# Please give a a reason, why 1 is a composite

Hans Aberg haberg-1 at telia.com
Wed Mar 25 10:20:14 UTC 2015

```> On 25 Mar 2015, at 07:32, yi lu <zhiwudazhanjiangshi at gmail.com> wrote:
>
> Please give a a reason, why 1 is a composite. This is a seriously bug.

As in the answer already given, it is a matter of providing a definition:

A reasonable modern definition that generalizes to (commutative) rings R (typically integral domains), is that it is a non-zero element that has no non-trival divisors. For an element x, the trivial divisors are the units u (the invertible elements) of R and u*x.

In the ring of integers Z, with this definition, if p > 0 is a prime, then also, -p becomes a prime. But the problem is that if you take other rings, like Z(i], when i is the imaginary unit, there may be no good way to define what positive numbers are.

Instead, one has an equivalence class of prime numbers: the elements of each class differ by multiplication of a unit of the ring. Then the condition p > 0 is a choice of representatives from these equivalence classes.

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