Difference of Two Squares cannot equal 2 modulo 4/Proof 2

Theorem

Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$.

Then $n$ cannot be expressed in the form:

$n = a^2 - b^2$

for $a, b \in \Z$.

Proof

Let $n_0 = c^2 - d^2$ for some $c, d \in \Z$.

Then:

$n_0 = \paren {c + d} \paren {c - d}$

and so:

$\paren {c + d} - \paren {c - d} = 2 d$

Therefore $n$ must be expressible as a product of two integers whose difference is even.

Now consider the integer $n \in \Z$ that satisfies $n \equiv 2 \pmod 4$.

$n$ is an even number that is not divisible by 4.

This implies that any even number that divides $n$ will produce an odd number.

Consider the ordered pairs of integers $\tuple {a, b}, a < b$ such that $ab = n$.

If $a$ is odd, then $b = \dfrac n a$ must in turn be even.

Hence the difference will not be even.

If $a$ is even, $b$ must be odd by the implication mentioned earlier.

Hence the difference will once again not be even.

Hence, $n$ cannot be expressed as a product of two integers whose difference is even.

In turn it cannot be expressed as a Difference of Two Squares.

$\blacksquare$