At the end of the proof of lemma 10, lemma 8 is cited. In order to use it and finish the contradiction, we need to show $n$ is not a multiple of $3.$ However, I don't see any contradiction in having $n \equiv \pm 2 \mod 8$ and $n \equiv 0 \mod 3.$ I also asked on MSE but not even a halfpenny of thoughts were given.
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$\begingroup$ The m.se question was math.stackexchange.com/questions/4557787/… $\endgroup$– Gerry MyersonCommented Oct 23, 2022 at 8:40
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Lemma 8 is used to conclude that the second factor on the left is $-1$. Note that this factor is $$ \left( \frac{5}{u_{2^s}}\right) $$ meaning you apply Lemma 8 for $m=2^s$ and not for $n$. It is obvious that $m$ is not a multiple of $3$.
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$\begingroup$ I didn't notice the superscript, my mistake $\endgroup$ Commented Oct 23, 2022 at 0:26