if and only if
is not of the form
.
Proof:
Using
as the
double factorial of
:
and therefore
if and only if
, where
satisfies
(that is,
, but
).
We'll use the fact that
(
prime) (Legendre's formula). The exponent of
in
is:
Where the equality signs of
are taken if and only if
is a power of two. Thus,
if and only if
is not a power of
, and so
if and only if
is not of the form
.
Q.E.D