On the Diophantine equation $ (12m^2 + 1)^x + (13m^2 - 1)^y = (5m)^z$

Question

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.