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Math Help - x^3 + y^3 = 35

  1. #1
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    Question x^3 + y^3 = 35

    Find all the solutions in positive integers to : x^3 + y^3 = 35 .

    Can you teach me how to solve this question? Thank you very much.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    Find all the solutions in positive integers to : x^3 + y^3 = 35 .

    Can you teach me how to solve this question? Thank you very much.
    Well, the numbers are small enough for trial and error. The only other way I can think of would be to factor:

    x^3 + y^3 = (x + y)(x^2 + xy + y^2) = 35.

    The only integer factors of 35 are (1, 35) or (5, 7). There are no integers x and y that add to 1, so x + y = 5 or 7.

    Using x + y = 5 we get that y = 5 - x, so y^3 = 125 - 75x + 15x^2 - x^3 so
    x^3 + y^3 = 15x^2 - 75x + 125 = 35
    or
    15x^2 - 75x + 90 = 0
    x^2 - 5x + 6 = 0
    (x - 3)(x - 2) = 0
    So x = 2 or x = 3
    Thus y = 3 or y = 2 respectively.

    Using x + y = 7 we get that y = 7 - x, so y^3 = 343 - 147x + 21x^2 - x^3 so
    x^3 + y^3 = 21x^2 - 147x + 343 = 35
    21x^2 - 147x + 308 = 0
    3x^2 - 21x + 44 = 0
    or
    x = (21 +/- sqrt(-87))/6
    Since x is complex, this can't be a solution.

    So (x, y) = (2, 3) or (x, y) = (3, 2)

    However, this is only going to be practical when there are a small number of possibilities for x + y.

    -Dan
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  3. #3
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    Hi topsquark,

    Thank you very much. I solved this question easily with your guideline.

    This is a typo in your factorization. It should be
    x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 35.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    Hi topsquark,

    Thank you very much. I solved this question easily with your guideline.

    This is a typo in your factorization. It should be
    x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 35.
    Oops! Thanks for the catch!

    -Dan
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