Square of n such that 2n-1 is Composite is not Sum of Square and Prime

Theorem
Let $n^2$ be a square such that $2 n - 1$ is composite.

Then $n^2$ cannot be expressed as the sum of a square and a prime.

Proof
The case where $n = 1$ is trivial, as there are no prime numbers less than $1$.

Let $n, m \in \Z$ be integers such that $n > 1$.

Let $n^2 = m^2 + p$ where $p$ is prime.

Then:

So if $2 n - 1$ is composite, there exists no prime $p$ such that $n^2 = m^2 + p$.

Further, if such a $p$ does exist, then $m = n - 1$, and so:
 * $n^2 = \left({n - 1}\right)^2 + p$