Is this an eigenfunction of a ladder operator?

Question

$a^{\dagger}$ and $a$ are ladder operators:

$a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle \,\,\text{and}\,\, a|n\rangle= \sqrt{n}|n-1\rangle $

Is the state $|n\rangle$ an eigenfunction of $a\dagger$ ?

Answer 1

An eigenstate of an operator is a state that is rescaled when the operator acts on it. $|n+1\rangle$ is not parallel to $|n\rangle$.