Posers and Puzzles

Posers and Puzzles

  1. Joined
    25 Aug '06
    Moves
    0
    04 Jun '08 15:042 edits
    For which positive integers x,y does the following equality hold?

    (x^2 + xy - y^2)^2 = 1
  2. Standard memberforkedknight
    Defend the Universe
    127.0.0.1
    Joined
    18 Dec '03
    Moves
    16180
    04 Jun '08 15:56
    is 0 a positive integer?
  3. Joined
    07 Sep '05
    Moves
    35068
    04 Jun '08 15:57
    Originally posted by forkedknight
    is 0 a positive integer?
    No
  4. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    04 Jun '08 15:59
    Originally posted by David113
    For which positive integers x,y does the following equality hold?

    (x^2 + xy - y^2)^2 = 1
    Weird!! Just tried a simple brute force sampling, but the pairings I get are:

    (1,1)
    (2,3)
    (5,8)
    (13,21)
    (34,55)

    which is just the Fibonacci numbers taken in pairs. Interesting! I'll have to take a look at the Fibonacci identities to see where this comes from.
  5. Joined
    07 Sep '05
    Moves
    35068
    04 Jun '08 16:01
    Originally posted by David113
    For which positive integers x,y does the following equality hold?

    (x^2 + xy - y^2)^2 = 1
    Nice! It's a Fibonacci series.
  6. Joined
    07 Sep '05
    Moves
    35068
    04 Jun '08 16:04
    Originally posted by PBE6
    Weird!! Just tried a simple brute force sampling, but the pairings I get are:

    (1,1)
    (2,3)
    (5,8)
    (13,21)
    (34,55)

    which is just the Fibonacci numbers taken in pairs. Interesting! I'll have to take a look at the Fibonacci identities to see where this comes from.
    Here's part of the explanation.

    If F(x, y) = x^2 + xy - y^2

    Then F(y, x + y) = -x^2 - xy + y^2 = -F(x, y)
    (just try the substitution)

    So once you have (1, 1) as a solution, you can construct the rest of the Fibonacci series from it: (1, 2), (2, 3), (3, 5), (5, 8) etc...

    So all you need to show is that there aren't any other solutions 🙂
  7. Standard memberPalynka
    Upward Spiral
    Halfway
    Joined
    02 Aug '04
    Moves
    8702
    04 Jun '08 16:081 edit
    Originally posted by PBE6
    Weird!! Just tried a simple brute force sampling, but the pairings I get are:

    (1,1)
    (2,3)
    (5,8)
    (13,21)
    (34,55)

    which is just the Fibonacci numbers taken in pairs. Interesting! I'll have to take a look at the Fibonacci identities to see where this comes from.
    Very interesting. I'm thinking the geometric representation (with the squares) of the Fibonacci sequence might be an easy way to find out why, but I can't pin it down just yet.

    Edit - Nice one, mtthw.
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