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.

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:

$\ds p$

$=$

$\ds n^2 - m^2$

$\ds $

$=$

$\ds \paren {n + m} \paren {n - m}$

$\ds \leadsto \ \ $

$\ds n - m$

$=$

$\ds 1$

Definition of Prime Number

$\ds \leadsto \ \ $

$\ds m$

$=$

$\ds n - 1$

Definition of Prime Number

$\ds \leadsto \ \ $

$\ds n + m$

$=$

$\ds 2 n - 1$

$\ds $

$=$

$\ds p$

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:

$n^2 = \paren {n - 1}^2 + p$

$\blacksquare$