Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!

Proof for odd numbers: Divide any odd number, $x$, in half, then let $C$ be the half rounded up and $B$ be the half rounded down. Then, $C^2 - B^2 = x$. Let's examine this algebraically:
$$C = \frac{x+1}{2}$$
$$B = \frac{x-1}{2}$$
$$C^2 - B^2 = \biggl(\frac{x+1}{2}\biggr)^2 - \biggl(\frac{x-1}{2}\biggr)^2 = \frac{x^2+2x+1}{4} - \frac{x^2-2x+1}{4} = \frac{x^2-x^2+2x+2x+1-1}{4} = \frac{4x}{4} = x$$
Then, because even/odd integers alternate, $x \pm 1$ will always result in an even integer when $x$ is odd and an odd integer when $x$ is even.
Because only even integers are divisible by two (without incurring a decimal), $x \pm 1$ must be even to be divisible by 2, so $x$ must be odd. Otherwise, if $x$ were even, then $C$ and $B$ would not be integers.
Example for odd numbers: Take the number 31. Let $C=\frac{31+1}{2}=16$ and $B=\frac{31-1}{2}=15$. Then, $16^2 - 15^2 = 256 - 225 = 31$.
Proof for all composite numbers: every pair of $x$'s factors, $p$ and $q$, including composite factor pairs, where the oddity of $p$ matches the oddity of $q$, corresponds to a separate representation as the difference of two squares:
$$p \times q = x$$
$$C = \frac{p + q}{2}$$
$$B = \frac{|p - q|}{2}$$
$$C^2 - B^2 = \biggl( \frac{p + q}{2} \biggl)^2 + \biggl( \frac{|p - q|}{2} \biggl)^2 = \frac{p^2 + 2pq + q^2}{4} - \frac{p^2 - 2pq + q^2}{4} = \frac{p^2 - p^2 + q^2 - q^2 + 4pq}{4} = pq = x$$
Example for every-other even numbers: Because half of every-other even number (that is, 4, 8, 12, etc.) has an even oddity, it can be shown that at least every-other even number can be expressed as the difference of two squares. For $x=48$, we get $p=2$ and $q=\frac{x}{p}=24$. Then, $C=\frac{2+24}{2}=13$ and $B=13-2=11$. We verify $C^2 - B^2 = 169 - 121 = 48$.
Conclusion: The only numbers that cannot be expressed as the difference of two squares are the sequence $4n+2$: 2, 6, 10, 14, 18, 22, etc.
