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Math Help - solve in ihtegers

  1. #1
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    solve in ihtegers

    x,y integers

    x^3 +(x+1)^3 +(x+2)^3+....+(x+7)^3=y^3
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by perash View Post
    x,y integers

    x^3 +(x+1)^3 +(x+2)^3+....+(x+7)^3=y^3
    There's quite a bit of work going on here! I would think that the simplest start to this would be to expand and simplify:
    x^3 + (x + 1)^3 + (x + 2)^3 +....+ (x + 7)^3 = y^3
    becomes (after a fair amount of work)
    8x^3 + 84x^2 + 420x + 784 = y^3

    8x^3 + 84x^2 + 420x + (784 - y^3) = 0

    x^3 + \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x + 98 - \frac{y^3}{8} = 0

    For this to be true for x and y integers we must have that
    \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x - \frac{y^3}{8}
    must be an integer. That might be a good starting point.

    -Dan
    Last edited by topsquark; July 23rd 2007 at 06:56 AM. Reason: Typo
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  3. #3
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    Thats about where i got, but isn't it 21/2 x (84/8).
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark View Post
    There's quite a bit of work going on here! I would think that the simplest start to this would be to expand and simplify:
    x^3 + (x + 1)^3 + (x + 2)^3 +....+ (x + 7)^3 = y^3
    becomes (after a fair amount of work)
    8x^3 + 84x^2 + 420x + 784 = y^3

    8x^3 + 84x^2 + 420x + (784 - y^3) = 0

    x^3 + \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x + 98 - \frac{y^3}{8} = 0

    For this to be true for x and y integers we must have that
    \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x - \frac{y^3}{8}
    must be an integer. That might be a good starting point.

    -Dan
    The typos have been fixed now.

    -Dan
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