Can negative integers be prime?

Question

Generally, the definition of prime numbers is all those natural numbers greater than 1, having only two divisiors, the number itself and 1. But, can the negative integers also be thought of in the same way?

For example: if we take the case of -1. Then it is divisible by 1 and itself. Can we call it prime? Why? Or why not?

score 5 · Answer 1 · answered Jul 12 '17 at 05:38

5

In the context of integer numbers, yes, $-3$, for instance, is also a prime. But note that it has four divisors, and not just two: $-3$, $-1$, $1$, and $3$.

An integer is prime if

it is not $0$;
it has no inverse (in $\mathbb Z$);
whenever it divides a product, it divides one of the factors.

So, again, $-3$ is a prime number. But $-1$ is not, since it has an inverse, which is itself.

answered Jul 12 '17 at 05:38

José Carlos Santos

440,053

It has no inverse with respect to which operation? – user160370 Jul 12 '17 at 05:57
@user160370 The product. We are talking about primes, right?! So, the operation that matters here is the product. – José Carlos Santos Jul 12 '17 at 06:00
Okay, sir. Thankyou. – user160370 Jul 12 '17 at 06:42
What do you mean has no multiplicative inverse? Isn't $1$ the identity of $(\mathbb {Z}, *)$, thus its not a group because integers percisely don't have inverses? (besides 1) – Andrew Tawfeek Jul 12 '17 at 07:52
@AndrewTawfeek You are so wrong here! For instance: no, it is not true that $1$ is the only integer which has an inverse (with respect to multiplication); $-1$ has the same property. – José Carlos Santos Jul 12 '17 at 08:33
Why do we have the second property? Why can't it hold an inverse? – user160370 Jul 12 '17 at 14:13
I don't understand your question. The second property is this: there are two integers which have an inverse in $\mathbb Z$: $1$ and $-1$. What is your doubt? – José Carlos Santos Jul 12 '17 at 14:25
My doubt is why can't 1 and -1 be also considered prime? – user160370 Jul 12 '17 at 17:02
@user160370 Within the natural, $1$ is not considered a prime. Why should it be considered a prime within the integers? Or do you want to know why $1$ is not considered a prime within the naturals? – José Carlos Santos Jul 12 '17 at 17:19

Can negative integers be prime?

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