who know how to solve this nonlinear diophantine equation

Discussion in 'Math Research' started by None, Nov 23, 2006.

  1. None

    None Guest

    ax^2+bx = cy^2+dy (a,b,c,d are Integer constants and x,y are
    variables)
    and this equation has little condition
    a+b = c+d
     
    None, Nov 23, 2006
    #1
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  2. a x^2 + b x = c y^2 + d y
    4 a^2 x^2 + 4 a b x = 4 a c y^2 + 4 a d y
    4 a^2 x^2 + 4 a b x + b^2 = 4 a c y^2 + 4 a d y + b^2
    (2 a x + b)^2 = 4 a c y^2 + 4 a d y + b^2
    u^2 = 4 a c y^2 + 4 a d y + b^2
    a c u^2 = 4 a^2 c^2 y^2 + 4 a^2 c d y + 4 a c b^2
    a c u^2 + a^2 d^2 = 4 a^2 c^2 y^2 + 4 a^2 c d y + a^2 d^2 + 4 a c b^2
    a c u^2 + a^2 d^2 = (2 a c y + a d)^2 + 4 a c b^2
    a c u^2 + a^2 d^2 = v^2 + 4 a c b^2
    v^2 - a c u^2 = a^2 d^2 - 4 a c b^2

    (*) v^2 - D u^2 = k,

    where v = 2 a c y + a d,
    D = a c,
    u = 2 a x + b,
    k = a^2 d^2 - 4 a c b^2.

    Now there are standard techniques for solving (*).
    See Pell's equation.
     
    Gerry Myerson, Nov 23, 2006
    #2
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  3. Gerry Myerson napisal(a):
    In 6th line should be:

    a c u^2 = 4 a^2 c^2 y^2 + 4 a^2 c d y + a c b^2

    Thus in the equation (*):

    k = a^2 d^2 - a c b^2 .
     
    Zbigniew Karno, Nov 27, 2006
    #3
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