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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?

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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

  1. it is not $0$;
  2. it has no inverse (in $\mathbb Z$);
  3. 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.

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  • $\begingroup$ It has no inverse with respect to which operation? $\endgroup$ – user160370 Jul 12 '17 at 5:57
  • $\begingroup$ @user160370 The product. We are talking about primes, right?! So, the operation that matters here is the product. $\endgroup$ – José Carlos Santos Jul 12 '17 at 6:00
  • $\begingroup$ Okay, sir. Thankyou. $\endgroup$ – user160370 Jul 12 '17 at 6:42
  • $\begingroup$ 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) $\endgroup$ – Andrew Tawfeek Jul 12 '17 at 7:52
  • $\begingroup$ @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. $\endgroup$ – José Carlos Santos Jul 12 '17 at 8:33

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