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Math Help - square = cube

  1. #1
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    square = cube

    A fun question.

    4^3 = 8^2

    Are there any more pairs of numbers (a, b) such that a^3 = b^2 - if so how many?

    I would be interested in how people go about finding the solution .
    Last edited by brennan; October 25th 2010 at 01:55 PM.
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  2. #2
    Member HappyJoe's Avatar
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    There's an infinite number of such pairs. Letting (a,b) = (n^2,n^3) works, where n is an integer. In fact, a has to be a square.

    Why? Consider prime factorizations of a and b, suppose a^3=b^2, and use uniqueness of prime factorizations:

    a = p_1^{x_1}\cdots p_k^{x_k},
    b = p_1^{y_1}\cdots p_k^{y_k},

    where the equation a^3=b^2 implies that

    p_i^{3x_i} = p_i^{2y_i}

    for all i=1,2,\ldots,k. Hence 3x_i = 2y_i, whence x_i must be even. In conclusion, each p_i^{x_i} is a square, and so all of a is square.
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  3. #3
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    I answered this by generating number by raising a base number to the powers 2 and 3.

    a = n^2
    b = n^3

    substitute into the expression

    a^3 = b^2

    gives

    (n^2)^3 = (n^3)^2 {\Rightarrow} n^6 = n^6

    Example

    n = 3, n^2 = 9, n^3 = 27

    so does 9^3 = 27^2 (yes it does)

    There are infinate solutions because n {\in} {\Re}
    Last edited by brennan; October 25th 2010 at 02:31 PM.
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