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

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Source Work

1971: George E. Andrews: Number Theory:

Chapter $2$: The Fundamental Theorem of Arithmetic
$\text {2-1}$ Euclid's Division Lemma
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.


Sources