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I have read a paper http://www.m-hikari.com/ija/ija-2015/ija-5-8-2015/p/teraiIJA5-8-2015.pdf concerning the Diophantine equation $$ (12m^2 + 1)^x + (13m^2 - 1)^y = (5m)^z$$ and the author have obtained one solution $(x,y,z)=(1,1,2)$ under the condition $m\not\equiv 17,33 \pmod{40}$. But I have used Maple software without using this condition and I obtained the same solution. My question is what is: the goal (role) of this condition? Thank you.

Sil
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    The last sentences of the linked paper (before the references): "Remark. When $m \equiv ± 2 (\mod 5)$ and $m \equiv 1 (\mod 8)$, i.e., $m \equiv 17, 33 (\mod 40)$, we could not prove Lemma 3. Hence the condition $m \not\equiv 17, 33 (\mod 40) $ is necessary to Theorem."

    The word "necessary" is a little weird there... I might speculate that they didn't find a specific counterexample (otherwise they would have probably listed it). But the condition was needed for the lemma to go through.

     – Mason Jul 06 '18 at 19:26

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