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RolyPoly
 2 years ago
Find all positive integer pairs (x,y) such that 4xyxy is a perfect square.
RolyPoly
 2 years ago
Find all positive integer pairs (x,y) such that 4xyxy is a perfect square.

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mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0without loss of generality we can suppose \[y\ge x \]because the expression is symmetric

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0assume that \[y=x+a\]with \(a\ge 0\)

RolyPoly
 2 years ago
Best ResponseYou've already chosen the best response.1Okay! I'll try from here!!

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0wait ... im afraid i made a mistake :/

RolyPoly
 2 years ago
Best ResponseYou've already chosen the best response.1Don't be afraid :P \[4xyxy=4x(x+a)x(x+a)=4x^2+(4a2)xa\]

RolyPoly
 2 years ago
Best ResponseYou've already chosen the best response.1\[\Delta=(4a2)^2  4(4)(a)=0\]\[16a^2  16a + 4 + 16a=0\]\[16a^2 + 4 =0\]Well well well...

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.04xyxy = (x+y)^2  (xy)^2  (x+y) = (x+y)(x+y1)  (xy)^2 i am not too sure if that'd help! :P

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0@RolyPoly ...Are you familiar with modular arithmetic??

RolyPoly
 2 years ago
Best ResponseYou've already chosen the best response.1I've never heard of it! :(

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.0i dont know much but i do have a little idea..will you please guide me/us through a solution @mukushla

RolyPoly
 2 years ago
Best ResponseYou've already chosen the best response.1Forgive me for closing this question, even though it's not yet solved!
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