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I've heard stories about why the number $1$ is neither a prime number nor a composite number, even on in the middle of this awesome math page.

Just scroll down to the middle to read about it. It's a bit short.

Anyway, let's cut to the chase. How come $1$ isn't a prime number or even a composite number? I want to know from your great-heard-of answers! At least I know that a prime number has only two factors, $1$ and itself. So, why isn't this the same for $1$? It has a factor of $1$, which is also itself, $1$. I want to hear about this, too.

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Because if it were prime, then the prime factorization of numbers wouldn't be unique. So it's defined as not-prime.

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  • $\begingroup$ Yeah, because if we talk about the prime factorization of 10 and we had one as a prime number, then it would be any amount of ones added to $2*5$, which is too silly. $\endgroup$ – Mathster Nov 3 '14 at 1:19
  • $\begingroup$ I think this is really the best answer. The fact 1 is a unit in Z doesn't really answer the question because it doesn't explain why that excludes it from either classification. $\endgroup$ – Mathemagician1234 Nov 3 '14 at 1:21
  • $\begingroup$ @Mathemagician1234 I think my answer is pretty clear now. Unique factorization is unique up to unit multipliers. This answer is just not giving the good definition of "factorization". This happens say in the unique factorization domain $\Bbb Q[X]$, and here you have lots of units: the rationals. $\endgroup$ – Pedro Tamaroff Nov 3 '14 at 1:22
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Because $1$ is a unit in $\Bbb Z$. In every ring, in particular in every UFD you have units and nonunits. It is the nonunits that are factored into irreducible (=prime) factors times units, but one doesn't factor units. Unique factorization is unique up to unit multipliers, since one can add an arbitrary unit times its inverse, and still get a good factorization.

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The main practical idea behind primes is to eliminate all their multiples as composites, see sieve of Eratosthenes. What would happen if we were to apply this reasoning to the number $1$ ?

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For a number to be a prime number it must have two factors one and it's self, but with 1 only 1 times one is 1, so it only has one factor, therefore making it not a prime number.

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