Difference between Odd Squares is Divisible by 8/Solution/Mistake

Source Work

 * Chapter $2$: The Fundamental Theorem of Arithmetic
 * $\text {2-1}$ Euclid's Division Lemma
 * Solution to Exercise $6$
 * Solution to Exercise $6$

Mistake

 * Since $a$ and $b$ are odd integers, $a = 2 r + 1$, and $b = 2 s + 1$. Thus
 * $a^2 - b^2 = \paren {4 r^2 + 4 r + 1} - \paren {4 s^2 + 4 s + 1} = 4 \paren {r - s} \paren {r - s + 1}$.
 * Now if $r - s$ is even, then $r - s = 2 m$ and $a^2 - b^2 = 8 m \paren {2 m + 1}$; if $r - s$ is odd, then $r - s = 2 n + 1$ and $a^2 - b^2 = 8 \paren {2 n + 1} \paren {n + 1}$. Thus in any case $a^2 - b^2$ is divisible by $8$ if $a$ and $b$ are odd integers.

Correction
The last expression should be:
 * $4 \paren {r - s} \paren {r + s + 1}$

The rest of the proof needs to be adjusted accordingly; for example, for odd $r - s$ we get this instead:


 * $a^2 - b^2 = 8 \paren {2 n + 1} \paren {r + n}$

See Difference between Odd Squares is Divisible by 8 for a correct working.