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I couldn't find this question with google, so maybe someone can help here :)

When looking at the definition of divisibility on wikipedia, it is written that $a,b \in \mathbb{Z}$ $a \mid b \iff \exists c \in \mathbb{Z}: b = a \cdot c$

Following this, negative integers should also be a divisor. But, one definition of prime numbers states:

$p$ is prime $\iff$ p can only be divided by 1 and p

So every prime number $p$ should also be divided by $-p$ hence the defintion of divisibility, but this is not the case?

What am I missing out?

EDIT: I know there exists other definitions where this isnt an issue, but this definition is used and seems to be correct but still is in conflict with other definitions...In other words, is there a reason to use $\mathbb{Z}$ and not $\mathbb{N}$?

Algebruh
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You omitted the $\color{#c00}{\rm context\!\!:}\ a\mid b\ \ \color{#c00}{{\rm in}\ M}\iff \exists\, c\ \ \color{#c00}{{\rm in}\ M\!:}\, ac = b.\ $

In your case $\,\color{#c00}{M = \Bbb N}\,$ but generally in divisibility theory it can be any (cancellative) monoid. In fact many well-known properties of domains (e.g. UFD,GCD,valuation) can be described purely in terms of their multiplicative monoid - see divisibility groups.

Bill Dubuque
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