Smallest Odd Number not of form 2 a squared plus p

Theorem
$17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form:
 * $n = 2 a^2 + p$

where:
 * $p$ is prime
 * $a \in \Z_{>0}$ is a (strictly) positive integer.

Proof
First note that $3$ is not so expressible:

and so $3$ cannot be written in such a form.

Then we have:

However, we have:

Also see

 * Definition:Stern Prime