumes the FTA. `````` Sort of, but not really. I'm not going to disagree with you, but make the following observation. People reading this article are likely to know about primes, and what you quote here is most likely the definition that they would be accustomed to. Introducing a new, technical term and then trying to describe the details of the difference would most likely derail the purpose, and abusing the terminology a little is perhaps justified, especially when it aligns with people's existing knowledge.But you are correct, and the reason we have these terms is exactly to avoid some of the "intuitively obvious" misconceptions.

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