Question

Find all positive integer pairs (x,y) such that 4xy-x-y is a perfect square.

Answer 1

without loss of generality we can suppose \[y\ge x \]because the expression is symmetric

Answer 2

assume that \[y=x+a\]with \(a\ge 0\)

Answer 3

Okay! I'll try from here!!

Answer 4

wait ... im afraid i made a mistake :/

Answer 5

Don't be afraid :P
\[4xy-x-y=4x(x+a)-x-(x+a)=4x^2+(4a-2)x-a\]

Answer 6

\[\Delta=(4a-2)^2 - 4(4)(-a)=0\]\[16a^2 - 16a + 4 + 16a=0\]\[16a^2 + 4 =0\]Well well well...

Answer 7

4xy-x-y = (x+y)^2 - (x-y)^2 - (x+y) = (x+y)(x+y-1) - (x-y)^2

i am not too sure if that'd help! :P

Answer 8

@RolyPoly ...Are you familiar with modular arithmetic??

Answer 9

I've never heard of it! :(

Answer 10

i dont know much but i do have a little idea..will you please guide me/us through a solution @mukushla

Answer 11

Forgive me for closing this question, even though it's not yet solved!