If one wanted to actually find examples of

, here is an approach that works for

.
I'll use the hyperbolic function

and its inverse, which aren't in the MX2 syllabus but their definitions are quite simple to understand:
)
and
Now, let
 = x^n + \frac 1{x^n})
,
be a real function where

.
If

is even, then
 = f(-x))
.
If

is odd, then
=-f(-x) )
.
That is, the parity of

is the parity of

.
So without loss of generality, assume

and make the substitution

.
If
 = k)
for some

, then

, so
Since

,

.
We require

, which should be expected, as
 \ge 2 )
for all positive

.
The original proposition
is equivalent (I believe) to
 \in \mathbb Z \implies 2 \cosh(nx) \in \mathbb Z )
.
We also find an interesting identity:
for all

.
EDIT: It's worth noting that this shows that if

, then

.
If
)
then

.
The gap between the second and third perfect square is

, and this gap will only increase as you go further out.
So

. Subbing in these values confirms that

.