# Why was 1 considered as prime years ago? [duplicate]

This question already has an answer here:

I've seen on Maths Is Fun that years ago, 1 was considered as prime, but now, it is not. How did this happen? I know that a prime number has only two factors, 1 and itself, and we have 1, which is also itself. Is this why? Tell me what you think. I also know that this would make the prime factorization too continuous: 1 x 1 x 1 x 1 x 1... x 3 x 3=9 and that would not be good.

## marked as duplicate by MJD, Mark Bennet, user147263, Aditya Hase, HakimNov 30 '14 at 18:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• A prime has exactly two distinct factors. And hence $1$ does not qualify – imranfat Nov 30 '14 at 17:06
• I believe it was stopped being considered a prime otherwise we would not be able to say that every integer can be expressed as a unique product of primes. e.g. $9=3^2$ this is unique. but if you allowed 1 to also be a prime then one could say: $9=3^2\times1$ or $9=3^2\times1^2$, etc – Mufasa Nov 30 '14 at 17:07
• With time poor $\;1\;$ became weaker and weaker until he couldn't hold his primality... Seriously: even now some people consider it prime. This position ignores the fact that $\;\Bbb Z\;$ is a ring and in it $\;1\;$ is a unit, so per definition it isn't prime. I guess for some uses it could be helpful to consider it a prime, but most people just don't. – Timbuc Nov 30 '14 at 17:07
• I went with the duplicate question option because the discussion linked by @MJD is high quality, with approaches at different mathematical levels and plenty of references, and although the question is not exactly the same, some of the answers are directly on the point. – Mark Bennet Nov 30 '14 at 17:26
• The reason given by Mark Bennet's is why I listed this as a duplicate. Although the question there isn't exactly the same as this one, several of the answers there address the question that Vlad is asking, which is why 1 was a prime number and isn't any more. – MJD Nov 30 '14 at 17:35

## 1 Answer

The Fundamental Theorem of Arithmetic states (in paraphrase) that all integers greater than $1$ have only one prime factorization, and that factorization is unique to that number. So $8$ has only one prime factorization, $2^3$, and no other positive integer has the prime factorization $2^3$. If we say $1$ is prime our statement falls apart. $8$ now has an infinite number of prime factorizations: $2^3$, $1*2^3$, $1^2*2^3$, etc... Thus $1$ is not prime because it violates the Fundamental Theorem of Arithmetic.

• I have added that it could make the prime factorization worse than it can be. – VladInTheTaylor Nov 30 '14 at 17:14
• I saw your addition just after posting this. Your statement is correct, this answer is simply a more formal way of stating the answer. – Ari Nov 30 '14 at 17:18