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RolyPoly

  • 2 years ago

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

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  1. mukushla
    • 2 years ago
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    without loss of generality we can suppose \[y\ge x \]because the expression is symmetric

  2. mukushla
    • 2 years ago
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    assume that \[y=x+a\]with \(a\ge 0\)

  3. RolyPoly
    • 2 years ago
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    Okay! I'll try from here!!

  4. mukushla
    • 2 years ago
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    wait ... im afraid i made a mistake :/

  5. RolyPoly
    • 2 years ago
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    Don't be afraid :P \[4xy-x-y=4x(x+a)-x-(x+a)=4x^2+(4a-2)x-a\]

  6. RolyPoly
    • 2 years ago
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    \[\Delta=(4a-2)^2 - 4(4)(-a)=0\]\[16a^2 - 16a + 4 + 16a=0\]\[16a^2 + 4 =0\]Well well well...

  7. shubhamsrg
    • 2 years ago
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    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

  8. mukushla
    • 2 years ago
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    @RolyPoly ...Are you familiar with modular arithmetic??

  9. RolyPoly
    • 2 years ago
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    I've never heard of it! :(

  10. shubhamsrg
    • 2 years ago
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    i dont know much but i do have a little idea..will you please guide me/us through a solution @mukushla

  11. RolyPoly
    • 2 years ago
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    Forgive me for closing this question, even though it's not yet solved!

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