What will accepting 1 as prime change? [duplicate]

How significant is the fact 1 isn't a prime number? What will happen if it is? What areas of Mathematics are affected by changing the fact? I know why and how 1 isn't a prime. My question is how significant is the fact.

• Since $1^n = 1$, there is no longer a unique prime factorization of any integer, and pretty much the whole of number theory will fall apart unless you change "prime" to "prime $\ne 1$" virtually everywhere. Feb 18 '19 at 16:53
• It will make theorems more annoying to state. Feb 18 '19 at 16:54
• See the answers in Why is $1$ not a prime number? Feb 18 '19 at 16:55
• It's analogous to asking what will change if we allow $0$ to be positive (as the French do). Many theorems using "positive" would need to be updated to remain correct. Feb 18 '19 at 17:10
• @usiro I can't make any sense of your remarks. In any case notions such as irreducible, prime and unit are independent of geometry. Feb 18 '19 at 21:27

That wouldn't be a disaster, but it would add some very annoying things. For example, instead of the uniqueness in the fundamental theorem of arithmetic we would have "uniqueness up to finitely many multiplications of $$1$$".

• We can have infinitely many 1's too, right? Feb 18 '19 at 17:02
• Actually yes, we can. But even if we want to say that every integer can be represented as a finite product of primes it would still not be a unique decomposition.
– Mark
Feb 18 '19 at 17:04
• There wouldn't be any uniqueness.Got your point. Feb 18 '19 at 17:05
• @Krishna No, infinite products don't make sense in general (one needs additional ring structure to make sense of them). Feb 18 '19 at 17:16
• additional ring structure? Feb 18 '19 at 19:27

We need factorisation of positive integers to be unique in many contexts, and this fails if we take $$1$$ to be prime, since then $$6,$$ for example, would factorise in the infinity of ways $$2×3=2×3×\underbrace{1×1×\cdots×1}_{n},$$ where $$n$$ is a nonnegative integer. This happens for the other positive integers too.

Actually, $$1$$ is different from the other positive integers in that it is neither composite nor prime. This distinguished position follows from the fact that $$1$$ is the multiplicative identity in $$\mathbf Z;$$ that is, for any integer $$m,$$ we have that $$1×m=m×1=m.$$