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    suppose that p and q are integers satisfying

    P^3 + PQ^2 + Q^3 = 0

    A, show this if one of P or Q is divisible by 2 then so is the other

    B, show that if p and q are both odd number then they cannot solve the above equation

    C, deduce that there is no rational number x satisfying

    x^3 + x + 1 =0

    So, what are your thoughts?

    A. Suppose that P = 2k.

    B. Prove that product of an odd number with an odd number is also odd. Hence notice that each of the three terms are odd. Is the sum of three odd numbers odd or even?

    C. Represent x by the ratio of two integers, which have no common divisors greater than 1.
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Updated: February 1, 2010
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