Mathematica: How to find Maximize of(m2+n2)/(5−mn)wherem,nand(m2+n2)/(5+mn)are integer numbers?

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How to find Maximize of(m2+n2)/(5−mn)wherem,nand(m2+n2)/(5+mn)are integer numbers?
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Thanks to all those great people for their contributions!

(mathematica.stackexchange.com/users/81746/john-paul-peter)John Paul Peter
(mathematica.stackexchange.com/users/26598/roman)Roman
(mathematica.stackexchange.com/users/72111/cvgmt)cvgmt
(mathematica.stackexchange.com/users/7152/user64494)user64494
A special thanks goes out to the (https://mathematica.stackexchange.com/questions/293584/how-to-find-maximize-of-m2-n2-5-m-n-where-m-n-and-m2-n2)Stackexchange community

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