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

This Question is Closed

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

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

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

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

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

RolyPoly Group TitleBest 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...
 2 years ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.0
4xyxy = (x+y)^2  (xy)^2  (x+y) = (x+y)(x+y1)  (xy)^2 i am not too sure if that'd help! :P
 2 years ago

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

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

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

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