Is there any good reason to define 1 as a non-prime number?
A prime is a number that has exactly two factors.
How many factors does 1 have?
I don't think any of you actually answered agro's question.
From what I understand, certain theorems (such as the one that states that all integers have a unique (up to the order of factors) decomposition into primes) become easier to state if you don't consider 1 to be prime.
Wow, Dr. Math says the same thing.
Unless doctor math says the following. The fundamental theorem of algebra states that all numbers greater than 1 can be decomposed into a unique product of prime numbers. If 1 was prime then it would not be possible to uniquely factorise numbers.
1 is a unit in the ring of integers, it cannot be a prime. It's the definition. It has its useful implications. It doesn't generate a prime ideal in Z for a start (it generates Z which is not a proper ideal). It's almost like asking 'is there any special reason not to define 3 as even as a special case?' No, it's not allowed by definition. It might seem a little silly just to say that, but units are excluded because they're invertible. If you like, stick with the idea that they are excluded because of the degeneracy of the ideal they generate, that at least seems the most interesting one. The other factorization ideas are always stated with an upto reordering the factors and units provise anyway.
Primes have been known for thousands of years, rings only for two hundred or so. Until there was a good reason to consider 1 as non-prime by way of units and whatnot, 1 was sometimes considered as a prime and sometimes not.
Maybe they should define 1 as being "hyper-prime" or something like that. In a way it kind is, more prime than a prime. ;)
Isn't there a theorem that states that every non-prime number can only be decomposed into primes in one single way? If you allow 1 to be prime, doesn't that mean that every number has infinitely many decompositions?
Decomposition into primes (in a UFD) is unique upto order of multiplication and multiplication by units. There is always more than one way to decompose a composite into primes, what matters is the essential uniqueness (order of factors and units)
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