Definition:Proper Divisor

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Definition

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $U$ be the group of units of $D$.

Let $x, y \in D$.


Then $x$ is a proper divisor of $y$ if and only if:

$(1): \quad x \divides y$
$(2): \quad y \nmid x$
$(3): \quad x \notin U$

That is:

$(1): \quad x$ is a divisor of $y$
$(2): \quad x$ is not an associate of $y$
$(3): \quad x$ is not a unit of $D$


Integers

As the set of integers form an integral domain, the concept of a proper divisor is fully applicable to the integers.


Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.

Let $x$ divides $y$ be defined as:

$x \divides y \iff \exists t \in \Z: y = t \times x$

in the conventional way.


Then $x$ is a proper divisor of $y$ if and only if:

$(1): \quad x \divides y$
$(2): \quad \size x \ne \size y$
$(3): \quad x \ne \pm 1$

That is:

$(1): \quad x$ is a divisor of $y$
$(2): \quad x$ and $y$ are not equal in absolute value
$(3): \quad x$ is not equal to either $1$ or $-1$.


Also known as

A proper divisor is also known as a proper factor.

An older name for a proper divisor of an integer is aliquot part.


Also see

  • Results about proper divisors can be found here.


Sources

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