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

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

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

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

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

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

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